a from scale-rotated-ellipse

Percentage Accurate: 2.6% → 58.4%

Time: 2.2min

Alternatives: 13

Speedup: 393.9×

• 30.6% 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: 2.6% 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: 58.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := y-scale\_m \cdot \sqrt{8}\\ \mathbf{if}\;x-scale\_m \leq 3.9 \cdot 10^{+29}:\\ \;\;\;\;0.25 \cdot \left(t\_2 \cdot \left(\mathsf{hypot}\left(a \cdot t\_1, b\right) \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot t\_2\right)\right) \cdot {\left(\sqrt[3]{\sqrt{2} \cdot \mathsf{hypot}\left(\left(a \cdot \cos t\_0\right) \cdot \frac{1}{y-scale\_m}, \frac{t\_1 \cdot b}{y-scale\_m}\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI))
        (t_1 (sin t_0))
        (t_2 (* y-scale_m (sqrt 8.0))))
   (if (<= x-scale_m 3.9e+29)
     (* 0.25 (* t_2 (* (hypot (* a t_1) b) (sqrt 2.0))))
     (*
      (* 0.25 (* x-scale_m t_2))
      (pow
       (cbrt
        (*
         (sqrt 2.0)
         (hypot
          (* (* a (cos t_0)) (/ 1.0 y-scale_m))
          (/ (* t_1 b) y-scale_m))))
       3.0)))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = y_45_scale_m * sqrt(8.0);
	double tmp;
	if (x_45_scale_m <= 3.9e+29) {
		tmp = 0.25 * (t_2 * (hypot((a * t_1), b) * sqrt(2.0)));
	} else {
		tmp = (0.25 * (x_45_scale_m * t_2)) * pow(cbrt((sqrt(2.0) * hypot(((a * cos(t_0)) * (1.0 / y_45_scale_m)), ((t_1 * b) / y_45_scale_m)))), 3.0);
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (0.005555555555555556 * angle) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = y_45_scale_m * Math.sqrt(8.0);
	double tmp;
	if (x_45_scale_m <= 3.9e+29) {
		tmp = 0.25 * (t_2 * (Math.hypot((a * t_1), b) * Math.sqrt(2.0)));
	} else {
		tmp = (0.25 * (x_45_scale_m * t_2)) * Math.pow(Math.cbrt((Math.sqrt(2.0) * Math.hypot(((a * Math.cos(t_0)) * (1.0 / y_45_scale_m)), ((t_1 * b) / y_45_scale_m)))), 3.0);
	}
	return tmp;
}

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * pi)
	t_1 = sin(t_0)
	t_2 = Float64(y_45_scale_m * sqrt(8.0))
	tmp = 0.0
	if (x_45_scale_m <= 3.9e+29)
		tmp = Float64(0.25 * Float64(t_2 * Float64(hypot(Float64(a * t_1), b) * sqrt(2.0))));
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * t_2)) * (cbrt(Float64(sqrt(2.0) * hypot(Float64(Float64(a * cos(t_0)) * Float64(1.0 / y_45_scale_m)), Float64(Float64(t_1 * b) / y_45_scale_m)))) ^ 3.0));
	end
	return tmp
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $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]}, If[LessEqual[x$45$scale$95$m, 3.9e+29], N[(0.25 * N[(t$95$2 * N[(N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(x$45$scale$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[(t$95$1 * b), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.89999999999999968e29

