a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 56.6%

Time: 2.4min

Alternatives: 23

Speedup: 2757.0×

• 29.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\\ t_1 := 0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\\ t_2 := \left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\\ t_3 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;y-scale\_m \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(t\_2 \cdot \mathsf{hypot}\left(t\_0, b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 2 \cdot 10^{-53}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 9.8 \cdot 10^{+17}:\\ \;\;\;\;0.25 \cdot \left(t\_2 \cdot \mathsf{hypot}\left(t\_0, b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 3.3 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_3, b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b\_m}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_3, b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (cos (* 0.005555555555555556 (* angle PI))) a))
        (t_1 (* 0.25 (* y-scale_m (sqrt 8.0))))
        (t_2 (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)))
        (t_3 (* PI (* 0.005555555555555556 angle))))
   (if (<= y-scale_m 3.5e-95)
     (*
      0.25
      (*
       t_2
       (hypot
        t_0
        (*
         b_m
         (sin
          (*
           0.005555555555555556
           (* angle (* (cbrt PI) (pow (cbrt PI) 2.0)))))))))
     (if (<= y-scale_m 2e-53)
       (*
        t_1
        (*
         (sqrt 2.0)
         (hypot
          (*
           a
           (sin
            (* (* 0.005555555555555556 angle) (pow (cbrt (pow PI 1.5)) 2.0))))
          b_m)))
       (if (<= y-scale_m 9.8e+17)
         (*
          0.25
          (*
           t_2
           (hypot
            t_0
            (*
             b_m
             (sin (* 0.005555555555555556 (* angle (pow (sqrt PI) 2.0))))))))
         (if (<= y-scale_m 3.3e+84)
           (*
            (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
            (* (sqrt 2.0) (hypot (* a t_3) b_m)))
           (if (<= y-scale_m 1.52e+140)
             (*
              -0.25
              (*
               (* x-scale_m (* y-scale_m (- (sqrt 8.0))))
               (/ (* (sqrt 2.0) b_m) x-scale_m)))
             (* t_1 (* (sqrt 2.0) (hypot (* a (sin t_3)) b_m))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI)))) * a;
	double t_1 = 0.25 * (y_45_scale_m * sqrt(8.0));
	double t_2 = (x_45_scale_m * sqrt(2.0)) * sqrt(8.0);
	double t_3 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 3.5e-95) {
		tmp = 0.25 * (t_2 * hypot(t_0, (b_m * sin((0.005555555555555556 * (angle * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))))));
	} else if (y_45_scale_m <= 2e-53) {
		tmp = t_1 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * pow(cbrt(pow(((double) M_PI), 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 9.8e+17) {
		tmp = 0.25 * (t_2 * hypot(t_0, (b_m * sin((0.005555555555555556 * (angle * pow(sqrt(((double) M_PI)), 2.0)))))));
	} else if (y_45_scale_m <= 3.3e+84) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_3), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_1 * (sqrt(2.0) * hypot((a * sin(t_3)), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.cos((0.005555555555555556 * (angle * Math.PI))) * a;
	double t_1 = 0.25 * (y_45_scale_m * Math.sqrt(8.0));
	double t_2 = (x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0);
	double t_3 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 3.5e-95) {
		tmp = 0.25 * (t_2 * Math.hypot(t_0, (b_m * Math.sin((0.005555555555555556 * (angle * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))))));
	} else if (y_45_scale_m <= 2e-53) {
		tmp = t_1 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.pow(Math.cbrt(Math.pow(Math.PI, 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 9.8e+17) {
		tmp = 0.25 * (t_2 * Math.hypot(t_0, (b_m * Math.sin((0.005555555555555556 * (angle * Math.pow(Math.sqrt(Math.PI), 2.0)))))));
	} else if (y_45_scale_m <= 3.3e+84) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_3), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -Math.sqrt(8.0))) * ((Math.sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_1 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_3)), b_m));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(cos(Float64(0.005555555555555556 * Float64(angle * pi))) * a)
	t_1 = Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0)))
	t_2 = Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0))
	t_3 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (y_45_scale_m <= 3.5e-95)
		tmp = Float64(0.25 * Float64(t_2 * hypot(t_0, Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))))))));
	elseif (y_45_scale_m <= 2e-53)
		tmp = Float64(t_1 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * (cbrt((pi ^ 1.5)) ^ 2.0)))), b_m)));
	elseif (y_45_scale_m <= 9.8e+17)
		tmp = Float64(0.25 * Float64(t_2 * hypot(t_0, Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * (sqrt(pi) ^ 2.0))))))));
	elseif (y_45_scale_m <= 3.3e+84)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_3), b_m)));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = Float64(-0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(-sqrt(8.0)))) * Float64(Float64(sqrt(2.0) * b_m) / x_45_scale_m)));
	else
		tmp = Float64(t_1 * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_3)), b_m)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 3.5e-95], N[(0.25 * N[(t$95$2 * N[Sqrt[t$95$0 ^ 2 + N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2e-53], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 9.8e+17], N[(0.25 * N[(t$95$2 * N[Sqrt[t$95$0 ^ 2 + N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.3e+84], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$3), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.52e+140], N[(-0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * b$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y-scale < 3.4999999999999997e-95

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

       1. add-cube-cbrt 26.0%

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

       2. pow2 26.0%

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

   12. Applied egg-rr 26.0%

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

   if 3.4999999999999997e-95 < y-scale < 2.00000000000000006e-53

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in x-scale around 0 32.1%

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

      1. associate-*r* 32.1%

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

      2. distribute-lft-out 32.1%

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

   7. Simplified 35.6%

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

   8. Taylor expanded in angle around 0 35.6%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 48.9%

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

       1. add-sqr-sqrt 49.1%

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

       2. pow2 49.1%

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

   12. Applied egg-rr 49.1%

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

       1. add-cbrt-cube 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       2. pow1/3 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       3. add-sqr-sqrt 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       4. pow1 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       5. pow1/2 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       6. pow-prod-up 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       7. metadata-eval 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   14. Applied egg-rr 49.1%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left({\pi}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]
   15. Step-by-step derivation

       1. unpow1/3 50.0%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   16. Simplified 50.0%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   if 2.00000000000000006e-53 < y-scale < 9.8e17

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in y-scale around 0 11.5%

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

      1. associate-*r* 11.5%

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

      2. distribute-lft-out 11.5%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in x-scale around 0 11.9%

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

   10. Simplified 12.9%

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

       1. add-sqr-sqrt 27.6%

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

       2. pow2 27.6%

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

   12. Applied egg-rr 12.9%

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

   if 9.8e17 < y-scale < 3.30000000000000017e84

   1. Initial program 0.5%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in x-scale around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. distribute-lft-out 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 45.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 54.5%

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

   11. Taylor expanded in angle around 0 53.5%

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

       1. associate-*r* 53.5%

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

   13. Simplified 53.5%

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

       1. add-sqr-sqrt 53.3%

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

       2. sqrt-unprod 53.5%

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

       3. swap-sqr 53.5%

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

       4. metadata-eval 53.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 53.5%

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

       6. swap-sqr 53.5%

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

       7. *-commutative 53.5%

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

       8. *-commutative 53.5%

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

       9. swap-sqr 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

       12. swap-sqr 53.2%

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

       13. rem-square-sqrt 54.0%

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

       14. pow2 54.0%

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

       15. metadata-eval 54.0%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   if 3.30000000000000017e84 < y-scale < 1.52e140

   1. Initial program 8.3%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in a around 0 0.2%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right)\right)} \]

   7. Simplified 0.2%

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

   8. Taylor expanded in angle around 0 34.7%

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

   if 1.52e140 < y-scale

   1. Initial program 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x-scale around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. distribute-lft-out 76.6%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in angle around 0 85.5%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 85.4%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 89.2%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 9.8 \cdot 10^{+17}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.3 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\\ t_1 := 0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\\ t_2 := x-scale\_m \cdot \sqrt{2}\\ \mathbf{if}\;y-scale\_m \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(t\_2 \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(t\_0, b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.5 \cdot 10^{-53}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 2.2 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{t\_2}\right)}^{2}\right) \cdot \mathsf{hypot}\left(t\_0, b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right), b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (cos (* 0.005555555555555556 (* angle PI))) a))
        (t_1 (* 0.25 (* y-scale_m (sqrt 8.0))))
        (t_2 (* x-scale_m (sqrt 2.0))))
   (if (<= y-scale_m 3.5e-95)
     (*
      0.25
      (*
       (* t_2 (sqrt 8.0))
       (hypot
        t_0
        (*
         b_m
         (sin
          (*
           0.005555555555555556
           (* angle (* (cbrt PI) (pow (cbrt PI) 2.0)))))))))
     (if (<= y-scale_m 1.5e-53)
       (*
        t_1
        (*
         (sqrt 2.0)
         (hypot
          (*
           a
           (sin
            (* (* 0.005555555555555556 angle) (pow (cbrt (pow PI 1.5)) 2.0))))
          b_m)))
       (if (<= y-scale_m 2.2e+18)
         (*
          0.25
          (*
           (* (sqrt 8.0) (pow (sqrt t_2) 2.0))
           (hypot
            t_0
            (*
             b_m
             (sin (* 0.005555555555555556 (* angle (pow (sqrt PI) 2.0))))))))
         (*
          t_1
          (*
           (sqrt 2.0)
           (hypot
            (* a (sin (* (* 0.005555555555555556 angle) (cbrt (pow PI 3.0)))))
            b_m))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI)))) * a;
	double t_1 = 0.25 * (y_45_scale_m * sqrt(8.0));
	double t_2 = x_45_scale_m * sqrt(2.0);
	double tmp;
	if (y_45_scale_m <= 3.5e-95) {
		tmp = 0.25 * ((t_2 * sqrt(8.0)) * hypot(t_0, (b_m * sin((0.005555555555555556 * (angle * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))))));
	} else if (y_45_scale_m <= 1.5e-53) {
		tmp = t_1 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * pow(cbrt(pow(((double) M_PI), 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 2.2e+18) {
		tmp = 0.25 * ((sqrt(8.0) * pow(sqrt(t_2), 2.0)) * hypot(t_0, (b_m * sin((0.005555555555555556 * (angle * pow(sqrt(((double) M_PI)), 2.0)))))));
	} else {
		tmp = t_1 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * cbrt(pow(((double) M_PI), 3.0))))), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.cos((0.005555555555555556 * (angle * Math.PI))) * a;
	double t_1 = 0.25 * (y_45_scale_m * Math.sqrt(8.0));
	double t_2 = x_45_scale_m * Math.sqrt(2.0);
	double tmp;
	if (y_45_scale_m <= 3.5e-95) {
		tmp = 0.25 * ((t_2 * Math.sqrt(8.0)) * Math.hypot(t_0, (b_m * Math.sin((0.005555555555555556 * (angle * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))))));
	} else if (y_45_scale_m <= 1.5e-53) {
		tmp = t_1 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.pow(Math.cbrt(Math.pow(Math.PI, 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 2.2e+18) {
		tmp = 0.25 * ((Math.sqrt(8.0) * Math.pow(Math.sqrt(t_2), 2.0)) * Math.hypot(t_0, (b_m * Math.sin((0.005555555555555556 * (angle * Math.pow(Math.sqrt(Math.PI), 2.0)))))));
	} else {
		tmp = t_1 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.cbrt(Math.pow(Math.PI, 3.0))))), b_m));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(cos(Float64(0.005555555555555556 * Float64(angle * pi))) * a)
	t_1 = Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0)))
	t_2 = Float64(x_45_scale_m * sqrt(2.0))
	tmp = 0.0
	if (y_45_scale_m <= 3.5e-95)
		tmp = Float64(0.25 * Float64(Float64(t_2 * sqrt(8.0)) * hypot(t_0, Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))))))));
	elseif (y_45_scale_m <= 1.5e-53)
		tmp = Float64(t_1 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * (cbrt((pi ^ 1.5)) ^ 2.0)))), b_m)));
	elseif (y_45_scale_m <= 2.2e+18)
		tmp = Float64(0.25 * Float64(Float64(sqrt(8.0) * (sqrt(t_2) ^ 2.0)) * hypot(t_0, Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * (sqrt(pi) ^ 2.0))))))));
	else
		tmp = Float64(t_1 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * cbrt((pi ^ 3.0))))), b_m)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 3.5e-95], N[(0.25 * N[(N[(t$95$2 * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$0 ^ 2 + N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.5e-53], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.2e+18], N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[Power[N[Sqrt[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$0 ^ 2 + N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y-scale < 3.4999999999999997e-95

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

       1. add-cube-cbrt 26.0%

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

       2. pow2 26.0%

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

   12. Applied egg-rr 26.0%

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

   if 3.4999999999999997e-95 < y-scale < 1.5000000000000001e-53

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in x-scale around 0 32.1%

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

      1. associate-*r* 32.1%

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

      2. distribute-lft-out 32.1%

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

   7. Simplified 35.6%

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

   8. Taylor expanded in angle around 0 35.6%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 48.9%

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

       1. add-sqr-sqrt 49.1%

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

       2. pow2 49.1%

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

   12. Applied egg-rr 49.1%

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

       1. add-cbrt-cube 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       2. pow1/3 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       3. add-sqr-sqrt 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       4. pow1 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       5. pow1/2 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       6. pow-prod-up 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       7. metadata-eval 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   14. Applied egg-rr 49.1%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left({\pi}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]
   15. Step-by-step derivation

       1. unpow1/3 50.0%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   16. Simplified 50.0%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   if 1.5000000000000001e-53 < y-scale < 2.2e18

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in y-scale around 0 11.5%

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

      1. associate-*r* 11.5%

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

      2. distribute-lft-out 11.5%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in x-scale around 0 11.9%

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

   10. Simplified 12.9%

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

       1. add-sqr-sqrt 12.4%

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

       2. pow2 12.4%

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

       3. *-commutative 12.4%

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

   12. Applied egg-rr 12.4%

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

       1. add-sqr-sqrt 27.6%

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

       2. pow2 27.6%

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

   14. Applied egg-rr 12.4%

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

   if 2.2e18 < y-scale

   1. Initial program 6.0%

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

   3. Simplified 4.2%

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

   5. Taylor expanded in x-scale around 0 67.5%

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

      1. associate-*r* 67.5%

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

      2. distribute-lft-out 67.5%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in angle around 0 72.7%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 78.2%

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

       1. add-cbrt-cube 78.3%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right), b\right) \cdot \sqrt{2}\right) \]

       2. pow3 78.3%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right), b\right) \cdot \sqrt{2}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right), b\right) \cdot \sqrt{2}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.5 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.2 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := 0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\\ t_2 := \left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\\ \mathbf{if}\;y-scale\_m \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(t\_2 \cdot \mathsf{hypot}\left(\cos t\_0 \cdot a, b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.55 \cdot 10^{-53}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 2.4 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(t\_2 \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\right)}^{2}\right)\right), b\_m \cdot \sin t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right), b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (* 0.25 (* y-scale_m (sqrt 8.0))))
        (t_2 (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0))))
   (if (<= y-scale_m 3.5e-95)
     (*
      0.25
      (*
       t_2
       (hypot
        (* (cos t_0) a)
        (*
         b_m
         (sin
          (*
           0.005555555555555556
           (* angle (* (cbrt PI) (pow (cbrt PI) 2.0)))))))))
     (if (<= y-scale_m 1.55e-53)
       (*
        t_1
        (*
         (sqrt 2.0)
         (hypot
          (*
           a
           (sin
            (* (* 0.005555555555555556 angle) (pow (cbrt (pow PI 1.5)) 2.0))))
          b_m)))
       (if (<= y-scale_m 2.4e+18)
         (*
          0.25
          (*
           t_2
           (hypot
            (*
             a
             (cos
              (*
               0.005555555555555556
               (* angle (pow (pow (cbrt (sqrt PI)) 3.0) 2.0)))))
            (* b_m (sin t_0)))))
         (*
          t_1
          (*
           (sqrt 2.0)
           (hypot
            (* a (sin (* (* 0.005555555555555556 angle) (cbrt (pow PI 3.0)))))
            b_m))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = 0.25 * (y_45_scale_m * sqrt(8.0));
	double t_2 = (x_45_scale_m * sqrt(2.0)) * sqrt(8.0);
	double tmp;
	if (y_45_scale_m <= 3.5e-95) {
		tmp = 0.25 * (t_2 * hypot((cos(t_0) * a), (b_m * sin((0.005555555555555556 * (angle * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0))))))));
	} else if (y_45_scale_m <= 1.55e-53) {
		tmp = t_1 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * pow(cbrt(pow(((double) M_PI), 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 2.4e+18) {
		tmp = 0.25 * (t_2 * hypot((a * cos((0.005555555555555556 * (angle * pow(pow(cbrt(sqrt(((double) M_PI))), 3.0), 2.0))))), (b_m * sin(t_0))));
	} else {
		tmp = t_1 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * cbrt(pow(((double) M_PI), 3.0))))), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = 0.25 * (y_45_scale_m * Math.sqrt(8.0));
	double t_2 = (x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0);
	double tmp;
	if (y_45_scale_m <= 3.5e-95) {
		tmp = 0.25 * (t_2 * Math.hypot((Math.cos(t_0) * a), (b_m * Math.sin((0.005555555555555556 * (angle * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0))))))));
	} else if (y_45_scale_m <= 1.55e-53) {
		tmp = t_1 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.pow(Math.cbrt(Math.pow(Math.PI, 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 2.4e+18) {
		tmp = 0.25 * (t_2 * Math.hypot((a * Math.cos((0.005555555555555556 * (angle * Math.pow(Math.pow(Math.cbrt(Math.sqrt(Math.PI)), 3.0), 2.0))))), (b_m * Math.sin(t_0))));
	} else {
		tmp = t_1 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.cbrt(Math.pow(Math.PI, 3.0))))), b_m));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0)))
	t_2 = Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0))
	tmp = 0.0
	if (y_45_scale_m <= 3.5e-95)
		tmp = Float64(0.25 * Float64(t_2 * hypot(Float64(cos(t_0) * a), Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))))))));
	elseif (y_45_scale_m <= 1.55e-53)
		tmp = Float64(t_1 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * (cbrt((pi ^ 1.5)) ^ 2.0)))), b_m)));
	elseif (y_45_scale_m <= 2.4e+18)
		tmp = Float64(0.25 * Float64(t_2 * hypot(Float64(a * cos(Float64(0.005555555555555556 * Float64(angle * ((cbrt(sqrt(pi)) ^ 3.0) ^ 2.0))))), Float64(b_m * sin(t_0)))));
	else
		tmp = Float64(t_1 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * cbrt((pi ^ 3.0))))), b_m)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 3.5e-95], N[(0.25 * N[(t$95$2 * N[Sqrt[N[(N[Cos[t$95$0], $MachinePrecision] * a), $MachinePrecision] ^ 2 + N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.55e-53], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.4e+18], N[(0.25 * N[(t$95$2 * N[Sqrt[N[(a * N[Cos[N[(0.005555555555555556 * N[(angle * N[Power[N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b$95$m * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[Pi, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y-scale < 3.4999999999999997e-95

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

       1. add-cube-cbrt 26.0%

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

       2. pow2 26.0%

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

   12. Applied egg-rr 26.0%

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

   if 3.4999999999999997e-95 < y-scale < 1.55000000000000008e-53

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in x-scale around 0 32.1%

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

      1. associate-*r* 32.1%

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

      2. distribute-lft-out 32.1%

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

   7. Simplified 35.6%

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

   8. Taylor expanded in angle around 0 35.6%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 48.9%

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

       1. add-sqr-sqrt 49.1%

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

       2. pow2 49.1%

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

   12. Applied egg-rr 49.1%

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

       1. add-cbrt-cube 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       2. pow1/3 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       3. add-sqr-sqrt 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       4. pow1 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       5. pow1/2 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       6. pow-prod-up 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       7. metadata-eval 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   14. Applied egg-rr 49.1%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left({\pi}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]
   15. Step-by-step derivation

       1. unpow1/3 50.0%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   16. Simplified 50.0%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   if 1.55000000000000008e-53 < y-scale < 2.4e18

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in y-scale around 0 11.5%

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

      1. associate-*r* 11.5%

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

      2. distribute-lft-out 11.5%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in x-scale around 0 11.9%

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

   10. Simplified 12.9%

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

       1. add-sqr-sqrt 27.6%

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

       2. pow2 27.6%

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

   12. Applied egg-rr 12.9%

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

       1. add-cube-cbrt 12.9%

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

       2. pow3 12.9%

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

   14. Applied egg-rr 12.9%

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

   if 2.4e18 < y-scale

   1. Initial program 6.0%

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

   3. Simplified 4.2%

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

   5. Taylor expanded in x-scale around 0 67.5%

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

      1. associate-*r* 67.5%

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

      2. distribute-lft-out 67.5%

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

   7. Simplified 71.1%

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

   8. Taylor expanded in angle around 0 72.7%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 78.2%

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

       1. add-cbrt-cube 78.3%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right), b\right) \cdot \sqrt{2}\right) \]

       2. pow3 78.3%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right), b\right) \cdot \sqrt{2}\right) \]