   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 3.5%

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

   5. Taylor expanded in x-scale around 0 21.2%

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

      1. distribute-lft-out 21.2%

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

   7. Simplified 22.3%

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

      1. pow1/2 22.3%

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

      2. *-commutative 22.3%

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

      3. unpow-prod-down 22.3%

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

   9. Applied egg-rr 24.9%

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

   10. Taylor expanded in angle around 0 25.3%

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

   if 3.89999999999999968e29 < x-scale

   1. Initial program 4.1%

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

   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 inf 27.7%

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

   7. Simplified 29.9%

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

      1. add-exp-log 29.4%

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

      2. div-inv 29.4%

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

      3. associate-*r* 29.3%

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

      4. pow-flip 29.3%

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

      5. metadata-eval 29.3%

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

   9. Applied egg-rr 29.3%

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

   11. Applied egg-rr 36.5%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 3.9 \cdot 10^{+29}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot {\left(\sqrt[3]{\sqrt{2} \cdot \mathsf{hypot}\left(\left(a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \frac{1}{y-scale}, \frac{\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b}{y-scale}\right)}\right)}^{3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(0.005555555555555556 \cdot angle\right) \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := y-scale\_m \cdot \sqrt{8}\\ \mathbf{if}\;x-scale\_m \leq 2.55 \cdot 10^{+29}:\\ \;\;\;\;0.25 \cdot \left(t\_2 \cdot \left(\mathsf{hypot}\left(a \cdot t\_1, b\right) \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot t\_2\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a \cdot \cos t\_0}{y-scale\_m}, t\_1 \cdot \frac{b}{y-scale\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* 0.005555555555555556 angle) PI))
        (t_1 (sin t_0))
        (t_2 (* y-scale_m (sqrt 8.0))))
   (if (<= x-scale_m 2.55e+29)
     (* 0.25 (* t_2 (* (hypot (* a t_1) b) (sqrt 2.0))))
     (*
      (* 0.25 (* x-scale_m t_2))
      (*
       (sqrt 2.0)
       (hypot (/ (* a (cos t_0)) y-scale_m) (* t_1 (/ b y-scale_m))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (0.005555555555555556 * angle) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = y_45_scale_m * sqrt(8.0);
	double tmp;
	if (x_45_scale_m <= 2.55e+29) {
		tmp = 0.25 * (t_2 * (hypot((a * t_1), b) * sqrt(2.0)));
	} else {
		tmp = (0.25 * (x_45_scale_m * t_2)) * (sqrt(2.0) * hypot(((a * cos(t_0)) / y_45_scale_m), (t_1 * (b / y_45_scale_m))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (0.005555555555555556 * angle) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = y_45_scale_m * Math.sqrt(8.0);
	double tmp;
	if (x_45_scale_m <= 2.55e+29) {
		tmp = 0.25 * (t_2 * (Math.hypot((a * t_1), b) * Math.sqrt(2.0)));
	} else {
		tmp = (0.25 * (x_45_scale_m * t_2)) * (Math.sqrt(2.0) * Math.hypot(((a * Math.cos(t_0)) / y_45_scale_m), (t_1 * (b / y_45_scale_m))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = (0.005555555555555556 * angle) * math.pi
	t_1 = math.sin(t_0)
	t_2 = y_45_scale_m * math.sqrt(8.0)
	tmp = 0
	if x_45_scale_m <= 2.55e+29:
		tmp = 0.25 * (t_2 * (math.hypot((a * t_1), b) * math.sqrt(2.0)))
	else:
		tmp = (0.25 * (x_45_scale_m * t_2)) * (math.sqrt(2.0) * math.hypot(((a * math.cos(t_0)) / y_45_scale_m), (t_1 * (b / y_45_scale_m))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(0.005555555555555556 * angle) * pi)
	t_1 = sin(t_0)
	t_2 = Float64(y_45_scale_m * sqrt(8.0))
	tmp = 0.0
	if (x_45_scale_m <= 2.55e+29)
		tmp = Float64(0.25 * Float64(t_2 * Float64(hypot(Float64(a * t_1), b) * sqrt(2.0))));
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * t_2)) * Float64(sqrt(2.0) * hypot(Float64(Float64(a * cos(t_0)) / y_45_scale_m), Float64(t_1 * Float64(b / y_45_scale_m)))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = (0.005555555555555556 * angle) * pi;
	t_1 = sin(t_0);
	t_2 = y_45_scale_m * sqrt(8.0);
	tmp = 0.0;
	if (x_45_scale_m <= 2.55e+29)
		tmp = 0.25 * (t_2 * (hypot((a * t_1), b) * sqrt(2.0)));
	else
		tmp = (0.25 * (x_45_scale_m * t_2)) * (sqrt(2.0) * hypot(((a * cos(t_0)) / y_45_scale_m), (t_1 * (b / y_45_scale_m))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $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]}, If[LessEqual[x$45$scale$95$m, 2.55e+29], N[(0.25 * N[(t$95$2 * N[(N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(x$45$scale$95$m * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] / y$45$scale$95$m), $MachinePrecision] ^ 2 + N[(t$95$1 * N[(b / y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.55e29