   12. Applied egg-rr 78.3%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right), b\right) \cdot \sqrt{2}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.55 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.4 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot {\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\right)}^{2}\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{{\pi}^{3}}\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t\_1 \cdot a\\ t_3 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;y-scale\_m \leq 2.9 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \left({2}^{0.25} \cdot \left(x-scale\_m \cdot {2}^{0.25}\right)\right)\right) \cdot \mathsf{hypot}\left(t\_2, b\_m \cdot \sin t\_1\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(t\_2, b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_3, b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b\_m}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_3, b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* y-scale_m (sqrt 8.0))))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (* (cos t_1) a))
        (t_3 (* PI (* 0.005555555555555556 angle))))
   (if (<= y-scale_m 2.9e-95)
     (*
      0.25
      (*
       (* (sqrt 8.0) (* (pow 2.0 0.25) (* x-scale_m (pow 2.0 0.25))))
       (hypot t_2 (* b_m (sin t_1)))))
     (if (<= y-scale_m 1.9e-53)
       (*
        t_0
        (*
         (sqrt 2.0)
         (hypot
          (*
           a
           (sin
            (* (* 0.005555555555555556 angle) (pow (cbrt (pow PI 1.5)) 2.0))))
          b_m)))
       (if (<= y-scale_m 6.5e+18)
         (*
          0.25
          (*
           (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0))
           (hypot
            t_2
            (*
             b_m
             (sin (* 0.005555555555555556 (* angle (pow (sqrt PI) 2.0))))))))
         (if (<= y-scale_m 4.2e+84)
           (*
            (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
            (* (sqrt 2.0) (hypot (* a t_3) b_m)))
           (if (<= y-scale_m 1.52e+140)
             (*
              -0.25
              (*
               (* x-scale_m (* y-scale_m (- (sqrt 8.0))))
               (/ (* (sqrt 2.0) b_m) x-scale_m)))
             (* t_0 (* (sqrt 2.0) (hypot (* a (sin t_3)) b_m))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * sqrt(8.0));
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = cos(t_1) * a;
	double t_3 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 2.9e-95) {
		tmp = 0.25 * ((sqrt(8.0) * (pow(2.0, 0.25) * (x_45_scale_m * pow(2.0, 0.25)))) * hypot(t_2, (b_m * sin(t_1))));
	} else if (y_45_scale_m <= 1.9e-53) {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * pow(cbrt(pow(((double) M_PI), 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 6.5e+18) {
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot(t_2, (b_m * sin((0.005555555555555556 * (angle * pow(sqrt(((double) M_PI)), 2.0)))))));
	} else if (y_45_scale_m <= 4.2e+84) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_3), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(t_3)), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * Math.sqrt(8.0));
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.cos(t_1) * a;
	double t_3 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 2.9e-95) {
		tmp = 0.25 * ((Math.sqrt(8.0) * (Math.pow(2.0, 0.25) * (x_45_scale_m * Math.pow(2.0, 0.25)))) * Math.hypot(t_2, (b_m * Math.sin(t_1))));
	} else if (y_45_scale_m <= 1.9e-53) {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.pow(Math.cbrt(Math.pow(Math.PI, 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 6.5e+18) {
		tmp = 0.25 * (((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * Math.hypot(t_2, (b_m * Math.sin((0.005555555555555556 * (angle * Math.pow(Math.sqrt(Math.PI), 2.0)))))));
	} else if (y_45_scale_m <= 4.2e+84) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_3), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -Math.sqrt(8.0))) * ((Math.sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_3)), b_m));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0)))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = Float64(cos(t_1) * a)
	t_3 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (y_45_scale_m <= 2.9e-95)
		tmp = Float64(0.25 * Float64(Float64(sqrt(8.0) * Float64((2.0 ^ 0.25) * Float64(x_45_scale_m * (2.0 ^ 0.25)))) * hypot(t_2, Float64(b_m * sin(t_1)))));
	elseif (y_45_scale_m <= 1.9e-53)
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * (cbrt((pi ^ 1.5)) ^ 2.0)))), b_m)));
	elseif (y_45_scale_m <= 6.5e+18)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot(t_2, Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * (sqrt(pi) ^ 2.0))))))));
	elseif (y_45_scale_m <= 4.2e+84)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_3), b_m)));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = Float64(-0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(-sqrt(8.0)))) * Float64(Float64(sqrt(2.0) * b_m) / x_45_scale_m)));
	else
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_3)), b_m)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t$95$1], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.9e-95], N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(x$45$scale$95$m * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2 ^ 2 + N[(b$95$m * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.9e-53], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 6.5e+18], N[(0.25 * N[(N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$2 ^ 2 + N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 4.2e+84], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$3), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.52e+140], N[(-0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * b$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y-scale < 2.90000000000000002e-95

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

       1. add-sqr-sqrt 25.0%

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

       2. pow2 25.0%

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

       3. *-commutative 25.0%

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

   12. Applied egg-rr 25.0%

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

       1. unpow2 25.0%

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

       2. add-sqr-sqrt 26.1%

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

       3. add-sqr-sqrt 26.2%

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

       4. associate-*l* 26.1%

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

       5. pow1/2 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       6. sqrt-pow1 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       7. metadata-eval 26.1%

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

       8. pow1/2 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left({2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       9. sqrt-pow1 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left({2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       10. metadata-eval 26.1%

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

   14. Applied egg-rr 26.1%

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

   if 2.90000000000000002e-95 < y-scale < 1.8999999999999999e-53

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in x-scale around 0 32.1%

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

      1. associate-*r* 32.1%

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

      2. distribute-lft-out 32.1%

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

   7. Simplified 35.6%

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

   8. Taylor expanded in angle around 0 35.6%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 48.9%

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

       1. add-sqr-sqrt 49.1%

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

       2. pow2 49.1%

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

   12. Applied egg-rr 49.1%

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

       1. add-cbrt-cube 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       2. pow1/3 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       3. add-sqr-sqrt 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       4. pow1 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       5. pow1/2 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       6. pow-prod-up 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       7. metadata-eval 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   14. Applied egg-rr 49.1%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left({\pi}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]
   15. Step-by-step derivation

       1. unpow1/3 50.0%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   16. Simplified 50.0%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   if 1.8999999999999999e-53 < y-scale < 6.5e18

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in y-scale around 0 11.5%

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

      1. associate-*r* 11.5%

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

      2. distribute-lft-out 11.5%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in x-scale around 0 11.9%

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

   10. Simplified 12.9%

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

       1. add-sqr-sqrt 27.6%

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

       2. pow2 27.6%

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

   12. Applied egg-rr 12.9%

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

   if 6.5e18 < y-scale < 4.20000000000000037e84

   1. Initial program 0.5%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in x-scale around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. distribute-lft-out 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 45.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 54.5%

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

   11. Taylor expanded in angle around 0 53.5%

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

       1. associate-*r* 53.5%

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

   13. Simplified 53.5%

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

       1. add-sqr-sqrt 53.3%

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

       2. sqrt-unprod 53.5%

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

       3. swap-sqr 53.5%

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

       4. metadata-eval 53.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 53.5%

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

       6. swap-sqr 53.5%

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

       7. *-commutative 53.5%

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

       8. *-commutative 53.5%

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

       9. swap-sqr 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

       12. swap-sqr 53.2%

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

       13. rem-square-sqrt 54.0%

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

       14. pow2 54.0%

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

       15. metadata-eval 54.0%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   if 4.20000000000000037e84 < y-scale < 1.52e140

   1. Initial program 8.3%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in a around 0 0.2%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right)\right)} \]

   7. Simplified 0.2%

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

   8. Taylor expanded in angle around 0 34.7%

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

   if 1.52e140 < y-scale

   1. Initial program 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x-scale around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. distribute-lft-out 76.6%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in angle around 0 85.5%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 85.4%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 89.2%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.9 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \left({2}^{0.25} \cdot \left(x-scale \cdot {2}^{0.25}\right)\right)\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 6.5 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot {\left(\sqrt{\pi}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := b\_m \cdot \sin t\_1\\ t_3 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;y-scale\_m \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \left({2}^{0.25} \cdot \left(x-scale\_m \cdot {2}^{0.25}\right)\right)\right) \cdot \mathsf{hypot}\left(\cos t\_1 \cdot a, t\_2\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 4.4 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale\_m \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(a, t\_2\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_3, b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b\_m}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_3, b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* y-scale_m (sqrt 8.0))))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (* b_m (sin t_1)))
        (t_3 (* PI (* 0.005555555555555556 angle))))
   (if (<= y-scale_m 3.5e-95)
     (*
      0.25
      (*
       (* (sqrt 8.0) (* (pow 2.0 0.25) (* x-scale_m (pow 2.0 0.25))))
       (hypot (* (cos t_1) a) t_2)))
     (if (<= y-scale_m 1.9e-53)
       (*
        t_0
        (*
         (sqrt 2.0)
         (hypot
          (*
           a
           (sin
            (* (* 0.005555555555555556 angle) (pow (cbrt (pow PI 1.5)) 2.0))))
          b_m)))
       (if (<= y-scale_m 4.4e+18)
         (*
          0.25
          (*
           (* (sqrt 8.0) (pow (sqrt (* x-scale_m (sqrt 2.0))) 2.0))
           (hypot a t_2)))
         (if (<= y-scale_m 4.2e+84)
           (*
            (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
            (* (sqrt 2.0) (hypot (* a t_3) b_m)))
           (if (<= y-scale_m 1.52e+140)
             (*
              -0.25
              (*
               (* x-scale_m (* y-scale_m (- (sqrt 8.0))))
               (/ (* (sqrt 2.0) b_m) x-scale_m)))
             (* t_0 (* (sqrt 2.0) (hypot (* a (sin t_3)) b_m))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * sqrt(8.0));
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = b_m * sin(t_1);
	double t_3 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 3.5e-95) {
		tmp = 0.25 * ((sqrt(8.0) * (pow(2.0, 0.25) * (x_45_scale_m * pow(2.0, 0.25)))) * hypot((cos(t_1) * a), t_2));
	} else if (y_45_scale_m <= 1.9e-53) {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * pow(cbrt(pow(((double) M_PI), 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 4.4e+18) {
		tmp = 0.25 * ((sqrt(8.0) * pow(sqrt((x_45_scale_m * sqrt(2.0))), 2.0)) * hypot(a, t_2));
	} else if (y_45_scale_m <= 4.2e+84) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_3), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(t_3)), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * Math.sqrt(8.0));
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = b_m * Math.sin(t_1);
	double t_3 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 3.5e-95) {
		tmp = 0.25 * ((Math.sqrt(8.0) * (Math.pow(2.0, 0.25) * (x_45_scale_m * Math.pow(2.0, 0.25)))) * Math.hypot((Math.cos(t_1) * a), t_2));
	} else if (y_45_scale_m <= 1.9e-53) {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.pow(Math.cbrt(Math.pow(Math.PI, 1.5)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 4.4e+18) {
		tmp = 0.25 * ((Math.sqrt(8.0) * Math.pow(Math.sqrt((x_45_scale_m * Math.sqrt(2.0))), 2.0)) * Math.hypot(a, t_2));
	} else if (y_45_scale_m <= 4.2e+84) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_3), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -Math.sqrt(8.0))) * ((Math.sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_3)), b_m));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0)))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = Float64(b_m * sin(t_1))
	t_3 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (y_45_scale_m <= 3.5e-95)
		tmp = Float64(0.25 * Float64(Float64(sqrt(8.0) * Float64((2.0 ^ 0.25) * Float64(x_45_scale_m * (2.0 ^ 0.25)))) * hypot(Float64(cos(t_1) * a), t_2)));
	elseif (y_45_scale_m <= 1.9e-53)
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * (cbrt((pi ^ 1.5)) ^ 2.0)))), b_m)));
	elseif (y_45_scale_m <= 4.4e+18)
		tmp = Float64(0.25 * Float64(Float64(sqrt(8.0) * (sqrt(Float64(x_45_scale_m * sqrt(2.0))) ^ 2.0)) * hypot(a, t_2)));
	elseif (y_45_scale_m <= 4.2e+84)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_3), b_m)));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = Float64(-0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(-sqrt(8.0)))) * Float64(Float64(sqrt(2.0) * b_m) / x_45_scale_m)));
	else
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_3)), b_m)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b$95$m * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 3.5e-95], N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(x$45$scale$95$m * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[t$95$1], $MachinePrecision] * a), $MachinePrecision] ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.9e-53], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Power[N[Power[Pi, 1.5], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 4.4e+18], N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[Power[N[Sqrt[N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[a ^ 2 + t$95$2 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 4.2e+84], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$3), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.52e+140], N[(-0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * b$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y-scale < 3.4999999999999997e-95

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

       1. add-sqr-sqrt 25.0%

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

       2. pow2 25.0%

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

       3. *-commutative 25.0%

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

   12. Applied egg-rr 25.0%

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

       1. unpow2 25.0%

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

       2. add-sqr-sqrt 26.1%

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

       3. add-sqr-sqrt 26.2%

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

       4. associate-*l* 26.1%

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

       5. pow1/2 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       6. sqrt-pow1 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       7. metadata-eval 26.1%

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

       8. pow1/2 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left({2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       9. sqrt-pow1 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left({2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       10. metadata-eval 26.1%

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

   14. Applied egg-rr 26.1%

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

   if 3.4999999999999997e-95 < y-scale < 1.8999999999999999e-53

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in x-scale around 0 32.1%

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

      1. associate-*r* 32.1%

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

      2. distribute-lft-out 32.1%

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

   7. Simplified 35.6%

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

   8. Taylor expanded in angle around 0 35.6%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 35.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 48.9%

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

       1. add-sqr-sqrt 49.1%

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

       2. pow2 49.1%

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

   12. Applied egg-rr 49.1%

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

       1. add-cbrt-cube 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       2. pow1/3 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       3. add-sqr-sqrt 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{\pi} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       4. pow1 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left(\color{blue}{{\pi}^{1}} \cdot \sqrt{\pi}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       5. pow1/2 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{1} \cdot \color{blue}{{\pi}^{0.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       6. pow-prod-up 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\color{blue}{\left({\pi}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

       7. metadata-eval 49.1%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left({\left({\pi}^{\color{blue}{1.5}}\right)}^{0.3333333333333333}\right)}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   14. Applied egg-rr 49.1%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left({\left({\pi}^{1.5}\right)}^{0.3333333333333333}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]
   15. Step-by-step derivation

       1. unpow1/3 50.0%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   16. Simplified 50.0%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\color{blue}{\left(\sqrt[3]{{\pi}^{1.5}}\right)}}^{2}\right), b\right) \cdot \sqrt{2}\right) \]

   if 1.8999999999999999e-53 < y-scale < 4.4e18

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in y-scale around 0 11.5%

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

      1. associate-*r* 11.5%

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

      2. distribute-lft-out 11.5%

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

   7. Simplified 11.6%

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

   8. Taylor expanded in x-scale around 0 11.9%

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

   10. Simplified 12.9%

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

       1. add-sqr-sqrt 12.4%

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

       2. pow2 12.4%

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

       3. *-commutative 12.4%

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

   12. Applied egg-rr 12.4%

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

   13. Taylor expanded in angle around 0 12.3%

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

   if 4.4e18 < y-scale < 4.20000000000000037e84

   1. Initial program 0.5%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in x-scale around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. distribute-lft-out 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 45.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 54.5%

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

   11. Taylor expanded in angle around 0 53.5%

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

       1. associate-*r* 53.5%

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

   13. Simplified 53.5%

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

       1. add-sqr-sqrt 53.3%

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

       2. sqrt-unprod 53.5%

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

       3. swap-sqr 53.5%

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

       4. metadata-eval 53.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 53.5%

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

       6. swap-sqr 53.5%

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

       7. *-commutative 53.5%

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

       8. *-commutative 53.5%

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

       9. swap-sqr 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

       12. swap-sqr 53.2%

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

       13. rem-square-sqrt 54.0%

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

       14. pow2 54.0%

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

       15. metadata-eval 54.0%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   if 4.20000000000000037e84 < y-scale < 1.52e140

   1. Initial program 8.3%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in a around 0 0.2%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right)\right)} \]

   7. Simplified 0.2%

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

   8. Taylor expanded in angle around 0 34.7%

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

   if 1.52e140 < y-scale

   1. Initial program 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x-scale around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. distribute-lft-out 76.6%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in angle around 0 85.5%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 85.4%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 89.2%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.5 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \left({2}^{0.25} \cdot \left(x-scale \cdot {2}^{0.25}\right)\right)\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt[3]{{\pi}^{1.5}}\right)}^{2}\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.4 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_3 := b\_m \cdot \sin t\_1\\ \mathbf{if}\;y-scale\_m \leq 2.2 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \left({2}^{0.25} \cdot \left(x-scale\_m \cdot {2}^{0.25}\right)\right)\right) \cdot \mathsf{hypot}\left(\cos t\_1 \cdot a, t\_3\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 3.9 \cdot 10^{-53}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{e^{3 \cdot \log \pi}}\right), b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 2 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale\_m \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(a, t\_3\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 3.3 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b\_m}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_2, b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* y-scale_m (sqrt 8.0))))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (* PI (* 0.005555555555555556 angle)))
        (t_3 (* b_m (sin t_1))))
   (if (<= y-scale_m 2.2e-99)
     (*
      0.25
      (*
       (* (sqrt 8.0) (* (pow 2.0 0.25) (* x-scale_m (pow 2.0 0.25))))
       (hypot (* (cos t_1) a) t_3)))
     (if (<= y-scale_m 3.9e-53)
       (*
        t_0
        (*
         (sqrt 2.0)
         (hypot
          (*
           a
           (sin
            (* (* 0.005555555555555556 angle) (cbrt (exp (* 3.0 (log PI)))))))
          b_m)))
       (if (<= y-scale_m 2e+18)
         (*
          0.25
          (*
           (* (sqrt 8.0) (pow (sqrt (* x-scale_m (sqrt 2.0))) 2.0))
           (hypot a t_3)))
         (if (<= y-scale_m 3.3e+84)
           (*
            (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
            (* (sqrt 2.0) (hypot (* a t_2) b_m)))
           (if (<= y-scale_m 1.52e+140)
             (*
              -0.25
              (*
               (* x-scale_m (* y-scale_m (- (sqrt 8.0))))
               (/ (* (sqrt 2.0) b_m) x-scale_m)))
             (* t_0 (* (sqrt 2.0) (hypot (* a (sin t_2)) b_m))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * sqrt(8.0));
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_3 = b_m * sin(t_1);
	double tmp;
	if (y_45_scale_m <= 2.2e-99) {
		tmp = 0.25 * ((sqrt(8.0) * (pow(2.0, 0.25) * (x_45_scale_m * pow(2.0, 0.25)))) * hypot((cos(t_1) * a), t_3));
	} else if (y_45_scale_m <= 3.9e-53) {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * cbrt(exp((3.0 * log(((double) M_PI)))))))), b_m));
	} else if (y_45_scale_m <= 2e+18) {
		tmp = 0.25 * ((sqrt(8.0) * pow(sqrt((x_45_scale_m * sqrt(2.0))), 2.0)) * hypot(a, t_3));
	} else if (y_45_scale_m <= 3.3e+84) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_2), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(t_2)), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * Math.sqrt(8.0));
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.PI * (0.005555555555555556 * angle);
	double t_3 = b_m * Math.sin(t_1);
	double tmp;
	if (y_45_scale_m <= 2.2e-99) {
		tmp = 0.25 * ((Math.sqrt(8.0) * (Math.pow(2.0, 0.25) * (x_45_scale_m * Math.pow(2.0, 0.25)))) * Math.hypot((Math.cos(t_1) * a), t_3));
	} else if (y_45_scale_m <= 3.9e-53) {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.cbrt(Math.exp((3.0 * Math.log(Math.PI))))))), b_m));
	} else if (y_45_scale_m <= 2e+18) {
		tmp = 0.25 * ((Math.sqrt(8.0) * Math.pow(Math.sqrt((x_45_scale_m * Math.sqrt(2.0))), 2.0)) * Math.hypot(a, t_3));
	} else if (y_45_scale_m <= 3.3e+84) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_2), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -Math.sqrt(8.0))) * ((Math.sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_2)), b_m));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0)))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_3 = Float64(b_m * sin(t_1))
	tmp = 0.0
	if (y_45_scale_m <= 2.2e-99)
		tmp = Float64(0.25 * Float64(Float64(sqrt(8.0) * Float64((2.0 ^ 0.25) * Float64(x_45_scale_m * (2.0 ^ 0.25)))) * hypot(Float64(cos(t_1) * a), t_3)));
	elseif (y_45_scale_m <= 3.9e-53)
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * cbrt(exp(Float64(3.0 * log(pi))))))), b_m)));
	elseif (y_45_scale_m <= 2e+18)
		tmp = Float64(0.25 * Float64(Float64(sqrt(8.0) * (sqrt(Float64(x_45_scale_m * sqrt(2.0))) ^ 2.0)) * hypot(a, t_3)));
	elseif (y_45_scale_m <= 3.3e+84)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_2), b_m)));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = Float64(-0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(-sqrt(8.0)))) * Float64(Float64(sqrt(2.0) * b_m) / x_45_scale_m)));
	else
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_2)), b_m)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b$95$m * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.2e-99], N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(x$45$scale$95$m * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[t$95$1], $MachinePrecision] * a), $MachinePrecision] ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.9e-53], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Exp[N[(3.0 * N[Log[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2e+18], N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[Power[N[Sqrt[N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[a ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.3e+84], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.52e+140], N[(-0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * b$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y-scale < 2.20000000000000004e-99

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

       1. add-sqr-sqrt 25.0%

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

       2. pow2 25.0%

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

       3. *-commutative 25.0%

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

   12. Applied egg-rr 25.0%

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

       1. unpow2 25.0%

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

       2. add-sqr-sqrt 26.1%

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

       3. add-sqr-sqrt 26.2%

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

       4. associate-*l* 26.1%

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

       5. pow1/2 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       6. sqrt-pow1 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       7. metadata-eval 26.1%

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

       8. pow1/2 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left({2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       9. sqrt-pow1 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left({2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       10. metadata-eval 26.1%

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

   14. Applied egg-rr 26.1%

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

   if 2.20000000000000004e-99 < y-scale < 3.9000000000000002e-53

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in x-scale around 0 29.2%

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

      1. associate-*r* 29.2%

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

      2. distribute-lft-out 29.2%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in angle around 0 32.3%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 46.7%

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

       1. add-cbrt-cube 46.5%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right), b\right) \cdot \sqrt{2}\right) \]

       2. pow3 46.5%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right), b\right) \cdot \sqrt{2}\right) \]

   12. Applied egg-rr 46.5%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right), b\right) \cdot \sqrt{2}\right) \]
   13. Step-by-step derivation

       1. add-exp-log 46.5%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\color{blue}{e^{\log \left({\pi}^{3}\right)}}}\right), b\right) \cdot \sqrt{2}\right) \]

       2. log-pow 46.9%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{e^{\color{blue}{3 \cdot \log \pi}}}\right), b\right) \cdot \sqrt{2}\right) \]

   14. Applied egg-rr 46.9%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\color{blue}{e^{3 \cdot \log \pi}}}\right), b\right) \cdot \sqrt{2}\right) \]

   if 3.9000000000000002e-53 < y-scale < 2e18

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in y-scale around 0 12.1%

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

      1. associate-*r* 12.1%

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

      2. distribute-lft-out 12.1%

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

   7. Simplified 12.2%

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

   8. Taylor expanded in x-scale around 0 12.2%

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

   10. Simplified 13.4%

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

       1. add-sqr-sqrt 12.8%

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

       2. pow2 12.8%

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

       3. *-commutative 12.8%

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

   12. Applied egg-rr 12.8%

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

   13. Taylor expanded in angle around 0 12.8%

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

   if 2e18 < y-scale < 3.30000000000000017e84

   1. Initial program 0.5%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in x-scale around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. distribute-lft-out 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 45.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 54.5%

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

   11. Taylor expanded in angle around 0 53.5%

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

       1. associate-*r* 53.5%

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

   13. Simplified 53.5%

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

       1. add-sqr-sqrt 53.3%

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

       2. sqrt-unprod 53.5%

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

       3. swap-sqr 53.5%

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

       4. metadata-eval 53.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 53.5%

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

       6. swap-sqr 53.5%

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

       7. *-commutative 53.5%

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

       8. *-commutative 53.5%

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

       9. swap-sqr 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

       12. swap-sqr 53.2%

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

       13. rem-square-sqrt 54.0%

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

       14. pow2 54.0%

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

       15. metadata-eval 54.0%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   if 3.30000000000000017e84 < y-scale < 1.52e140

   1. Initial program 8.3%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in a around 0 0.2%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right)\right)} \]

   7. Simplified 0.2%

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

   8. Taylor expanded in angle around 0 34.7%

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

   if 1.52e140 < y-scale

   1. Initial program 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x-scale around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. distribute-lft-out 76.6%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in angle around 0 85.5%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 85.4%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 89.2%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.2 \cdot 10^{-99}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \left({2}^{0.25} \cdot \left(x-scale \cdot {2}^{0.25}\right)\right)\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.9 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{e^{3 \cdot \log \pi}}\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 2 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.3 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 4.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_3 := b\_m \cdot \sin t\_2\\ \mathbf{if}\;y-scale\_m \leq 2.25 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \left({2}^{0.25} \cdot \left(x-scale\_m \cdot {2}^{0.25}\right)\right)\right) \cdot \mathsf{hypot}\left(\cos t\_2 \cdot a, t\_3\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 7.4 \cdot 10^{-53}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt{\pi}\right)}^{2}\right), b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale\_m \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(a, t\_3\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_1, b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b\_m}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_1, b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* y-scale_m (sqrt 8.0))))
        (t_1 (* PI (* 0.005555555555555556 angle)))
        (t_2 (* 0.005555555555555556 (* angle PI)))
        (t_3 (* b_m (sin t_2))))
   (if (<= y-scale_m 2.25e-95)
     (*
      0.25
      (*
       (* (sqrt 8.0) (* (pow 2.0 0.25) (* x-scale_m (pow 2.0 0.25))))
       (hypot (* (cos t_2) a) t_3)))
     (if (<= y-scale_m 7.4e-53)
       (*
        t_0
        (*
         (sqrt 2.0)
         (hypot
          (* a (sin (* (* 0.005555555555555556 angle) (pow (sqrt PI) 2.0))))
          b_m)))
       (if (<= y-scale_m 2.7e+18)
         (*
          0.25
          (*
           (* (sqrt 8.0) (pow (sqrt (* x-scale_m (sqrt 2.0))) 2.0))
           (hypot a t_3)))
         (if (<= y-scale_m 4.2e+84)
           (*
            (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
            (* (sqrt 2.0) (hypot (* a t_1) b_m)))
           (if (<= y-scale_m 1.52e+140)
             (*
              -0.25
              (*
               (* x-scale_m (* y-scale_m (- (sqrt 8.0))))
               (/ (* (sqrt 2.0) b_m) x-scale_m)))
             (* t_0 (* (sqrt 2.0) (hypot (* a (sin t_1)) b_m))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * sqrt(8.0));
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_2 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_3 = b_m * sin(t_2);
	double tmp;
	if (y_45_scale_m <= 2.25e-95) {
		tmp = 0.25 * ((sqrt(8.0) * (pow(2.0, 0.25) * (x_45_scale_m * pow(2.0, 0.25)))) * hypot((cos(t_2) * a), t_3));
	} else if (y_45_scale_m <= 7.4e-53) {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * pow(sqrt(((double) M_PI)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 2.7e+18) {
		tmp = 0.25 * ((sqrt(8.0) * pow(sqrt((x_45_scale_m * sqrt(2.0))), 2.0)) * hypot(a, t_3));
	} else if (y_45_scale_m <= 4.2e+84) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_1), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(t_1)), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * Math.sqrt(8.0));
	double t_1 = Math.PI * (0.005555555555555556 * angle);
	double t_2 = 0.005555555555555556 * (angle * Math.PI);
	double t_3 = b_m * Math.sin(t_2);
	double tmp;
	if (y_45_scale_m <= 2.25e-95) {
		tmp = 0.25 * ((Math.sqrt(8.0) * (Math.pow(2.0, 0.25) * (x_45_scale_m * Math.pow(2.0, 0.25)))) * Math.hypot((Math.cos(t_2) * a), t_3));
	} else if (y_45_scale_m <= 7.4e-53) {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.pow(Math.sqrt(Math.PI), 2.0)))), b_m));
	} else if (y_45_scale_m <= 2.7e+18) {
		tmp = 0.25 * ((Math.sqrt(8.0) * Math.pow(Math.sqrt((x_45_scale_m * Math.sqrt(2.0))), 2.0)) * Math.hypot(a, t_3));
	} else if (y_45_scale_m <= 4.2e+84) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_1), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -Math.sqrt(8.0))) * ((Math.sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_1)), b_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.25 * (y_45_scale_m * math.sqrt(8.0))
	t_1 = math.pi * (0.005555555555555556 * angle)
	t_2 = 0.005555555555555556 * (angle * math.pi)
	t_3 = b_m * math.sin(t_2)
	tmp = 0
	if y_45_scale_m <= 2.25e-95:
		tmp = 0.25 * ((math.sqrt(8.0) * (math.pow(2.0, 0.25) * (x_45_scale_m * math.pow(2.0, 0.25)))) * math.hypot((math.cos(t_2) * a), t_3))
	elif y_45_scale_m <= 7.4e-53:
		tmp = t_0 * (math.sqrt(2.0) * math.hypot((a * math.sin(((0.005555555555555556 * angle) * math.pow(math.sqrt(math.pi), 2.0)))), b_m))
	elif y_45_scale_m <= 2.7e+18:
		tmp = 0.25 * ((math.sqrt(8.0) * math.pow(math.sqrt((x_45_scale_m * math.sqrt(2.0))), 2.0)) * math.hypot(a, t_3))
	elif y_45_scale_m <= 4.2e+84:
		tmp = math.sqrt(((8.0 * math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (math.sqrt(2.0) * math.hypot((a * t_1), b_m))
	elif y_45_scale_m <= 1.52e+140:
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -math.sqrt(8.0))) * ((math.sqrt(2.0) * b_m) / x_45_scale_m))
	else:
		tmp = t_0 * (math.sqrt(2.0) * math.hypot((a * math.sin(t_1)), b_m))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0)))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_2 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_3 = Float64(b_m * sin(t_2))
	tmp = 0.0
	if (y_45_scale_m <= 2.25e-95)
		tmp = Float64(0.25 * Float64(Float64(sqrt(8.0) * Float64((2.0 ^ 0.25) * Float64(x_45_scale_m * (2.0 ^ 0.25)))) * hypot(Float64(cos(t_2) * a), t_3)));
	elseif (y_45_scale_m <= 7.4e-53)
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * (sqrt(pi) ^ 2.0)))), b_m)));
	elseif (y_45_scale_m <= 2.7e+18)
		tmp = Float64(0.25 * Float64(Float64(sqrt(8.0) * (sqrt(Float64(x_45_scale_m * sqrt(2.0))) ^ 2.0)) * hypot(a, t_3)));
	elseif (y_45_scale_m <= 4.2e+84)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_1), b_m)));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = Float64(-0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(-sqrt(8.0)))) * Float64(Float64(sqrt(2.0) * b_m) / x_45_scale_m)));
	else
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_1)), b_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.25 * (y_45_scale_m * sqrt(8.0));
	t_1 = pi * (0.005555555555555556 * angle);
	t_2 = 0.005555555555555556 * (angle * pi);
	t_3 = b_m * sin(t_2);
	tmp = 0.0;
	if (y_45_scale_m <= 2.25e-95)
		tmp = 0.25 * ((sqrt(8.0) * ((2.0 ^ 0.25) * (x_45_scale_m * (2.0 ^ 0.25)))) * hypot((cos(t_2) * a), t_3));
	elseif (y_45_scale_m <= 7.4e-53)
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * (sqrt(pi) ^ 2.0)))), b_m));
	elseif (y_45_scale_m <= 2.7e+18)
		tmp = 0.25 * ((sqrt(8.0) * (sqrt((x_45_scale_m * sqrt(2.0))) ^ 2.0)) * hypot(a, t_3));
	elseif (y_45_scale_m <= 4.2e+84)
		tmp = sqrt(((8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_1), b_m));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	else
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(t_1)), b_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b$95$m * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.25e-95], N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(x$45$scale$95$m * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[t$95$2], $MachinePrecision] * a), $MachinePrecision] ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 7.4e-53], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.7e+18], N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[Power[N[Sqrt[N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[a ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 4.2e+84], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.52e+140], N[(-0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * b$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y-scale < 2.25e-95

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

       1. add-sqr-sqrt 25.0%

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

       2. pow2 25.0%

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

       3. *-commutative 25.0%

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

   12. Applied egg-rr 25.0%

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

       1. unpow2 25.0%

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

       2. add-sqr-sqrt 26.1%

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

       3. add-sqr-sqrt 26.2%

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

       4. associate-*l* 26.1%

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

       5. pow1/2 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       6. sqrt-pow1 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       7. metadata-eval 26.1%

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

       8. pow1/2 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left({2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       9. sqrt-pow1 26.1%

          \[\leadsto 0.25 \cdot \left(\left(\left({2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x-scale\right)\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

       10. metadata-eval 26.1%

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

   14. Applied egg-rr 26.1%

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

   if 2.25e-95 < y-scale < 7.39999999999999965e-53

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in x-scale around 0 29.2%

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

      1. associate-*r* 29.2%

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

      2. distribute-lft-out 29.2%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in angle around 0 32.3%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 46.7%

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

       1. add-sqr-sqrt 46.9%

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

       2. pow2 46.9%

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

   12. Applied egg-rr 46.9%

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

   if 7.39999999999999965e-53 < y-scale < 2.7e18

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in y-scale around 0 12.1%

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

      1. associate-*r* 12.1%

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

      2. distribute-lft-out 12.1%

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

   7. Simplified 12.2%

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

   8. Taylor expanded in x-scale around 0 12.2%

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

   10. Simplified 13.4%

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

       1. add-sqr-sqrt 12.8%

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

       2. pow2 12.8%

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

       3. *-commutative 12.8%

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

   12. Applied egg-rr 12.8%

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

   13. Taylor expanded in angle around 0 12.8%

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

   if 2.7e18 < y-scale < 4.20000000000000037e84

   1. Initial program 0.5%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in x-scale around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. distribute-lft-out 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 45.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 54.5%

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

   11. Taylor expanded in angle around 0 53.5%

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

       1. associate-*r* 53.5%

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

   13. Simplified 53.5%

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

       1. add-sqr-sqrt 53.3%

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

       2. sqrt-unprod 53.5%

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

       3. swap-sqr 53.5%

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

       4. metadata-eval 53.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 53.5%

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

       6. swap-sqr 53.5%

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

       7. *-commutative 53.5%

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

       8. *-commutative 53.5%

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

       9. swap-sqr 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

       12. swap-sqr 53.2%

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

       13. rem-square-sqrt 54.0%

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

       14. pow2 54.0%

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

       15. metadata-eval 54.0%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   if 4.20000000000000037e84 < y-scale < 1.52e140

   1. Initial program 8.3%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in a around 0 0.2%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right)\right)} \]

   7. Simplified 0.2%

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

   8. Taylor expanded in angle around 0 34.7%

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

   if 1.52e140 < y-scale

   1. Initial program 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x-scale around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. distribute-lft-out 76.6%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in angle around 0 85.5%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 85.4%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 89.2%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.25 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot \left({2}^{0.25} \cdot \left(x-scale \cdot {2}^{0.25}\right)\right)\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 7.4 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt{\pi}\right)}^{2}\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.2 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\\ t_1 := 0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale\_m \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(a, b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ t_2 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;y-scale\_m \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y-scale\_m \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt{\pi}\right)}^{2}\right), b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 2.1 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y-scale\_m \leq 6.5 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b\_m}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_2, b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* y-scale_m (sqrt 8.0))))
        (t_1
         (*
          0.25
          (*
           (* (sqrt 8.0) (pow (sqrt (* x-scale_m (sqrt 2.0))) 2.0))
           (hypot a (* b_m (sin (* 0.005555555555555556 (* angle PI))))))))
        (t_2 (* PI (* 0.005555555555555556 angle))))
   (if (<= y-scale_m 1.1e-95)
     t_1
     (if (<= y-scale_m 4.5e-53)
       (*
        t_0
        (*
         (sqrt 2.0)
         (hypot
          (* a (sin (* (* 0.005555555555555556 angle) (pow (sqrt PI) 2.0))))
          b_m)))
       (if (<= y-scale_m 2.1e+18)
         t_1
         (if (<= y-scale_m 6.5e+83)
           (*
            (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
            (* (sqrt 2.0) (hypot (* a t_2) b_m)))
           (if (<= y-scale_m 1.52e+140)
             (*
              -0.25
              (*
               (* x-scale_m (* y-scale_m (- (sqrt 8.0))))
               (/ (* (sqrt 2.0) b_m) x-scale_m)))
             (* t_0 (* (sqrt 2.0) (hypot (* a (sin t_2)) b_m))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * sqrt(8.0));
	double t_1 = 0.25 * ((sqrt(8.0) * pow(sqrt((x_45_scale_m * sqrt(2.0))), 2.0)) * hypot(a, (b_m * sin((0.005555555555555556 * (angle * ((double) M_PI)))))));
	double t_2 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 1.1e-95) {
		tmp = t_1;
	} else if (y_45_scale_m <= 4.5e-53) {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * pow(sqrt(((double) M_PI)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 2.1e+18) {
		tmp = t_1;
	} else if (y_45_scale_m <= 6.5e+83) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_2), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(t_2)), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * Math.sqrt(8.0));
	double t_1 = 0.25 * ((Math.sqrt(8.0) * Math.pow(Math.sqrt((x_45_scale_m * Math.sqrt(2.0))), 2.0)) * Math.hypot(a, (b_m * Math.sin((0.005555555555555556 * (angle * Math.PI))))));
	double t_2 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 1.1e-95) {
		tmp = t_1;
	} else if (y_45_scale_m <= 4.5e-53) {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.pow(Math.sqrt(Math.PI), 2.0)))), b_m));
	} else if (y_45_scale_m <= 2.1e+18) {
		tmp = t_1;
	} else if (y_45_scale_m <= 6.5e+83) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_2), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -Math.sqrt(8.0))) * ((Math.sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_2)), b_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.25 * (y_45_scale_m * math.sqrt(8.0))
	t_1 = 0.25 * ((math.sqrt(8.0) * math.pow(math.sqrt((x_45_scale_m * math.sqrt(2.0))), 2.0)) * math.hypot(a, (b_m * math.sin((0.005555555555555556 * (angle * math.pi))))))
	t_2 = math.pi * (0.005555555555555556 * angle)
	tmp = 0
	if y_45_scale_m <= 1.1e-95:
		tmp = t_1
	elif y_45_scale_m <= 4.5e-53:
		tmp = t_0 * (math.sqrt(2.0) * math.hypot((a * math.sin(((0.005555555555555556 * angle) * math.pow(math.sqrt(math.pi), 2.0)))), b_m))
	elif y_45_scale_m <= 2.1e+18:
		tmp = t_1
	elif y_45_scale_m <= 6.5e+83:
		tmp = math.sqrt(((8.0 * math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (math.sqrt(2.0) * math.hypot((a * t_2), b_m))
	elif y_45_scale_m <= 1.52e+140:
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -math.sqrt(8.0))) * ((math.sqrt(2.0) * b_m) / x_45_scale_m))
	else:
		tmp = t_0 * (math.sqrt(2.0) * math.hypot((a * math.sin(t_2)), b_m))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0)))
	t_1 = Float64(0.25 * Float64(Float64(sqrt(8.0) * (sqrt(Float64(x_45_scale_m * sqrt(2.0))) ^ 2.0)) * hypot(a, Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * pi)))))))
	t_2 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (y_45_scale_m <= 1.1e-95)
		tmp = t_1;
	elseif (y_45_scale_m <= 4.5e-53)
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * (sqrt(pi) ^ 2.0)))), b_m)));
	elseif (y_45_scale_m <= 2.1e+18)
		tmp = t_1;
	elseif (y_45_scale_m <= 6.5e+83)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_2), b_m)));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = Float64(-0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(-sqrt(8.0)))) * Float64(Float64(sqrt(2.0) * b_m) / x_45_scale_m)));
	else
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_2)), b_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.25 * (y_45_scale_m * sqrt(8.0));
	t_1 = 0.25 * ((sqrt(8.0) * (sqrt((x_45_scale_m * sqrt(2.0))) ^ 2.0)) * hypot(a, (b_m * sin((0.005555555555555556 * (angle * pi))))));
	t_2 = pi * (0.005555555555555556 * angle);
	tmp = 0.0;
	if (y_45_scale_m <= 1.1e-95)
		tmp = t_1;
	elseif (y_45_scale_m <= 4.5e-53)
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * (sqrt(pi) ^ 2.0)))), b_m));
	elseif (y_45_scale_m <= 2.1e+18)
		tmp = t_1;
	elseif (y_45_scale_m <= 6.5e+83)
		tmp = sqrt(((8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_2), b_m));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	else
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(t_2)), b_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[Power[N[Sqrt[N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[a ^ 2 + N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 1.1e-95], t$95$1, If[LessEqual[y$45$scale$95$m, 4.5e-53], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.1e+18], t$95$1, If[LessEqual[y$45$scale$95$m, 6.5e+83], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.52e+140], N[(-0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * b$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y-scale < 1.0999999999999999e-95 or 4.49999999999999985e-53 < y-scale < 2.1e18

   1. Initial program 1.9%

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

   3. Simplified 2.0%

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

   5. Taylor expanded in y-scale around 0 9.3%

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

      1. associate-*r* 9.3%

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

      2. distribute-lft-out 9.3%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in x-scale around 0 23.9%