   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 3.5%

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

   5. Taylor expanded in x-scale around 0 21.2%

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

      1. distribute-lft-out 21.2%

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

   7. Simplified 22.3%

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

      1. pow1/2 22.3%

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

      2. *-commutative 22.3%

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

      3. unpow-prod-down 22.3%

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

   9. Applied egg-rr 24.9%

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

   10. Taylor expanded in angle around 0 25.3%

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

   if 2.55e29 < x-scale

   1. Initial program 4.1%

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

   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 inf 27.7%

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

   7. Simplified 29.9%

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

      1. add-exp-log 29.4%

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

      2. div-inv 29.4%

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

      3. associate-*r* 29.3%

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

      4. pow-flip 29.3%

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

      5. metadata-eval 29.3%

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

   9. Applied egg-rr 29.3%

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

       1. sqrt-prod 29.3%

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

       2. add-sqr-sqrt 29.3%

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

       3. add-sqr-sqrt 29.3%

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

       4. hypot-define 29.3%

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

   11. Applied egg-rr 36.6%

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

       1. associate-*r/ 36.6%

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

       2. *-rgt-identity 36.6%

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

       3. *-commutative 36.6%

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

       4. *-commutative 36.6%

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

       5. *-commutative 36.6%

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

       6. associate-/l* 36.6%

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

       7. *-commutative 36.6%

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

       8. *-commutative 36.6%

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

   13. Simplified 36.6%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.55 \cdot 10^{+29}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\frac{a \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}{y-scale}, \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \frac{b}{y-scale}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 10^{+30}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(4 \cdot \left(x-scale\_m \cdot \left(1 + {angle}^{2} \cdot \left(-1.54320987654321 \cdot 10^{-5} \cdot {\pi}^{2} + 3.969161205100849 \cdot 10^{-11} \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 1e+30)
   (*
    0.25
    (*
     (* y-scale_m (sqrt 8.0))
     (*
      (hypot (* a (sin (* (* 0.005555555555555556 angle) PI))) b)
      (sqrt 2.0))))
   (*
    (* 0.25 a)
    (*
     4.0
     (*
      x-scale_m
      (+
       1.0
       (*
        (pow angle 2.0)
        (+
         (* -1.54320987654321e-5 (pow PI 2.0))
         (* 3.969161205100849e-11 (* (pow angle 2.0) (pow PI 4.0)))))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 1e+30) {
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * (hypot((a * sin(((0.005555555555555556 * angle) * ((double) M_PI)))), b) * sqrt(2.0)));
	} else {
		tmp = (0.25 * a) * (4.0 * (x_45_scale_m * (1.0 + (pow(angle, 2.0) * ((-1.54320987654321e-5 * pow(((double) M_PI), 2.0)) + (3.969161205100849e-11 * (pow(angle, 2.0) * pow(((double) M_PI), 4.0))))))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 1e+30) {
		tmp = 0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * (Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.PI))), b) * Math.sqrt(2.0)));
	} else {
		tmp = (0.25 * a) * (4.0 * (x_45_scale_m * (1.0 + (Math.pow(angle, 2.0) * ((-1.54320987654321e-5 * Math.pow(Math.PI, 2.0)) + (3.969161205100849e-11 * (Math.pow(angle, 2.0) * Math.pow(Math.PI, 4.0))))))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 1e+30:
		tmp = 0.25 * ((y_45_scale_m * math.sqrt(8.0)) * (math.hypot((a * math.sin(((0.005555555555555556 * angle) * math.pi))), b) * math.sqrt(2.0)))
	else:
		tmp = (0.25 * a) * (4.0 * (x_45_scale_m * (1.0 + (math.pow(angle, 2.0) * ((-1.54320987654321e-5 * math.pow(math.pi, 2.0)) + (3.969161205100849e-11 * (math.pow(angle, 2.0) * math.pow(math.pi, 4.0))))))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 1e+30)
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * pi))), b) * sqrt(2.0))));
	else
		tmp = Float64(Float64(0.25 * a) * Float64(4.0 * Float64(x_45_scale_m * Float64(1.0 + Float64((angle ^ 2.0) * Float64(Float64(-1.54320987654321e-5 * (pi ^ 2.0)) + Float64(3.969161205100849e-11 * Float64((angle ^ 2.0) * (pi ^ 4.0)))))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 1e+30)
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * (hypot((a * sin(((0.005555555555555556 * angle) * pi))), b) * sqrt(2.0)));
	else
		tmp = (0.25 * a) * (4.0 * (x_45_scale_m * (1.0 + ((angle ^ 2.0) * ((-1.54320987654321e-5 * (pi ^ 2.0)) + (3.969161205100849e-11 * ((angle ^ 2.0) * (pi ^ 4.0))))))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 1e+30], N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * a), $MachinePrecision] * N[(4.0 * N[(x$45$scale$95$m * N[(1.0 + N[(N[Power[angle, 2.0], $MachinePrecision] * N[(N[(-1.54320987654321e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(3.969161205100849e-11 * N[(N[Power[angle, 2.0], $MachinePrecision] * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1e30