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

   10. Simplified 24.9%

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

       1. add-sqr-sqrt 23.9%

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

       2. pow2 23.9%

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

       3. *-commutative 23.9%

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

   12. Applied egg-rr 23.9%

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

   13. Taylor expanded in angle around 0 24.0%

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

   if 1.0999999999999999e-95 < y-scale < 4.49999999999999985e-53

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in x-scale around 0 29.2%

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

      1. associate-*r* 29.2%

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

      2. distribute-lft-out 29.2%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in angle around 0 32.3%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 46.7%

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

       1. add-sqr-sqrt 46.9%

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

       2. pow2 46.9%

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

   12. Applied egg-rr 46.9%

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

   if 2.1e18 < y-scale < 6.5000000000000003e83

   1. Initial program 0.5%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in x-scale around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. distribute-lft-out 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 45.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 54.5%

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

   11. Taylor expanded in angle around 0 53.5%

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

       1. associate-*r* 53.5%

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

   13. Simplified 53.5%

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

       1. add-sqr-sqrt 53.3%

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

       2. sqrt-unprod 53.5%

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

       3. swap-sqr 53.5%

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

       4. metadata-eval 53.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 53.5%

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

       6. swap-sqr 53.5%

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

       7. *-commutative 53.5%

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

       8. *-commutative 53.5%

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

       9. swap-sqr 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

       12. swap-sqr 53.2%

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

       13. rem-square-sqrt 54.0%

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

       14. pow2 54.0%

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

       15. metadata-eval 54.0%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   if 6.5000000000000003e83 < y-scale < 1.52e140

   1. Initial program 8.3%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in a around 0 0.2%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right)\right)} \]

   7. Simplified 0.2%

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

   8. Taylor expanded in angle around 0 34.7%

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

   if 1.52e140 < y-scale

   1. Initial program 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x-scale around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. distribute-lft-out 76.6%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in angle around 0 85.5%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 85.4%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 89.2%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.5 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt{\pi}\right)}^{2}\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.1 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{8} \cdot {\left(\sqrt{x-scale \cdot \sqrt{2}}\right)}^{2}\right) \cdot \mathsf{hypot}\left(a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 6.5 \cdot 10^{+83}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := y-scale\_m \cdot \sqrt{8}\\ t_1 := 0.25 \cdot t\_0\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_3 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_4 := \sin t\_3\\ \mathbf{if}\;y-scale\_m \leq 2.6 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos t\_2 \cdot a, b\_m \cdot \sin t\_2\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 5 \cdot 10^{-53}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt{\pi}\right)}^{2}\right), b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 8 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot t\_0\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a}{y-scale\_m}, t\_4 \cdot \frac{b\_m}{y-scale\_m}\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 3.8 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_3, b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b\_m}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_4, b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* y-scale_m (sqrt 8.0)))
        (t_1 (* 0.25 t_0))
        (t_2 (* 0.005555555555555556 (* angle PI)))
        (t_3 (* PI (* 0.005555555555555556 angle)))
        (t_4 (sin t_3)))
   (if (<= y-scale_m 2.6e-95)
     (*
      0.25
      (*
       (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0))
       (hypot (* (cos t_2) a) (* b_m (sin t_2)))))
     (if (<= y-scale_m 5e-53)
       (*
        t_1
        (*
         (sqrt 2.0)
         (hypot
          (* a (sin (* (* 0.005555555555555556 angle) (pow (sqrt PI) 2.0))))
          b_m)))
       (if (<= y-scale_m 8e+18)
         (*
          0.25
          (*
           (* x-scale_m t_0)
           (* (sqrt 2.0) (hypot (/ a y-scale_m) (* t_4 (/ b_m y-scale_m))))))
         (if (<= y-scale_m 3.8e+84)
           (*
            (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
            (* (sqrt 2.0) (hypot (* a t_3) b_m)))
           (if (<= y-scale_m 1.52e+140)
             (*
              -0.25
              (*
               (* x-scale_m (* y-scale_m (- (sqrt 8.0))))
               (/ (* (sqrt 2.0) b_m) x-scale_m)))
             (* t_1 (* (sqrt 2.0) (hypot (* a t_4) b_m))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * sqrt(8.0);
	double t_1 = 0.25 * t_0;
	double t_2 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_3 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_4 = sin(t_3);
	double tmp;
	if (y_45_scale_m <= 2.6e-95) {
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot((cos(t_2) * a), (b_m * sin(t_2))));
	} else if (y_45_scale_m <= 5e-53) {
		tmp = t_1 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * pow(sqrt(((double) M_PI)), 2.0)))), b_m));
	} else if (y_45_scale_m <= 8e+18) {
		tmp = 0.25 * ((x_45_scale_m * t_0) * (sqrt(2.0) * hypot((a / y_45_scale_m), (t_4 * (b_m / y_45_scale_m)))));
	} else if (y_45_scale_m <= 3.8e+84) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_3), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_1 * (sqrt(2.0) * hypot((a * t_4), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * Math.sqrt(8.0);
	double t_1 = 0.25 * t_0;
	double t_2 = 0.005555555555555556 * (angle * Math.PI);
	double t_3 = Math.PI * (0.005555555555555556 * angle);
	double t_4 = Math.sin(t_3);
	double tmp;
	if (y_45_scale_m <= 2.6e-95) {
		tmp = 0.25 * (((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * Math.hypot((Math.cos(t_2) * a), (b_m * Math.sin(t_2))));
	} else if (y_45_scale_m <= 5e-53) {
		tmp = t_1 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.pow(Math.sqrt(Math.PI), 2.0)))), b_m));
	} else if (y_45_scale_m <= 8e+18) {
		tmp = 0.25 * ((x_45_scale_m * t_0) * (Math.sqrt(2.0) * Math.hypot((a / y_45_scale_m), (t_4 * (b_m / y_45_scale_m)))));
	} else if (y_45_scale_m <= 3.8e+84) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_3), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -Math.sqrt(8.0))) * ((Math.sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_1 * (Math.sqrt(2.0) * Math.hypot((a * t_4), b_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = y_45_scale_m * math.sqrt(8.0)
	t_1 = 0.25 * t_0
	t_2 = 0.005555555555555556 * (angle * math.pi)
	t_3 = math.pi * (0.005555555555555556 * angle)
	t_4 = math.sin(t_3)
	tmp = 0
	if y_45_scale_m <= 2.6e-95:
		tmp = 0.25 * (((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0)) * math.hypot((math.cos(t_2) * a), (b_m * math.sin(t_2))))
	elif y_45_scale_m <= 5e-53:
		tmp = t_1 * (math.sqrt(2.0) * math.hypot((a * math.sin(((0.005555555555555556 * angle) * math.pow(math.sqrt(math.pi), 2.0)))), b_m))
	elif y_45_scale_m <= 8e+18:
		tmp = 0.25 * ((x_45_scale_m * t_0) * (math.sqrt(2.0) * math.hypot((a / y_45_scale_m), (t_4 * (b_m / y_45_scale_m)))))
	elif y_45_scale_m <= 3.8e+84:
		tmp = math.sqrt(((8.0 * math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (math.sqrt(2.0) * math.hypot((a * t_3), b_m))
	elif y_45_scale_m <= 1.52e+140:
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -math.sqrt(8.0))) * ((math.sqrt(2.0) * b_m) / x_45_scale_m))
	else:
		tmp = t_1 * (math.sqrt(2.0) * math.hypot((a * t_4), b_m))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(y_45_scale_m * sqrt(8.0))
	t_1 = Float64(0.25 * t_0)
	t_2 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_3 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_4 = sin(t_3)
	tmp = 0.0
	if (y_45_scale_m <= 2.6e-95)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot(Float64(cos(t_2) * a), Float64(b_m * sin(t_2)))));
	elseif (y_45_scale_m <= 5e-53)
		tmp = Float64(t_1 * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * (sqrt(pi) ^ 2.0)))), b_m)));
	elseif (y_45_scale_m <= 8e+18)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * t_0) * Float64(sqrt(2.0) * hypot(Float64(a / y_45_scale_m), Float64(t_4 * Float64(b_m / y_45_scale_m))))));
	elseif (y_45_scale_m <= 3.8e+84)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_3), b_m)));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = Float64(-0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(-sqrt(8.0)))) * Float64(Float64(sqrt(2.0) * b_m) / x_45_scale_m)));
	else
		tmp = Float64(t_1 * Float64(sqrt(2.0) * hypot(Float64(a * t_4), b_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = y_45_scale_m * sqrt(8.0);
	t_1 = 0.25 * t_0;
	t_2 = 0.005555555555555556 * (angle * pi);
	t_3 = pi * (0.005555555555555556 * angle);
	t_4 = sin(t_3);
	tmp = 0.0;
	if (y_45_scale_m <= 2.6e-95)
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot((cos(t_2) * a), (b_m * sin(t_2))));
	elseif (y_45_scale_m <= 5e-53)
		tmp = t_1 * (sqrt(2.0) * hypot((a * sin(((0.005555555555555556 * angle) * (sqrt(pi) ^ 2.0)))), b_m));
	elseif (y_45_scale_m <= 8e+18)
		tmp = 0.25 * ((x_45_scale_m * t_0) * (sqrt(2.0) * hypot((a / y_45_scale_m), (t_4 * (b_m / y_45_scale_m)))));
	elseif (y_45_scale_m <= 3.8e+84)
		tmp = sqrt(((8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_3), b_m));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	else
		tmp = t_1 * (sqrt(2.0) * hypot((a * t_4), b_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$3], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.6e-95], N[(0.25 * N[(N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[t$95$2], $MachinePrecision] * a), $MachinePrecision] ^ 2 + N[(b$95$m * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 5e-53], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 8e+18], N[(0.25 * N[(N[(x$45$scale$95$m * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a / y$45$scale$95$m), $MachinePrecision] ^ 2 + N[(t$95$4 * N[(b$95$m / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.8e+84], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$3), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.52e+140], N[(-0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * b$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$4), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y-scale < 2.60000000000000001e-95

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

   if 2.60000000000000001e-95 < y-scale < 5e-53

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in x-scale around 0 29.2%

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

      1. associate-*r* 29.2%

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

      2. distribute-lft-out 29.2%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in angle around 0 32.3%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 32.3%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 46.7%

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

       1. add-sqr-sqrt 46.9%

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

       2. pow2 46.9%

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

   12. Applied egg-rr 46.9%

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

   if 5e-53 < y-scale < 8e18

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in x-scale around inf 12.3%

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

   6. Taylor expanded in angle around 0 12.3%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\color{blue}{1}}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
   7. Step-by-step derivation

      1. pow1/2 12.3%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}}\right) \]

      2. distribute-lft-out 12.3%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(2 \cdot \left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)}}^{0.5}\right) \]

      3. unpow-prod-down 12.2%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({2}^{0.5} \cdot {\left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}\right)}\right) \]

      4. pow1/2 12.2%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot {\left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}\right)\right) \]

   8. Applied egg-rr 12.4%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot {\left({\left(\frac{a}{y-scale}\right)}^{2} + {\left(\frac{b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale}\right)}^{2}\right)}^{0.5}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow1/2 12.4%

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

      2. unpow2 12.4%

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

      3. unpow2 12.4%

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

      4. hypot-define 13.3%

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

      5. *-commutative 13.3%

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

      6. associate-/l* 13.3%

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

      7. associate-*r* 13.3%

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

   10. Simplified 13.3%

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

   if 8e18 < y-scale < 3.8000000000000001e84

   1. Initial program 0.5%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in x-scale around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. distribute-lft-out 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 45.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 54.5%

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

   11. Taylor expanded in angle around 0 53.5%

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

       1. associate-*r* 53.5%

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

   13. Simplified 53.5%

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

       1. add-sqr-sqrt 53.3%

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

       2. sqrt-unprod 53.5%

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

       3. swap-sqr 53.5%

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

       4. metadata-eval 53.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 53.5%

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

       6. swap-sqr 53.5%

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

       7. *-commutative 53.5%

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

       8. *-commutative 53.5%

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

       9. swap-sqr 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

       12. swap-sqr 53.2%

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

       13. rem-square-sqrt 54.0%

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

       14. pow2 54.0%

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

       15. metadata-eval 54.0%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   if 3.8000000000000001e84 < y-scale < 1.52e140

   1. Initial program 8.3%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in a around 0 0.2%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right)\right)} \]

   7. Simplified 0.2%

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

   8. Taylor expanded in angle around 0 34.7%

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

   if 1.52e140 < y-scale

   1. Initial program 6.9%

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

   3. Simplified 6.9%

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

   5. Taylor expanded in x-scale around 0 76.6%

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

      1. associate-*r* 76.6%

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

      2. distribute-lft-out 76.6%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in angle around 0 85.5%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 85.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 85.4%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 89.2%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.6 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 5 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot {\left(\sqrt{\pi}\right)}^{2}\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 8 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a}{y-scale}, \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.8 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := y-scale\_m \cdot \sqrt{8}\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_3 := \sin t\_2\\ t_4 := \left(0.25 \cdot t\_0\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_3, b\_m\right)\right)\\ \mathbf{if}\;y-scale\_m \leq 2.25 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos t\_1 \cdot a, b\_m \cdot \sin t\_1\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.5 \cdot 10^{-52}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y-scale\_m \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot t\_0\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a}{y-scale\_m}, t\_3 \cdot \frac{b\_m}{y-scale\_m}\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 3.8 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b\_m}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* y-scale_m (sqrt 8.0)))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (* PI (* 0.005555555555555556 angle)))
        (t_3 (sin t_2))
        (t_4 (* (* 0.25 t_0) (* (sqrt 2.0) (hypot (* a t_3) b_m)))))
   (if (<= y-scale_m 2.25e-95)
     (*
      0.25
      (*
       (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0))
       (hypot (* (cos t_1) a) (* b_m (sin t_1)))))
     (if (<= y-scale_m 1.5e-52)
       t_4
       (if (<= y-scale_m 1.4e+19)
         (*
          0.25
          (*
           (* x-scale_m t_0)
           (* (sqrt 2.0) (hypot (/ a y-scale_m) (* t_3 (/ b_m y-scale_m))))))
         (if (<= y-scale_m 3.8e+84)
           (*
            (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
            (* (sqrt 2.0) (hypot (* a t_2) b_m)))
           (if (<= y-scale_m 1.52e+140)
             (*
              -0.25
              (*
               (* x-scale_m (* y-scale_m (- (sqrt 8.0))))
               (/ (* (sqrt 2.0) b_m) x-scale_m)))
             t_4)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * sqrt(8.0);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_3 = sin(t_2);
	double t_4 = (0.25 * t_0) * (sqrt(2.0) * hypot((a * t_3), b_m));
	double tmp;
	if (y_45_scale_m <= 2.25e-95) {
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot((cos(t_1) * a), (b_m * sin(t_1))));
	} else if (y_45_scale_m <= 1.5e-52) {
		tmp = t_4;
	} else if (y_45_scale_m <= 1.4e+19) {
		tmp = 0.25 * ((x_45_scale_m * t_0) * (sqrt(2.0) * hypot((a / y_45_scale_m), (t_3 * (b_m / y_45_scale_m)))));
	} else if (y_45_scale_m <= 3.8e+84) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_2), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * Math.sqrt(8.0);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.PI * (0.005555555555555556 * angle);
	double t_3 = Math.sin(t_2);
	double t_4 = (0.25 * t_0) * (Math.sqrt(2.0) * Math.hypot((a * t_3), b_m));
	double tmp;
	if (y_45_scale_m <= 2.25e-95) {
		tmp = 0.25 * (((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * Math.hypot((Math.cos(t_1) * a), (b_m * Math.sin(t_1))));
	} else if (y_45_scale_m <= 1.5e-52) {
		tmp = t_4;
	} else if (y_45_scale_m <= 1.4e+19) {
		tmp = 0.25 * ((x_45_scale_m * t_0) * (Math.sqrt(2.0) * Math.hypot((a / y_45_scale_m), (t_3 * (b_m / y_45_scale_m)))));
	} else if (y_45_scale_m <= 3.8e+84) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_2), b_m));
	} else if (y_45_scale_m <= 1.52e+140) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -Math.sqrt(8.0))) * ((Math.sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = y_45_scale_m * math.sqrt(8.0)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.pi * (0.005555555555555556 * angle)
	t_3 = math.sin(t_2)
	t_4 = (0.25 * t_0) * (math.sqrt(2.0) * math.hypot((a * t_3), b_m))
	tmp = 0
	if y_45_scale_m <= 2.25e-95:
		tmp = 0.25 * (((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0)) * math.hypot((math.cos(t_1) * a), (b_m * math.sin(t_1))))
	elif y_45_scale_m <= 1.5e-52:
		tmp = t_4
	elif y_45_scale_m <= 1.4e+19:
		tmp = 0.25 * ((x_45_scale_m * t_0) * (math.sqrt(2.0) * math.hypot((a / y_45_scale_m), (t_3 * (b_m / y_45_scale_m)))))
	elif y_45_scale_m <= 3.8e+84:
		tmp = math.sqrt(((8.0 * math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (math.sqrt(2.0) * math.hypot((a * t_2), b_m))
	elif y_45_scale_m <= 1.52e+140:
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -math.sqrt(8.0))) * ((math.sqrt(2.0) * b_m) / x_45_scale_m))
	else:
		tmp = t_4
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(y_45_scale_m * sqrt(8.0))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_3 = sin(t_2)
	t_4 = Float64(Float64(0.25 * t_0) * Float64(sqrt(2.0) * hypot(Float64(a * t_3), b_m)))
	tmp = 0.0
	if (y_45_scale_m <= 2.25e-95)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot(Float64(cos(t_1) * a), Float64(b_m * sin(t_1)))));
	elseif (y_45_scale_m <= 1.5e-52)
		tmp = t_4;
	elseif (y_45_scale_m <= 1.4e+19)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * t_0) * Float64(sqrt(2.0) * hypot(Float64(a / y_45_scale_m), Float64(t_3 * Float64(b_m / y_45_scale_m))))));
	elseif (y_45_scale_m <= 3.8e+84)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_2), b_m)));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = Float64(-0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(-sqrt(8.0)))) * Float64(Float64(sqrt(2.0) * b_m) / x_45_scale_m)));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = y_45_scale_m * sqrt(8.0);
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = pi * (0.005555555555555556 * angle);
	t_3 = sin(t_2);
	t_4 = (0.25 * t_0) * (sqrt(2.0) * hypot((a * t_3), b_m));
	tmp = 0.0;
	if (y_45_scale_m <= 2.25e-95)
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot((cos(t_1) * a), (b_m * sin(t_1))));
	elseif (y_45_scale_m <= 1.5e-52)
		tmp = t_4;
	elseif (y_45_scale_m <= 1.4e+19)
		tmp = 0.25 * ((x_45_scale_m * t_0) * (sqrt(2.0) * hypot((a / y_45_scale_m), (t_3 * (b_m / y_45_scale_m)))));
	elseif (y_45_scale_m <= 3.8e+84)
		tmp = sqrt(((8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_2), b_m));
	elseif (y_45_scale_m <= 1.52e+140)
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.25 * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$3), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.25e-95], N[(0.25 * N[(N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[t$95$1], $MachinePrecision] * a), $MachinePrecision] ^ 2 + N[(b$95$m * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.5e-52], t$95$4, If[LessEqual[y$45$scale$95$m, 1.4e+19], N[(0.25 * N[(N[(x$45$scale$95$m * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a / y$45$scale$95$m), $MachinePrecision] ^ 2 + N[(t$95$3 * N[(b$95$m / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 3.8e+84], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.52e+140], N[(-0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * b$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y-scale < 2.25e-95

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

   if 2.25e-95 < y-scale < 1.5e-52 or 1.52e140 < y-scale

   1. Initial program 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in x-scale around 0 66.0%

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

      1. associate-*r* 66.0%

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

      2. distribute-lft-out 66.0%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in angle around 0 73.5%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 73.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 73.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 73.4%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 79.6%

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

   if 1.5e-52 < y-scale < 1.4e19

   1. Initial program 0.0%

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

   3. Simplified 0.2%

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

   5. Taylor expanded in x-scale around inf 12.3%

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

   6. Taylor expanded in angle around 0 12.3%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\color{blue}{1}}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
   7. Step-by-step derivation

      1. pow1/2 12.3%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}}\right) \]

      2. distribute-lft-out 12.3%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(2 \cdot \left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)}}^{0.5}\right) \]

      3. unpow-prod-down 12.2%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({2}^{0.5} \cdot {\left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}\right)}\right) \]

      4. pow1/2 12.2%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot {\left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}\right)\right) \]

   8. Applied egg-rr 12.4%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot {\left({\left(\frac{a}{y-scale}\right)}^{2} + {\left(\frac{b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale}\right)}^{2}\right)}^{0.5}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow1/2 12.4%

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

      2. unpow2 12.4%

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

      3. unpow2 12.4%

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

      4. hypot-define 13.3%

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

      5. *-commutative 13.3%

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

      6. associate-/l* 13.3%

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

      7. associate-*r* 13.3%

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

   10. Simplified 13.3%

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

   if 1.4e19 < y-scale < 3.8000000000000001e84

   1. Initial program 0.5%

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

   3. Simplified 0.8%

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

   5. Taylor expanded in x-scale around 0 45.8%

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

      1. associate-*r* 45.8%

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

      2. distribute-lft-out 45.8%

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

   7. Simplified 45.8%

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

   8. Taylor expanded in angle around 0 45.8%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 45.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 45.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 54.5%