   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 3.5%

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

   5. Taylor expanded in x-scale around 0 21.2%

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

      1. distribute-lft-out 21.2%

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

   7. Simplified 22.3%

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

      1. pow1/2 22.3%

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

      2. *-commutative 22.3%

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

      3. unpow-prod-down 22.3%

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

   9. Applied egg-rr 24.9%

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

   10. Taylor expanded in angle around 0 25.3%

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

   if 1e30 < x-scale

   1. Initial program 4.1%

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

   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 inf 27.7%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in a around inf 19.2%

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

      1. associate-*r* 19.2%

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

      2. associate-*r* 19.2%

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

      3. *-commutative 19.2%

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

      4. associate-*r* 20.4%

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

      5. *-commutative 20.4%

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

   10. Simplified 20.4%

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

       1. sqrt-unprod 20.5%

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

       2. metadata-eval 20.5%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \sqrt{\color{blue}{16}}\right) \]

       3. metadata-eval 20.5%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   12. Applied egg-rr 20.5%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   13. Taylor expanded in angle around 0 30.5%

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

4. Final simplification 26.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 10^{+30}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(4 \cdot \left(x-scale \cdot \left(1 + {angle}^{2} \cdot \left(-1.54320987654321 \cdot 10^{-5} \cdot {\pi}^{2} + 3.969161205100849 \cdot 10^{-11} \cdot \left({angle}^{2} \cdot {\pi}^{4}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 44.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 5.5 \cdot 10^{-74}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale\_m \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 5.5e-74)
   (* (* 0.25 a) (* x-scale_m 4.0))
   (*
    0.25
    (*
     (* y-scale_m (sqrt 8.0))
     (*
      (hypot (* a (sin (* (* 0.005555555555555556 angle) PI))) b)
      (sqrt 2.0))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 5.5e-74) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else {
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * (hypot((a * sin(((0.005555555555555556 * angle) * ((double) M_PI)))), b) * sqrt(2.0)));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 5.5e-74) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else {
		tmp = 0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * (Math.hypot((a * Math.sin(((0.005555555555555556 * angle) * Math.PI))), b) * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 5.5e-74:
		tmp = (0.25 * a) * (x_45_scale_m * 4.0)
	else:
		tmp = 0.25 * ((y_45_scale_m * math.sqrt(8.0)) * (math.hypot((a * math.sin(((0.005555555555555556 * angle) * math.pi))), b) * math.sqrt(2.0)))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 5.5e-74)
		tmp = Float64(Float64(0.25 * a) * Float64(x_45_scale_m * 4.0));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(hypot(Float64(a * sin(Float64(Float64(0.005555555555555556 * angle) * pi))), b) * sqrt(2.0))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 5.5e-74)
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	else
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * (hypot((a * sin(((0.005555555555555556 * angle) * pi))), b) * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 5.5e-74], N[(N[(0.25 * a), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(a * N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 5.5000000000000001e-74

   1. Initial program 2.6%

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

   3. Simplified 3.2%

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

   5. Taylor expanded in x-scale around inf 11.9%

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

   7. Simplified 13.7%

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

   8. Taylor expanded in a around inf 14.4%

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

      1. associate-*r* 14.4%

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

      2. associate-*r* 14.4%

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

      3. *-commutative 14.4%

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

      4. associate-*r* 14.4%

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

      5. *-commutative 14.4%

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

   10. Simplified 14.4%

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

       1. sqrt-unprod 14.4%

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

       2. metadata-eval 14.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \sqrt{\color{blue}{16}}\right) \]