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

   11. Taylor expanded in angle around 0 53.5%

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

       1. associate-*r* 53.5%

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

   13. Simplified 53.5%

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

       1. add-sqr-sqrt 53.3%

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

       2. sqrt-unprod 53.5%

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

       3. swap-sqr 53.5%

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

       4. metadata-eval 53.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 53.5%

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

       6. swap-sqr 53.5%

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

       7. *-commutative 53.5%

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

       8. *-commutative 53.5%

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

       9. swap-sqr 53.5%

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

       10. *-commutative 53.5%

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

       11. *-commutative 53.5%

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

       12. swap-sqr 53.2%

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

       13. rem-square-sqrt 54.0%

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

       14. pow2 54.0%

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

       15. metadata-eval 54.0%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 54.0%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   if 3.8000000000000001e84 < y-scale < 1.52e140

   1. Initial program 8.3%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in a around 0 0.2%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right)\right)} \]

   7. Simplified 0.2%

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

   8. Taylor expanded in angle around 0 34.7%

      \[\leadsto -0.25 \cdot \left(\left(-x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\frac{b \cdot \sqrt{2}}{x-scale}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.25 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.5 \cdot 10^{-52}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a}{y-scale}, \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.8 \cdot 10^{+84}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.52 \cdot 10^{+140}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := \sin t\_0\\ t_2 := y-scale\_m \cdot \sqrt{8}\\ t_3 := 0.25 \cdot \left(\left(x-scale\_m \cdot t\_2\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a}{y-scale\_m}, t\_1 \cdot \frac{b\_m}{y-scale\_m}\right)\right)\right)\\ t_4 := \left(0.25 \cdot t\_2\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_1, b\_m\right)\right)\\ \mathbf{if}\;y-scale\_m \leq 2.55 \cdot 10^{-223}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;y-scale\_m \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y-scale\_m \leq 1.75 \cdot 10^{-53}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y-scale\_m \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y-scale\_m \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_0, b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle)))
        (t_1 (sin t_0))
        (t_2 (* y-scale_m (sqrt 8.0)))
        (t_3
         (*
          0.25
          (*
           (* x-scale_m t_2)
           (* (sqrt 2.0) (hypot (/ a y-scale_m) (* t_1 (/ b_m y-scale_m)))))))
        (t_4 (* (* 0.25 t_2) (* (sqrt 2.0) (hypot (* a t_1) b_m)))))
   (if (<= y-scale_m 2.55e-223)
     (* 0.25 (* (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)) a))
     (if (<= y-scale_m 2.1e-95)
       t_3
       (if (<= y-scale_m 1.75e-53)
         t_4
         (if (<= y-scale_m 3.8e+18)
           t_3
           (if (<= y-scale_m 2e+81)
             (*
              (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
              (* (sqrt 2.0) (hypot (* a t_0) b_m)))
             t_4)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_1 = sin(t_0);
	double t_2 = y_45_scale_m * sqrt(8.0);
	double t_3 = 0.25 * ((x_45_scale_m * t_2) * (sqrt(2.0) * hypot((a / y_45_scale_m), (t_1 * (b_m / y_45_scale_m)))));
	double t_4 = (0.25 * t_2) * (sqrt(2.0) * hypot((a * t_1), b_m));
	double tmp;
	if (y_45_scale_m <= 2.55e-223) {
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	} else if (y_45_scale_m <= 2.1e-95) {
		tmp = t_3;
	} else if (y_45_scale_m <= 1.75e-53) {
		tmp = t_4;
	} else if (y_45_scale_m <= 3.8e+18) {
		tmp = t_3;
	} else if (y_45_scale_m <= 2e+81) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_0), b_m));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double t_1 = Math.sin(t_0);
	double t_2 = y_45_scale_m * Math.sqrt(8.0);
	double t_3 = 0.25 * ((x_45_scale_m * t_2) * (Math.sqrt(2.0) * Math.hypot((a / y_45_scale_m), (t_1 * (b_m / y_45_scale_m)))));
	double t_4 = (0.25 * t_2) * (Math.sqrt(2.0) * Math.hypot((a * t_1), b_m));
	double tmp;
	if (y_45_scale_m <= 2.55e-223) {
		tmp = 0.25 * (((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * a);
	} else if (y_45_scale_m <= 2.1e-95) {
		tmp = t_3;
	} else if (y_45_scale_m <= 1.75e-53) {
		tmp = t_4;
	} else if (y_45_scale_m <= 3.8e+18) {
		tmp = t_3;
	} else if (y_45_scale_m <= 2e+81) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_0), b_m));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.pi * (0.005555555555555556 * angle)
	t_1 = math.sin(t_0)
	t_2 = y_45_scale_m * math.sqrt(8.0)
	t_3 = 0.25 * ((x_45_scale_m * t_2) * (math.sqrt(2.0) * math.hypot((a / y_45_scale_m), (t_1 * (b_m / y_45_scale_m)))))
	t_4 = (0.25 * t_2) * (math.sqrt(2.0) * math.hypot((a * t_1), b_m))
	tmp = 0
	if y_45_scale_m <= 2.55e-223:
		tmp = 0.25 * (((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0)) * a)
	elif y_45_scale_m <= 2.1e-95:
		tmp = t_3
	elif y_45_scale_m <= 1.75e-53:
		tmp = t_4
	elif y_45_scale_m <= 3.8e+18:
		tmp = t_3
	elif y_45_scale_m <= 2e+81:
		tmp = math.sqrt(((8.0 * math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (math.sqrt(2.0) * math.hypot((a * t_0), b_m))
	else:
		tmp = t_4
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_1 = sin(t_0)
	t_2 = Float64(y_45_scale_m * sqrt(8.0))
	t_3 = Float64(0.25 * Float64(Float64(x_45_scale_m * t_2) * Float64(sqrt(2.0) * hypot(Float64(a / y_45_scale_m), Float64(t_1 * Float64(b_m / y_45_scale_m))))))
	t_4 = Float64(Float64(0.25 * t_2) * Float64(sqrt(2.0) * hypot(Float64(a * t_1), b_m)))
	tmp = 0.0
	if (y_45_scale_m <= 2.55e-223)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a));
	elseif (y_45_scale_m <= 2.1e-95)
		tmp = t_3;
	elseif (y_45_scale_m <= 1.75e-53)
		tmp = t_4;
	elseif (y_45_scale_m <= 3.8e+18)
		tmp = t_3;
	elseif (y_45_scale_m <= 2e+81)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_0), b_m)));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = pi * (0.005555555555555556 * angle);
	t_1 = sin(t_0);
	t_2 = y_45_scale_m * sqrt(8.0);
	t_3 = 0.25 * ((x_45_scale_m * t_2) * (sqrt(2.0) * hypot((a / y_45_scale_m), (t_1 * (b_m / y_45_scale_m)))));
	t_4 = (0.25 * t_2) * (sqrt(2.0) * hypot((a * t_1), b_m));
	tmp = 0.0;
	if (y_45_scale_m <= 2.55e-223)
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	elseif (y_45_scale_m <= 2.1e-95)
		tmp = t_3;
	elseif (y_45_scale_m <= 1.75e-53)
		tmp = t_4;
	elseif (y_45_scale_m <= 3.8e+18)
		tmp = t_3;
	elseif (y_45_scale_m <= 2e+81)
		tmp = sqrt(((8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_0), b_m));
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.25 * N[(N[(x$45$scale$95$m * t$95$2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a / y$45$scale$95$m), $MachinePrecision] ^ 2 + N[(t$95$1 * N[(b$95$m / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.25 * t$95$2), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.55e-223], N[(0.25 * N[(N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.1e-95], t$95$3, If[LessEqual[y$45$scale$95$m, 1.75e-53], t$95$4, If[LessEqual[y$45$scale$95$m, 3.8e+18], t$95$3, If[LessEqual[y$45$scale$95$m, 2e+81], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$0), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y-scale < 2.54999999999999987e-223

   1. Initial program 1.7%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in y-scale around 0 9.1%

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

      1. associate-*r* 9.1%

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

      2. distribute-lft-out 9.1%

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

   7. Simplified 9.8%

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

   8. Taylor expanded in angle around 0 16.2%

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

      1. associate-*r* 16.2%

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

   10. Simplified 16.2%

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

   if 2.54999999999999987e-223 < y-scale < 2.1e-95 or 1.74999999999999997e-53 < y-scale < 3.8e18

   1. Initial program 2.5%

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

   3. Simplified 2.5%

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

   5. Taylor expanded in x-scale around inf 12.4%

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

   6. Taylor expanded in angle around 0 12.4%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\color{blue}{1}}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
   7. Step-by-step derivation

      1. pow1/2 12.4%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}}\right) \]

      2. distribute-lft-out 12.4%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(2 \cdot \left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)}}^{0.5}\right) \]

      3. unpow-prod-down 12.4%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({2}^{0.5} \cdot {\left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}\right)}\right) \]

      4. pow1/2 12.4%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot {\left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}\right)\right) \]

   8. Applied egg-rr 16.9%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot {\left({\left(\frac{a}{y-scale}\right)}^{2} + {\left(\frac{b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale}\right)}^{2}\right)}^{0.5}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow1/2 16.9%

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

      2. unpow2 16.9%

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

      3. unpow2 16.9%

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

      4. hypot-define 19.7%

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

      5. *-commutative 19.7%

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

      6. associate-/l* 19.7%

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

      7. associate-*r* 19.7%

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

   10. Simplified 19.7%

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

   if 2.1e-95 < y-scale < 1.74999999999999997e-53 or 1.99999999999999984e81 < y-scale

   1. Initial program 6.1%

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

   3. Simplified 4.2%

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

   5. Taylor expanded in x-scale around 0 66.8%

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

      1. associate-*r* 66.8%

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

      2. distribute-lft-out 66.8%

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

   7. Simplified 71.0%

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

   8. Taylor expanded in angle around 0 72.7%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 72.6%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 78.6%

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

   if 3.8e18 < y-scale < 1.99999999999999984e81

   1. Initial program 0.6%

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

   3. Simplified 0.9%

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

   5. Taylor expanded in x-scale around 0 40.0%

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

      1. associate-*r* 40.0%

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

      2. distribute-lft-out 40.0%

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

   7. Simplified 40.0%

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

   8. Taylor expanded in angle around 0 39.9%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 39.9%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 39.9%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 39.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 49.6%

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

   11. Taylor expanded in angle around 0 48.5%

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

       1. associate-*r* 48.5%

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

   13. Simplified 48.5%

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

       1. add-sqr-sqrt 48.3%

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

       2. sqrt-unprod 48.5%

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

       3. swap-sqr 48.5%

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

       4. metadata-eval 48.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 48.5%

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

       6. swap-sqr 48.5%

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

       7. *-commutative 48.5%

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

       8. *-commutative 48.5%

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

       9. swap-sqr 48.5%

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

       10. *-commutative 48.5%

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

       11. *-commutative 48.5%

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

       12. swap-sqr 48.2%

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

       13. rem-square-sqrt 48.9%

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

       14. pow2 48.9%

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

       15. metadata-eval 48.9%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 48.9%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.55 \cdot 10^{-223}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;y-scale \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a}{y-scale}, \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.75 \cdot 10^{-53}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 3.8 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a}{y-scale}, \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := \sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_0, b\_m\right)\right)\\ \mathbf{if}\;y-scale\_m \leq 3.9 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;y-scale\_m \leq 7 \cdot 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y-scale\_m \leq 4.4 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y-scale\_m \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b\_m}{x-scale\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_0, b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle)))
        (t_1
         (*
          (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
          (* (sqrt 2.0) (hypot (* a t_0) b_m)))))
   (if (<= y-scale_m 3.9e-144)
     (* 0.25 (* (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)) a))
     (if (<= y-scale_m 7e-83)
       t_1
       (if (<= y-scale_m 4.4e+18)
         (* 0.25 (* (* x-scale_m (sqrt 8.0)) (* (sqrt 2.0) a)))
         (if (<= y-scale_m 1.5e+85)
           t_1
           (if (<= y-scale_m 1.1e+95)
             (*
              -0.25
              (*
               (* x-scale_m (* y-scale_m (- (sqrt 8.0))))
               (/ (* (sqrt 2.0) b_m) x-scale_m)))
             (*
              (* 0.25 (* y-scale_m (sqrt 8.0)))
              (* (sqrt 2.0) (hypot (* a (sin t_0)) b_m))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_1 = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_0), b_m));
	double tmp;
	if (y_45_scale_m <= 3.9e-144) {
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	} else if (y_45_scale_m <= 7e-83) {
		tmp = t_1;
	} else if (y_45_scale_m <= 4.4e+18) {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	} else if (y_45_scale_m <= 1.5e+85) {
		tmp = t_1;
	} else if (y_45_scale_m <= 1.1e+95) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * hypot((a * sin(t_0)), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double t_1 = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_0), b_m));
	double tmp;
	if (y_45_scale_m <= 3.9e-144) {
		tmp = 0.25 * (((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * a);
	} else if (y_45_scale_m <= 7e-83) {
		tmp = t_1;
	} else if (y_45_scale_m <= 4.4e+18) {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * a));
	} else if (y_45_scale_m <= 1.5e+85) {
		tmp = t_1;
	} else if (y_45_scale_m <= 1.1e+95) {
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -Math.sqrt(8.0))) * ((Math.sqrt(2.0) * b_m) / x_45_scale_m));
	} else {
		tmp = (0.25 * (y_45_scale_m * Math.sqrt(8.0))) * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_0)), b_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.pi * (0.005555555555555556 * angle)
	t_1 = math.sqrt(((8.0 * math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (math.sqrt(2.0) * math.hypot((a * t_0), b_m))
	tmp = 0
	if y_45_scale_m <= 3.9e-144:
		tmp = 0.25 * (((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0)) * a)
	elif y_45_scale_m <= 7e-83:
		tmp = t_1
	elif y_45_scale_m <= 4.4e+18:
		tmp = 0.25 * ((x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * a))
	elif y_45_scale_m <= 1.5e+85:
		tmp = t_1
	elif y_45_scale_m <= 1.1e+95:
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -math.sqrt(8.0))) * ((math.sqrt(2.0) * b_m) / x_45_scale_m))
	else:
		tmp = (0.25 * (y_45_scale_m * math.sqrt(8.0))) * (math.sqrt(2.0) * math.hypot((a * math.sin(t_0)), b_m))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_1 = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_0), b_m)))
	tmp = 0.0
	if (y_45_scale_m <= 3.9e-144)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a));
	elseif (y_45_scale_m <= 7e-83)
		tmp = t_1;
	elseif (y_45_scale_m <= 4.4e+18)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * a)));
	elseif (y_45_scale_m <= 1.5e+85)
		tmp = t_1;
	elseif (y_45_scale_m <= 1.1e+95)
		tmp = Float64(-0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * Float64(-sqrt(8.0)))) * Float64(Float64(sqrt(2.0) * b_m) / x_45_scale_m)));
	else
		tmp = Float64(Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_0)), b_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = pi * (0.005555555555555556 * angle);
	t_1 = sqrt(((8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_0), b_m));
	tmp = 0.0;
	if (y_45_scale_m <= 3.9e-144)
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	elseif (y_45_scale_m <= 7e-83)
		tmp = t_1;
	elseif (y_45_scale_m <= 4.4e+18)
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	elseif (y_45_scale_m <= 1.5e+85)
		tmp = t_1;
	elseif (y_45_scale_m <= 1.1e+95)
		tmp = -0.25 * ((x_45_scale_m * (y_45_scale_m * -sqrt(8.0))) * ((sqrt(2.0) * b_m) / x_45_scale_m));
	else
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * hypot((a * sin(t_0)), b_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$0), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 3.9e-144], N[(0.25 * N[(N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 7e-83], t$95$1, If[LessEqual[y$45$scale$95$m, 4.4e+18], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.5e+85], t$95$1, If[LessEqual[y$45$scale$95$m, 1.1e+95], N[(-0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * (-N[Sqrt[8.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * b$95$m), $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y-scale < 3.90000000000000015e-144

   1. Initial program 1.6%

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

   3. Simplified 1.7%

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

   5. Taylor expanded in y-scale around 0 8.4%

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

      1. associate-*r* 8.4%

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

      2. distribute-lft-out 8.4%

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

   7. Simplified 9.0%

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

   8. Taylor expanded in angle around 0 16.8%

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

      1. associate-*r* 16.9%

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

   10. Simplified 16.9%

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

   if 3.90000000000000015e-144 < y-scale < 7.00000000000000061e-83 or 4.4e18 < y-scale < 1.5e85

   1. Initial program 4.5%

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

   3. Simplified 4.6%

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

   5. Taylor expanded in x-scale around 0 44.0%

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

      1. associate-*r* 44.0%

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

      2. distribute-lft-out 44.0%

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

   7. Simplified 48.1%

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

   8. Taylor expanded in angle around 0 48.1%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 48.1%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 48.1%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 48.0%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 48.0%

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

   11. Taylor expanded in angle around 0 46.5%

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

       1. associate-*r* 46.5%

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

   13. Simplified 46.5%

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

       1. add-sqr-sqrt 46.4%

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

       2. sqrt-unprod 46.5%

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

       3. swap-sqr 46.5%

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

       4. metadata-eval 46.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 46.5%

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

       6. swap-sqr 46.5%

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

       7. *-commutative 46.5%

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

       8. *-commutative 46.5%

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

       9. swap-sqr 46.5%

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

       10. *-commutative 46.5%

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

       11. *-commutative 46.5%

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

       12. swap-sqr 46.4%

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

       13. rem-square-sqrt 46.8%

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

       14. pow2 46.8%

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

       15. metadata-eval 46.8%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 46.8%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   if 7.00000000000000061e-83 < y-scale < 4.4e18

   1. Initial program 0.2%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in y-scale around 0 9.6%

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

      1. distribute-lft-out 9.6%

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

   7. Simplified 9.7%

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

   8. Taylor expanded in angle around 0 18.0%

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

      1. *-commutative 18.0%

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

   10. Simplified 18.0%

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

   if 1.5e85 < y-scale < 1.0999999999999999e95

   1. Initial program 25.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in a around 0 0.5%

      \[\leadsto -0.25 \cdot \color{blue}{\left(-1 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\sqrt{4 \cdot \frac{{b}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}\right)\right)} \]