       3. metadata-eval 14.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   12. Applied egg-rr 14.4%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   13. Taylor expanded in angle around 0 16.6%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\color{blue}{1} \cdot x-scale\right) \cdot 4\right) \]

   if 5.5000000000000001e-74 < y-scale

   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 47.5%

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

      1. distribute-lft-out 47.5%

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

   7. Simplified 52.8%

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

      1. pow1/2 52.8%

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

      2. *-commutative 52.8%

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

      3. unpow-prod-down 52.7%

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

   9. Applied egg-rr 60.6%

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

   10. Taylor expanded in angle around 0 60.6%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 5.5 \cdot 10^{-74}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right) \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.9% accurate, 6.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 3.6 \cdot 10^{-72}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale\_m \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 3.6e-72)
   (* (* 0.25 a) (* x-scale_m 4.0))
   (*
    0.25
    (*
     (* y-scale_m (sqrt 8.0))
     (*
      (sqrt 2.0)
      (hypot (* a (sin (* 0.005555555555555556 (* angle PI)))) b))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 3.6e-72) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else {
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * sin((0.005555555555555556 * (angle * ((double) M_PI))))), b)));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 3.6e-72) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else {
		tmp = 0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * Math.hypot((a * Math.sin((0.005555555555555556 * (angle * Math.PI)))), b)));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 3.6e-72:
		tmp = (0.25 * a) * (x_45_scale_m * 4.0)
	else:
		tmp = 0.25 * ((y_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * math.hypot((a * math.sin((0.005555555555555556 * (angle * math.pi)))), b)))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 3.6e-72)
		tmp = Float64(Float64(0.25 * a) * Float64(x_45_scale_m * 4.0));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * hypot(Float64(a * sin(Float64(0.005555555555555556 * Float64(angle * pi)))), b))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 3.6e-72)
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	else
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * sin((0.005555555555555556 * (angle * pi)))), b)));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 3.6e-72], N[(N[(0.25 * a), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 3.6e-72

   1. Initial program 2.6%

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

   3. Simplified 3.2%

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

   5. Taylor expanded in x-scale around inf 11.9%

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

   7. Simplified 13.7%

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

   8. Taylor expanded in a around inf 14.4%

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

      1. associate-*r* 14.4%

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

      2. associate-*r* 14.4%

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

      3. *-commutative 14.4%

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

      4. associate-*r* 14.4%

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

      5. *-commutative 14.4%

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

   10. Simplified 14.4%

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

       1. sqrt-unprod 14.4%

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

       2. metadata-eval 14.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \sqrt{\color{blue}{16}}\right) \]

       3. metadata-eval 14.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   12. Applied egg-rr 14.4%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   13. Taylor expanded in angle around 0 16.6%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\color{blue}{1} \cdot x-scale\right) \cdot 4\right) \]

   if 3.6e-72 < y-scale

   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 47.5%

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

      1. distribute-lft-out 47.5%

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

   7. Simplified 52.8%

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

      1. pow1/2 52.8%

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

      2. *-commutative 52.8%

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

      3. unpow-prod-down 52.7%

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

   9. Applied egg-rr 60.6%

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

   10. Taylor expanded in angle around 0 60.6%

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

   11. Taylor expanded in angle around inf 47.5%

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

       1. *-commutative 47.5%

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

       2. unpow2 47.5%

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

       3. unpow2 47.5%

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

       4. swap-sqr 52.7%

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

       5. unpow2 52.7%

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

       6. hypot-undefine 59.1%

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

       7. *-commutative 59.1%

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

   13. Simplified 59.1%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 3.6 \cdot 10^{-72}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 24.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := y-scale\_m \cdot \sqrt{8}\\ \mathbf{if}\;a \leq 4.1 \cdot 10^{+105}:\\ \;\;\;\;\left(0.25 \cdot \left(t\_0 \cdot \left(x-scale\_m \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale\_m}\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+183}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale\_m \cdot 4\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+203}:\\ \;\;\;\;0.25 \cdot \left(t\_0 \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \sqrt{2}\right) \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* y-scale_m (sqrt 8.0))))
   (if (<= a 4.1e+105)
     (* (* 0.25 (* t_0 (* x-scale_m b))) (/ (sqrt 2.0) x-scale_m))
     (if (<= a 8.4e+183)
       (* (* 0.25 a) (* x-scale_m 4.0))
       (if (<= a 1.2e+203)
         (*
          0.25
          (*
           t_0
           (*
            (* (sin (* (* 0.005555555555555556 angle) PI)) (sqrt 2.0))
            (- a))))
         (* 0.25 (* a (* x-scale_m (* (sqrt 8.0) (sqrt 2.0))))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * sqrt(8.0);
	double tmp;
	if (a <= 4.1e+105) {
		tmp = (0.25 * (t_0 * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	} else if (a <= 8.4e+183) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else if (a <= 1.2e+203) {
		tmp = 0.25 * (t_0 * ((sin(((0.005555555555555556 * angle) * ((double) M_PI))) * sqrt(2.0)) * -a));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * Math.sqrt(8.0);
	double tmp;
	if (a <= 4.1e+105) {
		tmp = (0.25 * (t_0 * (x_45_scale_m * b))) * (Math.sqrt(2.0) / x_45_scale_m);
	} else if (a <= 8.4e+183) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else if (a <= 1.2e+203) {
		tmp = 0.25 * (t_0 * ((Math.sin(((0.005555555555555556 * angle) * Math.PI)) * Math.sqrt(2.0)) * -a));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (Math.sqrt(8.0) * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = y_45_scale_m * math.sqrt(8.0)
	tmp = 0
	if a <= 4.1e+105:
		tmp = (0.25 * (t_0 * (x_45_scale_m * b))) * (math.sqrt(2.0) / x_45_scale_m)
	elif a <= 8.4e+183:
		tmp = (0.25 * a) * (x_45_scale_m * 4.0)
	elif a <= 1.2e+203:
		tmp = 0.25 * (t_0 * ((math.sin(((0.005555555555555556 * angle) * math.pi)) * math.sqrt(2.0)) * -a))
	else:
		tmp = 0.25 * (a * (x_45_scale_m * (math.sqrt(8.0) * math.sqrt(2.0))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(y_45_scale_m * sqrt(8.0))
	tmp = 0.0
	if (a <= 4.1e+105)
		tmp = Float64(Float64(0.25 * Float64(t_0 * Float64(x_45_scale_m * b))) * Float64(sqrt(2.0) / x_45_scale_m));
	elseif (a <= 8.4e+183)
		tmp = Float64(Float64(0.25 * a) * Float64(x_45_scale_m * 4.0));
	elseif (a <= 1.2e+203)
		tmp = Float64(0.25 * Float64(t_0 * Float64(Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * sqrt(2.0)) * Float64(-a))));
	else
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * Float64(sqrt(8.0) * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = y_45_scale_m * sqrt(8.0);
	tmp = 0.0;
	if (a <= 4.1e+105)
		tmp = (0.25 * (t_0 * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	elseif (a <= 8.4e+183)
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	elseif (a <= 1.2e+203)
		tmp = 0.25 * (t_0 * ((sin(((0.005555555555555556 * angle) * pi)) * sqrt(2.0)) * -a));
	else
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 4.1e+105], N[(N[(0.25 * N[(t$95$0 * N[(x$45$scale$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.4e+183], N[(N[(0.25 * a), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.2e+203], N[(0.25 * N[(t$95$0 * N[(N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(a * N[(x$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < 4.1000000000000002e105

   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 3.3%

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

   5. Taylor expanded in b around inf 9.6%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in angle around 0 21.6%

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

   if 4.1000000000000002e105 < a < 8.4e183

   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 16.7%

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

   5. Taylor expanded in x-scale around inf 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in a around inf 36.9%

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

      1. associate-*r* 36.9%

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

      2. associate-*r* 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-*r* 36.4%

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

      5. *-commutative 36.4%

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

   10. Simplified 36.4%

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

       1. sqrt-unprod 36.4%

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

       2. metadata-eval 36.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \sqrt{\color{blue}{16}}\right) \]

       3. metadata-eval 36.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   12. Applied egg-rr 36.4%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   13. Taylor expanded in angle around 0 68.2%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\color{blue}{1} \cdot x-scale\right) \cdot 4\right) \]