   7. Simplified 0.5%

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

   8. Taylor expanded in angle around 0 54.1%

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

   if 1.0999999999999999e95 < y-scale

   1. Initial program 5.6%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in x-scale around 0 75.8%

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

      1. associate-*r* 75.8%

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

      2. distribute-lft-out 75.8%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in angle around 0 83.1%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 83.1%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 83.1%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 83.1%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 86.2%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.9 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;y-scale \leq 7 \cdot 10^{-83}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.4 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.5 \cdot 10^{+85}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;-0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \left(-\sqrt{8}\right)\right)\right) \cdot \frac{\sqrt{2} \cdot b}{x-scale}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := y-scale\_m \cdot \sqrt{8}\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_2 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_3 := \sin t\_2\\ t_4 := \left(0.25 \cdot t\_0\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_3, b\_m\right)\right)\\ \mathbf{if}\;y-scale\_m \leq 2.6 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos t\_1 \cdot a, t\_1 \cdot b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 1.95 \cdot 10^{-57}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y-scale\_m \leq 2.1 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot t\_0\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a}{y-scale\_m}, t\_3 \cdot \frac{b\_m}{y-scale\_m}\right)\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 2.4 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale\_m}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* y-scale_m (sqrt 8.0)))
        (t_1 (* 0.005555555555555556 (* angle PI)))
        (t_2 (* PI (* 0.005555555555555556 angle)))
        (t_3 (sin t_2))
        (t_4 (* (* 0.25 t_0) (* (sqrt 2.0) (hypot (* a t_3) b_m)))))
   (if (<= y-scale_m 2.6e-95)
     (*
      0.25
      (*
       (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0))
       (hypot (* (cos t_1) a) (* t_1 b_m))))
     (if (<= y-scale_m 1.95e-57)
       t_4
       (if (<= y-scale_m 2.1e+18)
         (*
          0.25
          (*
           (* x-scale_m t_0)
           (* (sqrt 2.0) (hypot (/ a y-scale_m) (* t_3 (/ b_m y-scale_m))))))
         (if (<= y-scale_m 2.4e+81)
           (*
            (sqrt (* (* 8.0 (pow y-scale_m 2.0)) 0.0625))
            (* (sqrt 2.0) (hypot (* a t_2) b_m)))
           t_4))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * sqrt(8.0);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_2 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_3 = sin(t_2);
	double t_4 = (0.25 * t_0) * (sqrt(2.0) * hypot((a * t_3), b_m));
	double tmp;
	if (y_45_scale_m <= 2.6e-95) {
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot((cos(t_1) * a), (t_1 * b_m)));
	} else if (y_45_scale_m <= 1.95e-57) {
		tmp = t_4;
	} else if (y_45_scale_m <= 2.1e+18) {
		tmp = 0.25 * ((x_45_scale_m * t_0) * (sqrt(2.0) * hypot((a / y_45_scale_m), (t_3 * (b_m / y_45_scale_m)))));
	} else if (y_45_scale_m <= 2.4e+81) {
		tmp = sqrt(((8.0 * pow(y_45_scale_m, 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_2), b_m));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * Math.sqrt(8.0);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double t_2 = Math.PI * (0.005555555555555556 * angle);
	double t_3 = Math.sin(t_2);
	double t_4 = (0.25 * t_0) * (Math.sqrt(2.0) * Math.hypot((a * t_3), b_m));
	double tmp;
	if (y_45_scale_m <= 2.6e-95) {
		tmp = 0.25 * (((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * Math.hypot((Math.cos(t_1) * a), (t_1 * b_m)));
	} else if (y_45_scale_m <= 1.95e-57) {
		tmp = t_4;
	} else if (y_45_scale_m <= 2.1e+18) {
		tmp = 0.25 * ((x_45_scale_m * t_0) * (Math.sqrt(2.0) * Math.hypot((a / y_45_scale_m), (t_3 * (b_m / y_45_scale_m)))));
	} else if (y_45_scale_m <= 2.4e+81) {
		tmp = Math.sqrt(((8.0 * Math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (Math.sqrt(2.0) * Math.hypot((a * t_2), b_m));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = y_45_scale_m * math.sqrt(8.0)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	t_2 = math.pi * (0.005555555555555556 * angle)
	t_3 = math.sin(t_2)
	t_4 = (0.25 * t_0) * (math.sqrt(2.0) * math.hypot((a * t_3), b_m))
	tmp = 0
	if y_45_scale_m <= 2.6e-95:
		tmp = 0.25 * (((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0)) * math.hypot((math.cos(t_1) * a), (t_1 * b_m)))
	elif y_45_scale_m <= 1.95e-57:
		tmp = t_4
	elif y_45_scale_m <= 2.1e+18:
		tmp = 0.25 * ((x_45_scale_m * t_0) * (math.sqrt(2.0) * math.hypot((a / y_45_scale_m), (t_3 * (b_m / y_45_scale_m)))))
	elif y_45_scale_m <= 2.4e+81:
		tmp = math.sqrt(((8.0 * math.pow(y_45_scale_m, 2.0)) * 0.0625)) * (math.sqrt(2.0) * math.hypot((a * t_2), b_m))
	else:
		tmp = t_4
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(y_45_scale_m * sqrt(8.0))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_2 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_3 = sin(t_2)
	t_4 = Float64(Float64(0.25 * t_0) * Float64(sqrt(2.0) * hypot(Float64(a * t_3), b_m)))
	tmp = 0.0
	if (y_45_scale_m <= 2.6e-95)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot(Float64(cos(t_1) * a), Float64(t_1 * b_m))));
	elseif (y_45_scale_m <= 1.95e-57)
		tmp = t_4;
	elseif (y_45_scale_m <= 2.1e+18)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * t_0) * Float64(sqrt(2.0) * hypot(Float64(a / y_45_scale_m), Float64(t_3 * Float64(b_m / y_45_scale_m))))));
	elseif (y_45_scale_m <= 2.4e+81)
		tmp = Float64(sqrt(Float64(Float64(8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * Float64(sqrt(2.0) * hypot(Float64(a * t_2), b_m)));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = y_45_scale_m * sqrt(8.0);
	t_1 = 0.005555555555555556 * (angle * pi);
	t_2 = pi * (0.005555555555555556 * angle);
	t_3 = sin(t_2);
	t_4 = (0.25 * t_0) * (sqrt(2.0) * hypot((a * t_3), b_m));
	tmp = 0.0;
	if (y_45_scale_m <= 2.6e-95)
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * hypot((cos(t_1) * a), (t_1 * b_m)));
	elseif (y_45_scale_m <= 1.95e-57)
		tmp = t_4;
	elseif (y_45_scale_m <= 2.1e+18)
		tmp = 0.25 * ((x_45_scale_m * t_0) * (sqrt(2.0) * hypot((a / y_45_scale_m), (t_3 * (b_m / y_45_scale_m)))));
	elseif (y_45_scale_m <= 2.4e+81)
		tmp = sqrt(((8.0 * (y_45_scale_m ^ 2.0)) * 0.0625)) * (sqrt(2.0) * hypot((a * t_2), b_m));
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.25 * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$3), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.6e-95], N[(0.25 * N[(N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Cos[t$95$1], $MachinePrecision] * a), $MachinePrecision] ^ 2 + N[(t$95$1 * b$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 1.95e-57], t$95$4, If[LessEqual[y$45$scale$95$m, 2.1e+18], N[(0.25 * N[(N[(x$45$scale$95$m * t$95$0), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a / y$45$scale$95$m), $MachinePrecision] ^ 2 + N[(t$95$3 * N[(b$95$m / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 2.4e+81], N[(N[Sqrt[N[(N[(8.0 * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] * 0.0625), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y-scale < 2.60000000000000001e-95

   1. Initial program 2.1%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in y-scale around 0 9.0%

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

      1. associate-*r* 9.0%

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

      2. distribute-lft-out 9.0%

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

   7. Simplified 9.6%

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

   8. Taylor expanded in x-scale around 0 25.1%

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

   10. Simplified 26.1%

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

   11. Taylor expanded in angle around 0 24.9%

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

   if 2.60000000000000001e-95 < y-scale < 1.95000000000000003e-57 or 2.3999999999999999e81 < y-scale

   1. Initial program 6.2%

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

   3. Simplified 4.3%

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

   5. Taylor expanded in x-scale around 0 67.7%

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

      1. associate-*r* 67.7%

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

      2. distribute-lft-out 67.7%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in angle around 0 73.7%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 73.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 73.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 73.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 79.7%

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

   if 1.95000000000000003e-57 < y-scale < 2.1e18

   1. Initial program 0.0%

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

   3. Simplified 0.1%

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

   5. Taylor expanded in x-scale around inf 11.5%

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

   6. Taylor expanded in angle around 0 11.5%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot {\color{blue}{1}}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}}\right) \]
   7. Step-by-step derivation

      1. pow1/2 11.5%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}}\right) \]

      2. distribute-lft-out 11.5%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(2 \cdot \left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)}}^{0.5}\right) \]

      3. unpow-prod-down 11.4%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({2}^{0.5} \cdot {\left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}\right)}\right) \]

      4. pow1/2 11.4%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\color{blue}{\sqrt{2}} \cdot {\left(\frac{{a}^{2} \cdot {1}^{2}}{{y-scale}^{2}} + \frac{{b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{0.5}\right)\right) \]

   8. Applied egg-rr 11.6%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\sqrt{2} \cdot {\left({\left(\frac{a}{y-scale}\right)}^{2} + {\left(\frac{b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale}\right)}^{2}\right)}^{0.5}\right)}\right) \]
   9. Step-by-step derivation

      1. unpow1/2 11.6%

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

      2. unpow2 11.6%

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

      3. unpow2 11.6%

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

      4. hypot-define 12.4%

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

      5. *-commutative 12.4%

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

      6. associate-/l* 12.4%

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

      7. associate-*r* 12.4%

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

   10. Simplified 12.4%

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

   if 2.1e18 < y-scale < 2.3999999999999999e81

   1. Initial program 0.6%

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

   3. Simplified 0.9%

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

   5. Taylor expanded in x-scale around 0 40.0%

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

      1. associate-*r* 40.0%

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

      2. distribute-lft-out 40.0%

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

   7. Simplified 40.0%

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

   8. Taylor expanded in angle around 0 39.9%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 39.9%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 39.9%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 39.8%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 49.6%

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

   11. Taylor expanded in angle around 0 48.5%

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

       1. associate-*r* 48.5%

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

   13. Simplified 48.5%

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

       1. add-sqr-sqrt 48.3%

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

       2. sqrt-unprod 48.5%

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

       3. swap-sqr 48.5%

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

       4. metadata-eval 48.5%

          \[\leadsto \sqrt{\color{blue}{0.0625} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

       5. metadata-eval 48.5%

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

       6. swap-sqr 48.5%

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

       7. *-commutative 48.5%

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

       8. *-commutative 48.5%

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

       9. swap-sqr 48.5%

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

       10. *-commutative 48.5%

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

       11. *-commutative 48.5%

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

       12. swap-sqr 48.2%

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

       13. rem-square-sqrt 48.9%

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

       14. pow2 48.9%

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

       15. metadata-eval 48.9%

           \[\leadsto \sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot \color{blue}{0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]

   15. Applied egg-rr 48.9%

       \[\leadsto \color{blue}{\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625}} \cdot \left(\mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.6 \cdot 10^{-95}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \mathsf{hypot}\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\\ \mathbf{elif}\;y-scale \leq 1.95 \cdot 10^{-57}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.1 \cdot 10^{+18}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a}{y-scale}, \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \frac{b}{y-scale}\right)\right)\right)\\ \mathbf{elif}\;y-scale \leq 2.4 \cdot 10^{+81}:\\ \;\;\;\;\sqrt{\left(8 \cdot {y-scale}^{2}\right) \cdot 0.0625} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\\ t_1 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ \mathbf{if}\;y-scale\_m \leq 3.9 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;y-scale\_m \leq 7 \cdot 10^{-83}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_1, b\_m\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 10500000:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_1, b\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.25 (* y-scale_m (sqrt 8.0))))
        (t_1 (* PI (* 0.005555555555555556 angle))))
   (if (<= y-scale_m 3.9e-144)
     (* 0.25 (* (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)) a))
     (if (<= y-scale_m 7e-83)
       (* t_0 (* (sqrt 2.0) (hypot (* a t_1) b_m)))
       (if (<= y-scale_m 10500000.0)
         (* 0.25 (* (* x-scale_m (sqrt 8.0)) (* (sqrt 2.0) a)))
         (* t_0 (* (sqrt 2.0) (hypot (* a (sin t_1)) b_m))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * sqrt(8.0));
	double t_1 = ((double) M_PI) * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 3.9e-144) {
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	} else if (y_45_scale_m <= 7e-83) {
		tmp = t_0 * (sqrt(2.0) * hypot((a * t_1), b_m));
	} else if (y_45_scale_m <= 10500000.0) {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	} else {
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(t_1)), b_m));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.25 * (y_45_scale_m * Math.sqrt(8.0));
	double t_1 = Math.PI * (0.005555555555555556 * angle);
	double tmp;
	if (y_45_scale_m <= 3.9e-144) {
		tmp = 0.25 * (((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * a);
	} else if (y_45_scale_m <= 7e-83) {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * t_1), b_m));
	} else if (y_45_scale_m <= 10500000.0) {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * a));
	} else {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_1)), b_m));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.25 * (y_45_scale_m * math.sqrt(8.0))
	t_1 = math.pi * (0.005555555555555556 * angle)
	tmp = 0
	if y_45_scale_m <= 3.9e-144:
		tmp = 0.25 * (((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0)) * a)
	elif y_45_scale_m <= 7e-83:
		tmp = t_0 * (math.sqrt(2.0) * math.hypot((a * t_1), b_m))
	elif y_45_scale_m <= 10500000.0:
		tmp = 0.25 * ((x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * a))
	else:
		tmp = t_0 * (math.sqrt(2.0) * math.hypot((a * math.sin(t_1)), b_m))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0)))
	t_1 = Float64(pi * Float64(0.005555555555555556 * angle))
	tmp = 0.0
	if (y_45_scale_m <= 3.9e-144)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a));
	elseif (y_45_scale_m <= 7e-83)
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * t_1), b_m)));
	elseif (y_45_scale_m <= 10500000.0)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * a)));
	else
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_1)), b_m)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.25 * (y_45_scale_m * sqrt(8.0));
	t_1 = pi * (0.005555555555555556 * angle);
	tmp = 0.0;
	if (y_45_scale_m <= 3.9e-144)
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	elseif (y_45_scale_m <= 7e-83)
		tmp = t_0 * (sqrt(2.0) * hypot((a * t_1), b_m));
	elseif (y_45_scale_m <= 10500000.0)
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	else
		tmp = t_0 * (sqrt(2.0) * hypot((a * sin(t_1)), b_m));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 3.9e-144], N[(0.25 * N[(N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 7e-83], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 10500000.0], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y-scale < 3.90000000000000015e-144

   1. Initial program 1.6%

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

   3. Simplified 1.7%

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

   5. Taylor expanded in y-scale around 0 8.4%

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

      1. associate-*r* 8.4%

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

      2. distribute-lft-out 8.4%

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

   7. Simplified 9.0%

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

   8. Taylor expanded in angle around 0 16.8%

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

      1. associate-*r* 16.9%

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

   10. Simplified 16.9%

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

   if 3.90000000000000015e-144 < y-scale < 7.00000000000000061e-83

   1. Initial program 8.2%

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

   3. Simplified 8.2%

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

   5. Taylor expanded in x-scale around 0 37.8%

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

      1. associate-*r* 37.8%

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

      2. distribute-lft-out 37.8%

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

   7. Simplified 45.7%

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

   8. Taylor expanded in angle around 0 45.7%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 45.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 45.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 45.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 38.3%

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

   11. Taylor expanded in angle around 0 36.2%

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

       1. associate-*r* 36.2%

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

   13. Simplified 36.2%

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

   if 7.00000000000000061e-83 < y-scale < 1.05e7

   1. Initial program 0.2%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.3%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 9.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 9.6%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \]

   7. Simplified 9.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)} \]

   8. Taylor expanded in angle around 0 18.0%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 18.0%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)}\right) \]

   10. Simplified 18.0%

       \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)}\right) \]

   if 1.05e7 < y-scale

   1. Initial program 6.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 4.2%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x-scale around 0 67.5%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 67.5%

         \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}} \]

      2. distribute-lft-out 67.5%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}} \]

   7. Simplified 71.1%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}} \]

   8. Taylor expanded in angle around 0 72.7%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 72.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 78.2%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.9 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;y-scale \leq 7 \cdot 10^{-83}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{elif}\;y-scale \leq 10500000:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.1% accurate, 8.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 3.2 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;y-scale\_m \leq 7 \cdot 10^{-83} \lor \neg \left(y-scale\_m \leq 400000000\right):\\ \;\;\;\;\left(0.25 \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 3.2e-144)
   (* 0.25 (* (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)) a))
   (if (or (<= y-scale_m 7e-83) (not (<= y-scale_m 400000000.0)))
     (*
      (* 0.25 (* y-scale_m (sqrt 8.0)))
      (* (sqrt 2.0) (hypot (* a (* PI (* 0.005555555555555556 angle))) b_m)))
     (* 0.25 (* (* x-scale_m (sqrt 8.0)) (* (sqrt 2.0) a))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 3.2e-144) {
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	} else if ((y_45_scale_m <= 7e-83) || !(y_45_scale_m <= 400000000.0)) {
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * hypot((a * (((double) M_PI) * (0.005555555555555556 * angle))), b_m));
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 3.2e-144) {
		tmp = 0.25 * (((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * a);
	} else if ((y_45_scale_m <= 7e-83) || !(y_45_scale_m <= 400000000.0)) {
		tmp = (0.25 * (y_45_scale_m * Math.sqrt(8.0))) * (Math.sqrt(2.0) * Math.hypot((a * (Math.PI * (0.005555555555555556 * angle))), b_m));
	} else {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 3.2e-144:
		tmp = 0.25 * (((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0)) * a)
	elif (y_45_scale_m <= 7e-83) or not (y_45_scale_m <= 400000000.0):
		tmp = (0.25 * (y_45_scale_m * math.sqrt(8.0))) * (math.sqrt(2.0) * math.hypot((a * (math.pi * (0.005555555555555556 * angle))), b_m))
	else:
		tmp = 0.25 * ((x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * a))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 3.2e-144)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a));
	elseif ((y_45_scale_m <= 7e-83) || !(y_45_scale_m <= 400000000.0))
		tmp = Float64(Float64(0.25 * Float64(y_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * hypot(Float64(a * Float64(pi * Float64(0.005555555555555556 * angle))), b_m)));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 3.2e-144)
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	elseif ((y_45_scale_m <= 7e-83) || ~((y_45_scale_m <= 400000000.0)))
		tmp = (0.25 * (y_45_scale_m * sqrt(8.0))) * (sqrt(2.0) * hypot((a * (pi * (0.005555555555555556 * angle))), b_m));
	else
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 3.2e-144], N[(0.25 * N[(N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$45$scale$95$m, 7e-83], N[Not[LessEqual[y$45$scale$95$m, 400000000.0]], $MachinePrecision]], N[(N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + b$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 3.19999999999999973e-144

   1. Initial program 1.6%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 1.7%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 8.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
   6. Step-by-step derivation

      1. associate-*r* 8.4%

         \[\leadsto \color{blue}{\left(\left(0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

      2. distribute-lft-out 8.4%

         \[\leadsto \left(\left(0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   7. Simplified 9.0%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   8. Taylor expanded in angle around 0 16.8%

      \[\leadsto \color{blue}{0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 16.9%

         \[\leadsto 0.25 \cdot \left(a \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)}\right) \]

   10. Simplified 16.9%

       \[\leadsto \color{blue}{0.25 \cdot \left(a \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)} \]

   if 3.19999999999999973e-144 < y-scale < 7.00000000000000061e-83 or 4e8 < y-scale

   1. Initial program 6.4%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 4.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x-scale around 0 62.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 62.0%

         \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}} \]

      2. distribute-lft-out 62.0%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}} \]

   7. Simplified 66.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}} \]

   8. Taylor expanded in angle around 0 67.7%

      \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. pow1/2 67.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left(2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)}^{0.5}} \]

      2. *-commutative 67.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot {\color{blue}{\left(\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right) \cdot 2\right)}}^{0.5} \]

      3. unpow-prod-down 67.7%

         \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left({\left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)}^{0.5} \cdot {2}^{0.5}\right)} \]

   10. Applied egg-rr 70.8%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{\left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)} \]

   11. Taylor expanded in angle around 0 68.5%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}, b\right) \cdot \sqrt{2}\right) \]
   12. Step-by-step derivation

       1. associate-*r* 68.5%

          \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}, b\right) \cdot \sqrt{2}\right) \]

   13. Simplified 68.5%

       \[\leadsto \left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}, b\right) \cdot \sqrt{2}\right) \]

   if 7.00000000000000061e-83 < y-scale < 4e8

   1. Initial program 0.2%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.3%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 9.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 9.6%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \]

   7. Simplified 9.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)} \]

   8. Taylor expanded in angle around 0 18.0%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 18.0%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)}\right) \]

   10. Simplified 18.0%

       \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.2 \cdot 10^{-144}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;y-scale \leq 7 \cdot 10^{-83} \lor \neg \left(y-scale \leq 400000000\right):\\ \;\;\;\;\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.3% accurate, 12.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := b\_m \cdot \left(y-scale\_m \cdot 4\right)\\ \mathbf{if}\;b\_m \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{elif}\;b\_m \leq 6.2 \cdot 10^{+219}:\\ \;\;\;\;0.25 \cdot \sqrt{{t\_0}^{2}}\\ \mathbf{elif}\;b\_m \leq 1.38 \cdot 10^{+243}:\\ \;\;\;\;0.25 \cdot t\_0\\ \mathbf{elif}\;b\_m \leq 1.55 \cdot 10^{+265}:\\ \;\;\;\;0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + \left|\mathsf{fma}\left(b\_m, y-scale\_m \cdot 4, -1\right)\right|\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* b_m (* y-scale_m 4.0))))
   (if (<= b_m 3.2e+37)
     (* 0.25 (* (* x-scale_m (sqrt 8.0)) (* (sqrt 2.0) a)))
     (if (<= b_m 6.2e+219)
       (* 0.25 (sqrt (pow t_0 2.0)))
       (if (<= b_m 1.38e+243)
         (* 0.25 t_0)
         (if (<= b_m 1.55e+265)
           (* 0.25 (log1p (expm1 t_0)))
           (* 0.25 (+ 1.0 (fabs (fma b_m (* y-scale_m 4.0) -1.0))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if (b_m <= 3.2e+37) {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	} else if (b_m <= 6.2e+219) {
		tmp = 0.25 * sqrt(pow(t_0, 2.0));
	} else if (b_m <= 1.38e+243) {
		tmp = 0.25 * t_0;
	} else if (b_m <= 1.55e+265) {
		tmp = 0.25 * log1p(expm1(t_0));
	} else {
		tmp = 0.25 * (1.0 + fabs(fma(b_m, (y_45_scale_m * 4.0), -1.0)));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(b_m * Float64(y_45_scale_m * 4.0))
	tmp = 0.0
	if (b_m <= 3.2e+37)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * a)));
	elseif (b_m <= 6.2e+219)
		tmp = Float64(0.25 * sqrt((t_0 ^ 2.0)));
	elseif (b_m <= 1.38e+243)
		tmp = Float64(0.25 * t_0);
	elseif (b_m <= 1.55e+265)
		tmp = Float64(0.25 * log1p(expm1(t_0)));
	else
		tmp = Float64(0.25 * Float64(1.0 + abs(fma(b_m, Float64(y_45_scale_m * 4.0), -1.0))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 3.2e+37], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 6.2e+219], N[(0.25 * N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.38e+243], N[(0.25 * t$95$0), $MachinePrecision], If[LessEqual[b$95$m, 1.55e+265], N[(0.25 * N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(1.0 + N[Abs[N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < 3.20000000000000014e37