   if 8.4e183 < a < 1.2000000000000001e203

   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 x-scale around 0 2.4%

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

      1. distribute-lft-out 2.4%

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

   7. Simplified 2.9%

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

   8. Taylor expanded in a around -inf 8.2%

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

      1. mul-1-neg 8.2%

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

      2. *-commutative 8.2%

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

      3. associate-*r* 8.2%

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

      4. *-commutative 8.2%

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

   10. Simplified 8.2%

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

   if 1.2000000000000001e203 < a

   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 4.5%

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

   5. Taylor expanded in x-scale around inf 14.2%

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

   7. Simplified 18.2%

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

   8. Taylor expanded in angle around 0 18.1%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+105}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale}\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+183}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+203}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \sqrt{2}\right) \cdot \left(-a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 24.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := y-scale\_m \cdot \sqrt{8}\\ \mathbf{if}\;a \leq 4 \cdot 10^{+105}:\\ \;\;\;\;\left(0.25 \cdot \left(t\_0 \cdot \left(x-scale\_m \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale\_m}\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+183}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale\_m \cdot 4\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+203}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(t\_0 \cdot \left(a \cdot \left(-\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* y-scale_m (sqrt 8.0))))
   (if (<= a 4e+105)
     (* (* 0.25 (* t_0 (* x-scale_m b))) (/ (sqrt 2.0) x-scale_m))
     (if (<= a 8.4e+183)
       (* (* 0.25 a) (* x-scale_m 4.0))
       (if (<= a 1.2e+203)
         (*
          0.25
          (*
           (sqrt 2.0)
           (* t_0 (* a (- (sin (* 0.005555555555555556 (* angle PI))))))))
         (* 0.25 (* a (* x-scale_m (* (sqrt 8.0) (sqrt 2.0))))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * sqrt(8.0);
	double tmp;
	if (a <= 4e+105) {
		tmp = (0.25 * (t_0 * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	} else if (a <= 8.4e+183) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else if (a <= 1.2e+203) {
		tmp = 0.25 * (sqrt(2.0) * (t_0 * (a * -sin((0.005555555555555556 * (angle * ((double) M_PI)))))));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = y_45_scale_m * Math.sqrt(8.0);
	double tmp;
	if (a <= 4e+105) {
		tmp = (0.25 * (t_0 * (x_45_scale_m * b))) * (Math.sqrt(2.0) / x_45_scale_m);
	} else if (a <= 8.4e+183) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else if (a <= 1.2e+203) {
		tmp = 0.25 * (Math.sqrt(2.0) * (t_0 * (a * -Math.sin((0.005555555555555556 * (angle * Math.PI))))));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (Math.sqrt(8.0) * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = y_45_scale_m * math.sqrt(8.0)
	tmp = 0
	if a <= 4e+105:
		tmp = (0.25 * (t_0 * (x_45_scale_m * b))) * (math.sqrt(2.0) / x_45_scale_m)
	elif a <= 8.4e+183:
		tmp = (0.25 * a) * (x_45_scale_m * 4.0)
	elif a <= 1.2e+203:
		tmp = 0.25 * (math.sqrt(2.0) * (t_0 * (a * -math.sin((0.005555555555555556 * (angle * math.pi))))))
	else:
		tmp = 0.25 * (a * (x_45_scale_m * (math.sqrt(8.0) * math.sqrt(2.0))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(y_45_scale_m * sqrt(8.0))
	tmp = 0.0
	if (a <= 4e+105)
		tmp = Float64(Float64(0.25 * Float64(t_0 * Float64(x_45_scale_m * b))) * Float64(sqrt(2.0) / x_45_scale_m));
	elseif (a <= 8.4e+183)
		tmp = Float64(Float64(0.25 * a) * Float64(x_45_scale_m * 4.0));
	elseif (a <= 1.2e+203)
		tmp = Float64(0.25 * Float64(sqrt(2.0) * Float64(t_0 * Float64(a * Float64(-sin(Float64(0.005555555555555556 * Float64(angle * pi))))))));
	else
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * Float64(sqrt(8.0) * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = y_45_scale_m * sqrt(8.0);
	tmp = 0.0;
	if (a <= 4e+105)
		tmp = (0.25 * (t_0 * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	elseif (a <= 8.4e+183)
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	elseif (a <= 1.2e+203)
		tmp = 0.25 * (sqrt(2.0) * (t_0 * (a * -sin((0.005555555555555556 * (angle * pi))))));
	else
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 4e+105], N[(N[(0.25 * N[(t$95$0 * N[(x$45$scale$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.4e+183], N[(N[(0.25 * a), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.2e+203], N[(0.25 * N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[(a * (-N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(a * N[(x$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < 3.9999999999999998e105

   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 3.3%

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

   5. Taylor expanded in b around inf 9.6%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in angle around 0 21.6%

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

   if 3.9999999999999998e105 < a < 8.4e183

   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 16.7%

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

   5. Taylor expanded in x-scale around inf 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in a around inf 36.9%

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

      1. associate-*r* 36.9%

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

      2. associate-*r* 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-*r* 36.4%

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

      5. *-commutative 36.4%

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

   10. Simplified 36.4%

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

       1. sqrt-unprod 36.4%

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

       2. metadata-eval 36.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \sqrt{\color{blue}{16}}\right) \]