   1. Initial program 1.9%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 23.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 23.6%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \]

   7. Simplified 24.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)} \]

   8. Taylor expanded in angle around 0 18.0%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 18.0%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)}\right) \]

   10. Simplified 18.0%

       \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)}\right) \]

   if 3.20000000000000014e37 < b < 6.19999999999999938e219

   1. Initial program 6.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 6.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 28.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 28.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 28.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 28.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 28.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 28.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 28.8%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 27.5%

          \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

   11. Applied egg-rr 27.5%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 28.8%

          \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

       2. add-sqr-sqrt 28.4%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\sqrt{b \cdot \left(y-scale \cdot 4\right)} \cdot \sqrt{b \cdot \left(y-scale \cdot 4\right)}\right)} \]

       3. sqrt-unprod 62.0%

          \[\leadsto 0.25 \cdot \color{blue}{\sqrt{\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

       4. pow2 62.0%

          \[\leadsto 0.25 \cdot \sqrt{\color{blue}{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}} \]

   13. Applied egg-rr 62.0%

       \[\leadsto 0.25 \cdot \color{blue}{\sqrt{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}} \]

   if 6.19999999999999938e219 < b < 1.38000000000000003e243

   1. Initial program 16.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 50.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 50.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 50.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 50.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 50.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 50.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 50.8%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   if 1.38000000000000003e243 < b < 1.55000000000000004e265

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 34.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 34.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 34.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 35.2%

         \[\leadsto 0.25 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)} \]

      2. *-commutative 35.2%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \color{blue}{\left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot y-scale\right)}\right)\right) \]

      3. sqrt-unprod 35.2%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\color{blue}{\sqrt{8 \cdot 2}} \cdot y-scale\right)\right)\right) \]

      4. metadata-eval 35.2%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\sqrt{\color{blue}{16}} \cdot y-scale\right)\right)\right) \]

      5. metadata-eval 35.2%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\color{blue}{4} \cdot y-scale\right)\right)\right) \]

   9. Applied egg-rr 35.2%

      \[\leadsto 0.25 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(4 \cdot y-scale\right)\right)\right)} \]

   if 1.55000000000000004e265 < b

   1. Initial program 0.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 15.2%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 15.2%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 15.2%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 15.4%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 15.4%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 15.4%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 15.4%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 14.3%

          \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

   11. Applied egg-rr 14.3%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 15.4%

          \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

       2. expm1-log1p-u 14.3%

          \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)} \]

       3. expm1-undefine 14.3%

          \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]

   13. Applied egg-rr 14.3%

       \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. log1p-undefine 14.3%

          \[\leadsto 0.25 \cdot \left(e^{\color{blue}{\log \left(1 + b \cdot \left(y-scale \cdot 4\right)\right)}} - 1\right) \]

       2. rem-exp-log 15.4%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(1 + b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right) \]

       3. associate-+r- 15.4%

          \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \left(b \cdot \left(y-scale \cdot 4\right) - 1\right)\right)} \]

       4. fma-neg 15.4%

          \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(b, y-scale \cdot 4, -1\right)}\right) \]

       5. metadata-eval 15.4%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, y-scale \cdot 4, \color{blue}{-1}\right)\right) \]

       6. *-commutative 15.4%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, \color{blue}{4 \cdot y-scale}, -1\right)\right) \]

   15. Simplified 15.4%

       \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)\right)} \]
   16. Step-by-step derivation

       1. add-sqr-sqrt 15.2%

          \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\sqrt{\mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)} \cdot \sqrt{\mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)}}\right) \]

       2. sqrt-unprod 54.1%

          \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\sqrt{\mathsf{fma}\left(b, 4 \cdot y-scale, -1\right) \cdot \mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)}}\right) \]

       3. pow2 54.1%

          \[\leadsto 0.25 \cdot \left(1 + \sqrt{\color{blue}{{\left(\mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)\right)}^{2}}}\right) \]

       4. *-commutative 54.1%

          \[\leadsto 0.25 \cdot \left(1 + \sqrt{{\left(\mathsf{fma}\left(b, \color{blue}{y-scale \cdot 4}, -1\right)\right)}^{2}}\right) \]

   17. Applied egg-rr 54.1%

       \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\sqrt{{\left(\mathsf{fma}\left(b, y-scale \cdot 4, -1\right)\right)}^{2}}}\right) \]
   18. Step-by-step derivation

       1. unpow2 54.1%

          \[\leadsto 0.25 \cdot \left(1 + \sqrt{\color{blue}{\mathsf{fma}\left(b, y-scale \cdot 4, -1\right) \cdot \mathsf{fma}\left(b, y-scale \cdot 4, -1\right)}}\right) \]

       2. rem-sqrt-square 70.0%

          \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\left|\mathsf{fma}\left(b, y-scale \cdot 4, -1\right)\right|}\right) \]

       3. *-commutative 70.0%

          \[\leadsto 0.25 \cdot \left(1 + \left|\mathsf{fma}\left(b, \color{blue}{4 \cdot y-scale}, -1\right)\right|\right) \]

   19. Simplified 70.0%

       \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\left|\mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)\right|}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.2 \cdot 10^{+37}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{+219}:\\ \;\;\;\;0.25 \cdot \sqrt{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}\\ \mathbf{elif}\;b \leq 1.38 \cdot 10^{+243}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+265}:\\ \;\;\;\;0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + \left|\mathsf{fma}\left(b, y-scale \cdot 4, -1\right)\right|\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 33.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := b\_m \cdot \left(y-scale\_m \cdot 4\right)\\ \mathbf{if}\;b\_m \leq 1.15 \cdot 10^{+38}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{elif}\;b\_m \leq 7.6 \cdot 10^{+219}:\\ \;\;\;\;0.25 \cdot \sqrt{{t\_0}^{2}}\\ \mathbf{elif}\;b\_m \leq 4.8 \cdot 10^{+242}:\\ \;\;\;\;0.25 \cdot t\_0\\ \mathbf{elif}\;b\_m \leq 1.8 \cdot 10^{+265}:\\ \;\;\;\;0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* b_m (* y-scale_m 4.0))))
   (if (<= b_m 1.15e+38)
     (* 0.25 (* (* x-scale_m (sqrt 8.0)) (* (sqrt 2.0) a)))
     (if (<= b_m 7.6e+219)
       (* 0.25 (sqrt (pow t_0 2.0)))
       (if (<= b_m 4.8e+242)
         (* 0.25 t_0)
         (if (<= b_m 1.8e+265)
           (* 0.25 (log1p (expm1 t_0)))
           (* 0.25 (+ 1.0 t_0))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if (b_m <= 1.15e+38) {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	} else if (b_m <= 7.6e+219) {
		tmp = 0.25 * sqrt(pow(t_0, 2.0));
	} else if (b_m <= 4.8e+242) {
		tmp = 0.25 * t_0;
	} else if (b_m <= 1.8e+265) {
		tmp = 0.25 * log1p(expm1(t_0));
	} else {
		tmp = 0.25 * (1.0 + t_0);
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if (b_m <= 1.15e+38) {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * a));
	} else if (b_m <= 7.6e+219) {
		tmp = 0.25 * Math.sqrt(Math.pow(t_0, 2.0));
	} else if (b_m <= 4.8e+242) {
		tmp = 0.25 * t_0;
	} else if (b_m <= 1.8e+265) {
		tmp = 0.25 * Math.log1p(Math.expm1(t_0));
	} else {
		tmp = 0.25 * (1.0 + t_0);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = b_m * (y_45_scale_m * 4.0)
	tmp = 0
	if b_m <= 1.15e+38:
		tmp = 0.25 * ((x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * a))
	elif b_m <= 7.6e+219:
		tmp = 0.25 * math.sqrt(math.pow(t_0, 2.0))
	elif b_m <= 4.8e+242:
		tmp = 0.25 * t_0
	elif b_m <= 1.8e+265:
		tmp = 0.25 * math.log1p(math.expm1(t_0))
	else:
		tmp = 0.25 * (1.0 + t_0)
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(b_m * Float64(y_45_scale_m * 4.0))
	tmp = 0.0
	if (b_m <= 1.15e+38)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * a)));
	elseif (b_m <= 7.6e+219)
		tmp = Float64(0.25 * sqrt((t_0 ^ 2.0)));
	elseif (b_m <= 4.8e+242)
		tmp = Float64(0.25 * t_0);
	elseif (b_m <= 1.8e+265)
		tmp = Float64(0.25 * log1p(expm1(t_0)));
	else
		tmp = Float64(0.25 * Float64(1.0 + t_0));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 1.15e+38], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 7.6e+219], N[(0.25 * N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.8e+242], N[(0.25 * t$95$0), $MachinePrecision], If[LessEqual[b$95$m, 1.8e+265], N[(0.25 * N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < 1.1500000000000001e38

   1. Initial program 1.9%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 23.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 23.6%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \]

   7. Simplified 24.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)} \]

   8. Taylor expanded in angle around 0 18.0%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)}\right) \]
   9. Step-by-step derivation

      1. *-commutative 18.0%

         \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)}\right) \]

   10. Simplified 18.0%

       \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\left(\sqrt{2} \cdot a\right)}\right) \]

   if 1.1500000000000001e38 < b < 7.59999999999999992e219

   1. Initial program 6.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 6.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 28.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 28.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 28.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 28.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 28.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 28.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 28.8%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 27.5%

          \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

   11. Applied egg-rr 27.5%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 28.8%

          \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

       2. add-sqr-sqrt 28.4%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\sqrt{b \cdot \left(y-scale \cdot 4\right)} \cdot \sqrt{b \cdot \left(y-scale \cdot 4\right)}\right)} \]

       3. sqrt-unprod 62.0%

          \[\leadsto 0.25 \cdot \color{blue}{\sqrt{\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

       4. pow2 62.0%

          \[\leadsto 0.25 \cdot \sqrt{\color{blue}{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}} \]

   13. Applied egg-rr 62.0%

       \[\leadsto 0.25 \cdot \color{blue}{\sqrt{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}} \]

   if 7.59999999999999992e219 < b < 4.80000000000000024e242

   1. Initial program 16.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 50.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 50.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 50.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 50.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 50.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 50.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 50.8%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   if 4.80000000000000024e242 < b < 1.80000000000000001e265

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 34.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 34.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 34.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 35.2%

         \[\leadsto 0.25 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)} \]

      2. *-commutative 35.2%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \color{blue}{\left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot y-scale\right)}\right)\right) \]

      3. sqrt-unprod 35.2%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\color{blue}{\sqrt{8 \cdot 2}} \cdot y-scale\right)\right)\right) \]

      4. metadata-eval 35.2%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\sqrt{\color{blue}{16}} \cdot y-scale\right)\right)\right) \]

      5. metadata-eval 35.2%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\color{blue}{4} \cdot y-scale\right)\right)\right) \]

   9. Applied egg-rr 35.2%

      \[\leadsto 0.25 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(4 \cdot y-scale\right)\right)\right)} \]

   if 1.80000000000000001e265 < b

   1. Initial program 0.7%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 15.2%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 15.2%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 15.2%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 15.4%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 15.4%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 15.4%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 15.4%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 14.3%

          \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

   11. Applied egg-rr 14.3%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 15.4%

          \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

       2. expm1-log1p-u 14.3%

          \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)} \]

       3. expm1-undefine 14.3%

          \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]

   13. Applied egg-rr 14.3%

       \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. log1p-undefine 14.3%

          \[\leadsto 0.25 \cdot \left(e^{\color{blue}{\log \left(1 + b \cdot \left(y-scale \cdot 4\right)\right)}} - 1\right) \]

       2. rem-exp-log 15.4%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(1 + b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right) \]

       3. associate-+r- 15.4%

          \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \left(b \cdot \left(y-scale \cdot 4\right) - 1\right)\right)} \]

       4. fma-neg 15.4%

          \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(b, y-scale \cdot 4, -1\right)}\right) \]

       5. metadata-eval 15.4%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, y-scale \cdot 4, \color{blue}{-1}\right)\right) \]

       6. *-commutative 15.4%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, \color{blue}{4 \cdot y-scale}, -1\right)\right) \]

   15. Simplified 15.4%

       \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)\right)} \]

   16. Taylor expanded in b around inf 15.4%

       \[\leadsto 0.25 \cdot \left(1 + \color{blue}{b \cdot \left(4 \cdot y-scale - \frac{1}{b}\right)}\right) \]

   17. Taylor expanded in y-scale around inf 15.6%

       \[\leadsto 0.25 \cdot \left(1 + b \cdot \color{blue}{\left(4 \cdot y-scale\right)}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 24.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.15 \cdot 10^{+38}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 7.6 \cdot 10^{+219}:\\ \;\;\;\;0.25 \cdot \sqrt{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{+242}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{+265}:\\ \;\;\;\;0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + b \cdot \left(y-scale \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 35.0% accurate, 12.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := b\_m \cdot \left(y-scale\_m \cdot 4\right)\\ \mathbf{if}\;a \leq 1.26 \cdot 10^{+26} \lor \neg \left(a \leq 1.3 \cdot 10^{+173}\right) \land a \leq 6 \cdot 10^{+173}:\\ \;\;\;\;0.25 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* b_m (* y-scale_m 4.0))))
   (if (or (<= a 1.26e+26) (and (not (<= a 1.3e+173)) (<= a 6e+173)))
     (* 0.25 t_0)
     (* 0.25 (log1p (expm1 t_0))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if ((a <= 1.26e+26) || (!(a <= 1.3e+173) && (a <= 6e+173))) {
		tmp = 0.25 * t_0;
	} else {
		tmp = 0.25 * log1p(expm1(t_0));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if ((a <= 1.26e+26) || (!(a <= 1.3e+173) && (a <= 6e+173))) {
		tmp = 0.25 * t_0;
	} else {
		tmp = 0.25 * Math.log1p(Math.expm1(t_0));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = b_m * (y_45_scale_m * 4.0)
	tmp = 0
	if (a <= 1.26e+26) or (not (a <= 1.3e+173) and (a <= 6e+173)):
		tmp = 0.25 * t_0
	else:
		tmp = 0.25 * math.log1p(math.expm1(t_0))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(b_m * Float64(y_45_scale_m * 4.0))
	tmp = 0.0
	if ((a <= 1.26e+26) || (!(a <= 1.3e+173) && (a <= 6e+173)))
		tmp = Float64(0.25 * t_0);
	else
		tmp = Float64(0.25 * log1p(expm1(t_0)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[a, 1.26e+26], And[N[Not[LessEqual[a, 1.3e+173]], $MachinePrecision], LessEqual[a, 6e+173]]], N[(0.25 * t$95$0), $MachinePrecision], N[(0.25 * N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25999999999999995e26 or 1.2999999999999999e173 < a < 5.9999999999999995e173

   1. Initial program 3.0%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 2.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 25.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 25.1%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 25.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 25.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 25.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 25.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 25.3%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   if 1.25999999999999995e26 < a < 1.2999999999999999e173 or 5.9999999999999995e173 < a

   1. Initial program 1.9%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 2.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 12.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 12.0%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 12.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 18.6%

         \[\leadsto 0.25 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\right)} \]

      2. *-commutative 18.6%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \color{blue}{\left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot y-scale\right)}\right)\right) \]

      3. sqrt-unprod 18.6%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\color{blue}{\sqrt{8 \cdot 2}} \cdot y-scale\right)\right)\right) \]

      4. metadata-eval 18.6%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\sqrt{\color{blue}{16}} \cdot y-scale\right)\right)\right) \]

      5. metadata-eval 18.6%

         \[\leadsto 0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(\color{blue}{4} \cdot y-scale\right)\right)\right) \]

   9. Applied egg-rr 18.6%

      \[\leadsto 0.25 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(4 \cdot y-scale\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.26 \cdot 10^{+26} \lor \neg \left(a \leq 1.3 \cdot 10^{+173}\right) \land a \leq 6 \cdot 10^{+173}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 32.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := b\_m \cdot \left(y-scale\_m \cdot 4\right)\\ \mathbf{if}\;b\_m \leq 3.5 \cdot 10^{+70}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale\_m \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;b\_m \leq 6.6 \cdot 10^{+205}:\\ \;\;\;\;0.25 \cdot \sqrt{{t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* b_m (* y-scale_m 4.0))))
   (if (<= b_m 3.5e+70)
     (* 0.25 (* (* (* x-scale_m (sqrt 2.0)) (sqrt 8.0)) a))
     (if (<= b_m 6.6e+205)
       (* 0.25 (sqrt (pow t_0 2.0)))
       (* 0.25 (+ 1.0 t_0))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if (b_m <= 3.5e+70) {
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	} else if (b_m <= 6.6e+205) {
		tmp = 0.25 * sqrt(pow(t_0, 2.0));
	} else {
		tmp = 0.25 * (1.0 + t_0);
	}
	return tmp;
}

Fortran

b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b_m * (y_45scale_m * 4.0d0)
    if (b_m <= 3.5d+70) then
        tmp = 0.25d0 * (((x_45scale_m * sqrt(2.0d0)) * sqrt(8.0d0)) * a)
    else if (b_m <= 6.6d+205) then
        tmp = 0.25d0 * sqrt((t_0 ** 2.0d0))
    else
        tmp = 0.25d0 * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if (b_m <= 3.5e+70) {
		tmp = 0.25 * (((x_45_scale_m * Math.sqrt(2.0)) * Math.sqrt(8.0)) * a);
	} else if (b_m <= 6.6e+205) {
		tmp = 0.25 * Math.sqrt(Math.pow(t_0, 2.0));
	} else {
		tmp = 0.25 * (1.0 + t_0);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = b_m * (y_45_scale_m * 4.0)
	tmp = 0
	if b_m <= 3.5e+70:
		tmp = 0.25 * (((x_45_scale_m * math.sqrt(2.0)) * math.sqrt(8.0)) * a)
	elif b_m <= 6.6e+205:
		tmp = 0.25 * math.sqrt(math.pow(t_0, 2.0))
	else:
		tmp = 0.25 * (1.0 + t_0)
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(b_m * Float64(y_45_scale_m * 4.0))
	tmp = 0.0
	if (b_m <= 3.5e+70)
		tmp = Float64(0.25 * Float64(Float64(Float64(x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a));
	elseif (b_m <= 6.6e+205)
		tmp = Float64(0.25 * sqrt((t_0 ^ 2.0)));
	else
		tmp = Float64(0.25 * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = b_m * (y_45_scale_m * 4.0);
	tmp = 0.0;
	if (b_m <= 3.5e+70)
		tmp = 0.25 * (((x_45_scale_m * sqrt(2.0)) * sqrt(8.0)) * a);
	elseif (b_m <= 6.6e+205)
		tmp = 0.25 * sqrt((t_0 ^ 2.0));
	else
		tmp = 0.25 * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 3.5e+70], N[(0.25 * N[(N[(N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 6.6e+205], N[(0.25 * N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.50000000000000002e70

   1. Initial program 1.9%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 8.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
   6. Step-by-step derivation

      1. associate-*r* 8.4%

         \[\leadsto \color{blue}{\left(\left(0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

      2. distribute-lft-out 8.4%

         \[\leadsto \left(\left(0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   7. Simplified 8.9%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   8. Taylor expanded in angle around 0 17.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 17.8%

         \[\leadsto 0.25 \cdot \left(a \cdot \color{blue}{\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)}\right) \]

   10. Simplified 17.8%

       \[\leadsto \color{blue}{0.25 \cdot \left(a \cdot \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)} \]

   if 3.50000000000000002e70 < b < 6.6000000000000004e205

   1. Initial program 9.1%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 9.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 33.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 33.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 33.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 33.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 33.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 33.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 33.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 32.3%

          \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

   11. Applied egg-rr 32.3%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 33.9%

          \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

       2. add-sqr-sqrt 33.6%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\sqrt{b \cdot \left(y-scale \cdot 4\right)} \cdot \sqrt{b \cdot \left(y-scale \cdot 4\right)}\right)} \]

       3. sqrt-unprod 61.1%

          \[\leadsto 0.25 \cdot \color{blue}{\sqrt{\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

       4. pow2 61.1%

          \[\leadsto 0.25 \cdot \sqrt{\color{blue}{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}} \]

   13. Applied egg-rr 61.1%

       \[\leadsto 0.25 \cdot \color{blue}{\sqrt{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}} \]

   if 6.6000000000000004e205 < b

   1. Initial program 3.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 24.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 24.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 24.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 24.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 24.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 24.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 24.8%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 23.6%

          \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

   11. Applied egg-rr 23.6%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 24.8%

          \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

       2. expm1-log1p-u 23.7%

          \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)} \]

       3. expm1-undefine 23.6%

          \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]

   13. Applied egg-rr 23.6%

       \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. log1p-undefine 23.6%

          \[\leadsto 0.25 \cdot \left(e^{\color{blue}{\log \left(1 + b \cdot \left(y-scale \cdot 4\right)\right)}} - 1\right) \]

       2. rem-exp-log 24.8%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(1 + b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right) \]

       3. associate-+r- 24.8%

          \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \left(b \cdot \left(y-scale \cdot 4\right) - 1\right)\right)} \]

       4. fma-neg 24.8%

          \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(b, y-scale \cdot 4, -1\right)}\right) \]

       5. metadata-eval 24.8%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, y-scale \cdot 4, \color{blue}{-1}\right)\right) \]

       6. *-commutative 24.8%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, \color{blue}{4 \cdot y-scale}, -1\right)\right) \]

   15. Simplified 24.8%

       \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)\right)} \]

   16. Taylor expanded in b around inf 24.8%

       \[\leadsto 0.25 \cdot \left(1 + \color{blue}{b \cdot \left(4 \cdot y-scale - \frac{1}{b}\right)}\right) \]

   17. Taylor expanded in y-scale around inf 25.0%

       \[\leadsto 0.25 \cdot \left(1 + b \cdot \color{blue}{\left(4 \cdot y-scale\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+70}:\\ \;\;\;\;0.25 \cdot \left(\left(\left(x-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right)\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{+205}:\\ \;\;\;\;0.25 \cdot \sqrt{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + b \cdot \left(y-scale \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 32.4% accurate, 12.6× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := b\_m \cdot \left(y-scale\_m \cdot 4\right)\\ \mathbf{if}\;b\_m \leq 3.4 \cdot 10^{+70}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;b\_m \leq 10^{+206}:\\ \;\;\;\;0.25 \cdot \sqrt{{t\_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* b_m (* y-scale_m 4.0))))
   (if (<= b_m 3.4e+70)
     (* 0.25 (* a (* x-scale_m (* (sqrt 2.0) (sqrt 8.0)))))
     (if (<= b_m 1e+206) (* 0.25 (sqrt (pow t_0 2.0))) (* 0.25 (+ 1.0 t_0))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if (b_m <= 3.4e+70) {
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(2.0) * sqrt(8.0))));
	} else if (b_m <= 1e+206) {
		tmp = 0.25 * sqrt(pow(t_0, 2.0));
	} else {
		tmp = 0.25 * (1.0 + t_0);
	}
	return tmp;
}

Fortran

b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = b_m * (y_45scale_m * 4.0d0)
    if (b_m <= 3.4d+70) then
        tmp = 0.25d0 * (a * (x_45scale_m * (sqrt(2.0d0) * sqrt(8.0d0))))
    else if (b_m <= 1d+206) then
        tmp = 0.25d0 * sqrt((t_0 ** 2.0d0))
    else
        tmp = 0.25d0 * (1.0d0 + t_0)
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if (b_m <= 3.4e+70) {
		tmp = 0.25 * (a * (x_45_scale_m * (Math.sqrt(2.0) * Math.sqrt(8.0))));
	} else if (b_m <= 1e+206) {
		tmp = 0.25 * Math.sqrt(Math.pow(t_0, 2.0));
	} else {
		tmp = 0.25 * (1.0 + t_0);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = b_m * (y_45_scale_m * 4.0)
	tmp = 0
	if b_m <= 3.4e+70:
		tmp = 0.25 * (a * (x_45_scale_m * (math.sqrt(2.0) * math.sqrt(8.0))))
	elif b_m <= 1e+206:
		tmp = 0.25 * math.sqrt(math.pow(t_0, 2.0))
	else:
		tmp = 0.25 * (1.0 + t_0)
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(b_m * Float64(y_45_scale_m * 4.0))
	tmp = 0.0
	if (b_m <= 3.4e+70)
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * Float64(sqrt(2.0) * sqrt(8.0)))));
	elseif (b_m <= 1e+206)
		tmp = Float64(0.25 * sqrt((t_0 ^ 2.0)));
	else
		tmp = Float64(0.25 * Float64(1.0 + t_0));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = b_m * (y_45_scale_m * 4.0);
	tmp = 0.0;
	if (b_m <= 3.4e+70)
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(2.0) * sqrt(8.0))));
	elseif (b_m <= 1e+206)
		tmp = 0.25 * sqrt((t_0 ^ 2.0));
	else
		tmp = 0.25 * (1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 3.4e+70], N[(0.25 * N[(a * N[(x$45$scale$95$m * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1e+206], N[(0.25 * N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 3.4000000000000001e70

   1. Initial program 1.9%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 2.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in y-scale around 0 8.4%

      \[\leadsto \color{blue}{\left(0.25 \cdot \left(\frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}} \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]
   6. Step-by-step derivation

      1. associate-*r* 8.4%

         \[\leadsto \color{blue}{\left(\left(0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}}\right) \cdot \sqrt{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

      2. distribute-lft-out 8.4%

         \[\leadsto \left(\left(0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}}\right) \cdot \sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}}\right) \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   7. Simplified 8.9%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \frac{\sqrt{8}}{x-scale \cdot {y-scale}^{2}}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2} \]

   8. Taylor expanded in angle around 0 17.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]

   if 3.4000000000000001e70 < b < 1e206

   1. Initial program 9.1%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 9.1%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 33.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 33.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 33.7%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 33.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 33.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 33.9%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 33.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 32.3%

          \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

   11. Applied egg-rr 32.3%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 33.9%

          \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

       2. add-sqr-sqrt 33.6%

          \[\leadsto 0.25 \cdot \color{blue}{\left(\sqrt{b \cdot \left(y-scale \cdot 4\right)} \cdot \sqrt{b \cdot \left(y-scale \cdot 4\right)}\right)} \]

       3. sqrt-unprod 61.1%

          \[\leadsto 0.25 \cdot \color{blue}{\sqrt{\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

       4. pow2 61.1%

          \[\leadsto 0.25 \cdot \sqrt{\color{blue}{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}} \]

   13. Applied egg-rr 61.1%

       \[\leadsto 0.25 \cdot \color{blue}{\sqrt{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}} \]

   if 1e206 < b

   1. Initial program 3.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 0.0%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 24.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 24.6%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 24.6%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 24.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 24.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 24.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 24.8%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 23.6%

          \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

   11. Applied egg-rr 23.6%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 24.8%

          \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

       2. expm1-log1p-u 23.7%

          \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)} \]

       3. expm1-undefine 23.6%

          \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]

   13. Applied egg-rr 23.6%

       \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. log1p-undefine 23.6%

          \[\leadsto 0.25 \cdot \left(e^{\color{blue}{\log \left(1 + b \cdot \left(y-scale \cdot 4\right)\right)}} - 1\right) \]

       2. rem-exp-log 24.8%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(1 + b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right) \]

       3. associate-+r- 24.8%

          \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \left(b \cdot \left(y-scale \cdot 4\right) - 1\right)\right)} \]

       4. fma-neg 24.8%

          \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(b, y-scale \cdot 4, -1\right)}\right) \]

       5. metadata-eval 24.8%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, y-scale \cdot 4, \color{blue}{-1}\right)\right) \]

       6. *-commutative 24.8%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, \color{blue}{4 \cdot y-scale}, -1\right)\right) \]

   15. Simplified 24.8%

       \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)\right)} \]

   16. Taylor expanded in b around inf 24.8%

       \[\leadsto 0.25 \cdot \left(1 + \color{blue}{b \cdot \left(4 \cdot y-scale - \frac{1}{b}\right)}\right) \]

   17. Taylor expanded in y-scale around inf 25.0%

       \[\leadsto 0.25 \cdot \left(1 + b \cdot \color{blue}{\left(4 \cdot y-scale\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+70}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\\ \mathbf{elif}\;b \leq 10^{+206}:\\ \;\;\;\;0.25 \cdot \sqrt{{\left(b \cdot \left(y-scale \cdot 4\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + b \cdot \left(y-scale \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 33.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + b\_m \cdot \frac{{y-scale\_m}^{2} \cdot 16 - \frac{-1}{b\_m} \cdot \frac{-1}{b\_m}}{y-scale\_m \cdot 4 - \frac{-1}{b\_m}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 4.8e+78)
   (* 0.25 (* b_m (* y-scale_m 4.0)))
   (*
    0.25
    (+
     1.0
     (*
      b_m
      (/
       (- (* (pow y-scale_m 2.0) 16.0) (* (/ -1.0 b_m) (/ -1.0 b_m)))
       (- (* y-scale_m 4.0) (/ -1.0 b_m))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 4.8e+78) {
		tmp = 0.25 * (b_m * (y_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * (1.0 + (b_m * (((pow(y_45_scale_m, 2.0) * 16.0) - ((-1.0 / b_m) * (-1.0 / b_m))) / ((y_45_scale_m * 4.0) - (-1.0 / b_m)))));
	}
	return tmp;
}

Fortran

b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 4.8d+78) then
        tmp = 0.25d0 * (b_m * (y_45scale_m * 4.0d0))
    else
        tmp = 0.25d0 * (1.0d0 + (b_m * ((((y_45scale_m ** 2.0d0) * 16.0d0) - (((-1.0d0) / b_m) * ((-1.0d0) / b_m))) / ((y_45scale_m * 4.0d0) - ((-1.0d0) / b_m)))))
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 4.8e+78) {
		tmp = 0.25 * (b_m * (y_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * (1.0 + (b_m * (((Math.pow(y_45_scale_m, 2.0) * 16.0) - ((-1.0 / b_m) * (-1.0 / b_m))) / ((y_45_scale_m * 4.0) - (-1.0 / b_m)))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 4.8e+78:
		tmp = 0.25 * (b_m * (y_45_scale_m * 4.0))
	else:
		tmp = 0.25 * (1.0 + (b_m * (((math.pow(y_45_scale_m, 2.0) * 16.0) - ((-1.0 / b_m) * (-1.0 / b_m))) / ((y_45_scale_m * 4.0) - (-1.0 / b_m)))))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 4.8e+78)
		tmp = Float64(0.25 * Float64(b_m * Float64(y_45_scale_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(1.0 + Float64(b_m * Float64(Float64(Float64((y_45_scale_m ^ 2.0) * 16.0) - Float64(Float64(-1.0 / b_m) * Float64(-1.0 / b_m))) / Float64(Float64(y_45_scale_m * 4.0) - Float64(-1.0 / b_m))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 4.8e+78)
		tmp = 0.25 * (b_m * (y_45_scale_m * 4.0));
	else
		tmp = 0.25 * (1.0 + (b_m * ((((y_45_scale_m ^ 2.0) * 16.0) - ((-1.0 / b_m) * (-1.0 / b_m))) / ((y_45_scale_m * 4.0) - (-1.0 / b_m)))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 4.8e+78], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(1.0 + N[(b$95$m * N[(N[(N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * 16.0), $MachinePrecision] - N[(N[(-1.0 / b$95$m), $MachinePrecision] * N[(-1.0 / b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] - N[(-1.0 / b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.7999999999999997e78

   1. Initial program 2.8%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 2.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 24.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 24.1%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 24.1%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 24.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 24.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 24.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 24.3%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   if 4.7999999999999997e78 < a

   1. Initial program 2.2%

      \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   3. Simplified 2.4%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 12.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative 12.3%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

   7. Simplified 12.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
   8. Step-by-step derivation

      1. sqrt-unprod 12.4%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

      2. metadata-eval 12.4%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 12.4%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 12.4%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
   10. Step-by-step derivation

       1. add-exp-log 11.5%

          \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

   11. Applied egg-rr 11.5%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
   12. Step-by-step derivation

       1. rem-exp-log 12.4%

          \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

       2. expm1-log1p-u 11.8%

          \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)} \]

       3. expm1-undefine 11.8%

          \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]

   13. Applied egg-rr 11.8%

       \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]
   14. Step-by-step derivation

       1. log1p-undefine 11.8%

          \[\leadsto 0.25 \cdot \left(e^{\color{blue}{\log \left(1 + b \cdot \left(y-scale \cdot 4\right)\right)}} - 1\right) \]

       2. rem-exp-log 12.4%

          \[\leadsto 0.25 \cdot \left(\color{blue}{\left(1 + b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right) \]

       3. associate-+r- 12.4%

          \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \left(b \cdot \left(y-scale \cdot 4\right) - 1\right)\right)} \]

       4. fma-neg 12.4%

          \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(b, y-scale \cdot 4, -1\right)}\right) \]

       5. metadata-eval 12.4%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, y-scale \cdot 4, \color{blue}{-1}\right)\right) \]

       6. *-commutative 12.4%

          \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, \color{blue}{4 \cdot y-scale}, -1\right)\right) \]

   15. Simplified 12.4%

       \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)\right)} \]

   16. Taylor expanded in b around inf 12.5%

       \[\leadsto 0.25 \cdot \left(1 + \color{blue}{b \cdot \left(4 \cdot y-scale - \frac{1}{b}\right)}\right) \]
   17. Step-by-step derivation

       1. sub-neg 12.5%

          \[\leadsto 0.25 \cdot \left(1 + b \cdot \color{blue}{\left(4 \cdot y-scale + \left(-\frac{1}{b}\right)\right)}\right) \]

       2. flip-+ 15.9%

          \[\leadsto 0.25 \cdot \left(1 + b \cdot \color{blue}{\frac{\left(4 \cdot y-scale\right) \cdot \left(4 \cdot y-scale\right) - \left(-\frac{1}{b}\right) \cdot \left(-\frac{1}{b}\right)}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}}\right) \]

       3. *-commutative 15.9%

          \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{\color{blue}{\left(y-scale \cdot 4\right)} \cdot \left(4 \cdot y-scale\right) - \left(-\frac{1}{b}\right) \cdot \left(-\frac{1}{b}\right)}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}\right) \]

       4. *-commutative 15.9%

          \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{\left(y-scale \cdot 4\right) \cdot \color{blue}{\left(y-scale \cdot 4\right)} - \left(-\frac{1}{b}\right) \cdot \left(-\frac{1}{b}\right)}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}\right) \]

       5. swap-sqr 15.9%

          \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{\color{blue}{\left(y-scale \cdot y-scale\right) \cdot \left(4 \cdot 4\right)} - \left(-\frac{1}{b}\right) \cdot \left(-\frac{1}{b}\right)}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}\right) \]

       6. pow2 15.9%

          \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{\color{blue}{{y-scale}^{2}} \cdot \left(4 \cdot 4\right) - \left(-\frac{1}{b}\right) \cdot \left(-\frac{1}{b}\right)}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}\right) \]

       7. metadata-eval 15.9%

          \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{{y-scale}^{2} \cdot \color{blue}{16} - \left(-\frac{1}{b}\right) \cdot \left(-\frac{1}{b}\right)}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}\right) \]

       8. distribute-neg-frac 15.9%

          \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{{y-scale}^{2} \cdot 16 - \color{blue}{\frac{-1}{b}} \cdot \left(-\frac{1}{b}\right)}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}\right) \]

       9. metadata-eval 15.9%

          \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{{y-scale}^{2} \cdot 16 - \frac{\color{blue}{-1}}{b} \cdot \left(-\frac{1}{b}\right)}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}\right) \]

       10. distribute-neg-frac 15.9%

           \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{{y-scale}^{2} \cdot 16 - \frac{-1}{b} \cdot \color{blue}{\frac{-1}{b}}}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}\right) \]

       11. metadata-eval 15.9%

           \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{{y-scale}^{2} \cdot 16 - \frac{-1}{b} \cdot \frac{\color{blue}{-1}}{b}}{4 \cdot y-scale - \left(-\frac{1}{b}\right)}\right) \]

       12. distribute-neg-frac 15.9%

           \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{{y-scale}^{2} \cdot 16 - \frac{-1}{b} \cdot \frac{-1}{b}}{4 \cdot y-scale - \color{blue}{\frac{-1}{b}}}\right) \]

       13. metadata-eval 15.9%

           \[\leadsto 0.25 \cdot \left(1 + b \cdot \frac{{y-scale}^{2} \cdot 16 - \frac{-1}{b} \cdot \frac{-1}{b}}{4 \cdot y-scale - \frac{\color{blue}{-1}}{b}}\right) \]

   18. Applied egg-rr 15.9%

       \[\leadsto 0.25 \cdot \left(1 + b \cdot \color{blue}{\frac{{y-scale}^{2} \cdot 16 - \frac{-1}{b} \cdot \frac{-1}{b}}{4 \cdot y-scale - \frac{-1}{b}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(1 + b \cdot \frac{{y-scale}^{2} \cdot 16 - \frac{-1}{b} \cdot \frac{-1}{b}}{y-scale \cdot 4 - \frac{-1}{b}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 32.8% accurate, 393.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ 0.25 \cdot \left(b\_m \cdot \left(y-scale\_m \cdot 4\right)\right) \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (* 0.25 (* b_m (* y-scale_m 4.0))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.25 * (b_m * (y_45_scale_m * 4.0));
}

Fortran

b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = 0.25d0 * (b_m * (y_45scale_m * 4.0d0))
end function

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.25 * (b_m * (y_45_scale_m * 4.0));
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	return 0.25 * (b_m * (y_45_scale_m * 4.0))

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(0.25 * Float64(b_m * Float64(y_45_scale_m * 4.0)))
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.25 * (b_m * (y_45_scale_m * 4.0));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.7%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

3. Simplified 2.4%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 21.7%

   \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 21.7%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

7. Simplified 21.7%

   \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
8. Step-by-step derivation

   1. sqrt-unprod 21.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

   2. metadata-eval 21.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

   3. metadata-eval 21.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

9. Applied egg-rr 21.9%

   \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
10. Add Preprocessing

Alternative 23: 3.7% accurate, 2757.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ 0 \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m) :precision binary64 0.0)

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.0;
}

Fortran

b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = 0.0d0
end function

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.0;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	return 0.0

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	return 0.0
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := 0.0

Derivation

1. Initial program 2.7%

   \[\frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

3. Simplified 2.4%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(2 \cdot \left(\left(4 \cdot \frac{a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)\right)\right) \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \left(\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \sqrt{{\left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2} + {\left(\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{x-scale \cdot y-scale}\right)}^{2}}\right)\right)}}{4 \cdot \left(a \cdot \left(b \cdot \left(b \cdot \left(-a\right)\right)\right)\right)} \cdot {\left(x-scale \cdot y-scale\right)}^{2}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 21.7%

   \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]
6. Step-by-step derivation

   1. *-commutative 21.7%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\left(\sqrt{8} \cdot \sqrt{2}\right)}\right)\right) \]

7. Simplified 21.7%

   \[\leadsto \color{blue}{0.25 \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)} \]
8. Step-by-step derivation

   1. sqrt-unprod 21.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{8 \cdot 2}}\right)\right) \]

   2. metadata-eval 21.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

   3. metadata-eval 21.9%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]

9. Applied egg-rr 21.9%

   \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \]
10. Step-by-step derivation

    1. add-exp-log 20.6%

       \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]

11. Applied egg-rr 20.6%

    \[\leadsto 0.25 \cdot \color{blue}{e^{\log \left(b \cdot \left(y-scale \cdot 4\right)\right)}} \]
12. Step-by-step derivation

    1. rem-exp-log 21.9%

       \[\leadsto 0.25 \cdot \color{blue}{\left(b \cdot \left(y-scale \cdot 4\right)\right)} \]

    2. expm1-log1p-u 21.1%

       \[\leadsto 0.25 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)\right)} \]

    3. expm1-undefine 19.0%

       \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]

13. Applied egg-rr 19.0%

    \[\leadsto 0.25 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right)} \]
14. Step-by-step derivation

    1. log1p-undefine 19.0%

       \[\leadsto 0.25 \cdot \left(e^{\color{blue}{\log \left(1 + b \cdot \left(y-scale \cdot 4\right)\right)}} - 1\right) \]

    2. rem-exp-log 19.8%

       \[\leadsto 0.25 \cdot \left(\color{blue}{\left(1 + b \cdot \left(y-scale \cdot 4\right)\right)} - 1\right) \]

    3. associate-+r- 19.8%

       \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \left(b \cdot \left(y-scale \cdot 4\right) - 1\right)\right)} \]

    4. fma-neg 19.8%

       \[\leadsto 0.25 \cdot \left(1 + \color{blue}{\mathsf{fma}\left(b, y-scale \cdot 4, -1\right)}\right) \]

    5. metadata-eval 19.8%

       \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, y-scale \cdot 4, \color{blue}{-1}\right)\right) \]

    6. *-commutative 19.8%

       \[\leadsto 0.25 \cdot \left(1 + \mathsf{fma}\left(b, \color{blue}{4 \cdot y-scale}, -1\right)\right) \]

15. Simplified 19.8%

    \[\leadsto 0.25 \cdot \color{blue}{\left(1 + \mathsf{fma}\left(b, 4 \cdot y-scale, -1\right)\right)} \]

16. Taylor expanded in b around 0 3.8%

    \[\leadsto 0.25 \cdot \left(1 + \color{blue}{-1}\right) \]

17. Final simplification 3.8%

    \[\leadsto 0 \]
18. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))))