       3. metadata-eval 36.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   12. Applied egg-rr 36.4%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   13. Taylor expanded in angle around 0 68.2%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\color{blue}{1} \cdot x-scale\right) \cdot 4\right) \]

   if 8.4e183 < a < 1.2000000000000001e203

   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 x-scale around 0 2.4%

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

      1. distribute-lft-out 2.4%

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

   7. Simplified 2.9%

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

      1. log1p-expm1-u 2.6%

         \[\leadsto 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)\right)}\right) \]

      2. log1p-undefine 2.6%

         \[\leadsto 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}\right)\right)}\right) \]

      3. sqrt-prod 2.6%

         \[\leadsto 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \log \left(1 + \mathsf{expm1}\left(\color{blue}{\sqrt{2} \cdot \sqrt{{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}}\right)\right)\right) \]

      4. unpow2 2.6%

         \[\leadsto 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \log \left(1 + \mathsf{expm1}\left(\sqrt{2} \cdot \sqrt{\color{blue}{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)} + {\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right)\right)\right) \]

      5. unpow2 2.6%

         \[\leadsto 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \log \left(1 + \mathsf{expm1}\left(\sqrt{2} \cdot \sqrt{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) + \color{blue}{\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}\right)\right)\right) \]

      6. hypot-define 2.6%

         \[\leadsto 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \log \left(1 + \mathsf{expm1}\left(\sqrt{2} \cdot \color{blue}{\mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)\right)\right) \]

   9. Applied egg-rr 2.6%

      \[\leadsto 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \color{blue}{\log \left(1 + \mathsf{expm1}\left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b \cdot \cos \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)\right)}\right) \]

   10. Taylor expanded in a around -inf 8.2%

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

       1. mul-1-neg 8.2%

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

       2. *-commutative 8.2%

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

       3. distribute-rgt-neg-in 8.2%

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

       4. *-commutative 8.2%

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

       5. associate-*r* 8.2%

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

       6. *-commutative 8.2%

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

       7. *-commutative 8.2%

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

   12. Simplified 8.2%

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

       1. associate-*r* 8.3%

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

       2. distribute-rgt-neg-out 8.3%

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

       3. add-sqr-sqrt 8.3%

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

       4. sqrt-unprod 2.4%

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

       5. sqr-neg 2.4%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 22.7%

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

       8. associate-*r* 22.9%

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

       9. *-commutative 22.9%

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

       10. associate-*l* 22.9%

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

       11. associate-*l* 22.9%

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

   14. Applied egg-rr 8.2%

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

   if 1.2000000000000001e203 < a

   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 4.5%

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

   5. Taylor expanded in x-scale around inf 14.2%

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

   7. Simplified 18.2%

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

   8. Taylor expanded in angle around 0 18.1%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4 \cdot 10^{+105}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale}\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+183}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+203}:\\ \;\;\;\;0.25 \cdot \left(\sqrt{2} \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(a \cdot \left(-\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 24.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \sqrt{8} \cdot \sqrt{2}\\ \mathbf{if}\;a \leq 2.7 \cdot 10^{+105}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(x-scale\_m \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale\_m}\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+183}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale\_m \cdot 4\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+203}:\\ \;\;\;\;\left(a \cdot -0.25\right) \cdot \left(y-scale\_m \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (sqrt 8.0) (sqrt 2.0))))
   (if (<= a 2.7e+105)
     (*
      (* 0.25 (* (* y-scale_m (sqrt 8.0)) (* x-scale_m b)))
      (/ (sqrt 2.0) x-scale_m))
     (if (<= a 8.4e+183)
       (* (* 0.25 a) (* x-scale_m 4.0))
       (if (<= a 1.2e+203)
         (*
          (* a -0.25)
          (* y-scale_m (* (sin (* (* 0.005555555555555556 angle) PI)) t_0)))
         (* 0.25 (* a (* x-scale_m t_0))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = sqrt(8.0) * sqrt(2.0);
	double tmp;
	if (a <= 2.7e+105) {
		tmp = (0.25 * ((y_45_scale_m * sqrt(8.0)) * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	} else if (a <= 8.4e+183) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else if (a <= 1.2e+203) {
		tmp = (a * -0.25) * (y_45_scale_m * (sin(((0.005555555555555556 * angle) * ((double) M_PI))) * t_0));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * t_0));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.sqrt(8.0) * Math.sqrt(2.0);
	double tmp;
	if (a <= 2.7e+105) {
		tmp = (0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * (x_45_scale_m * b))) * (Math.sqrt(2.0) / x_45_scale_m);
	} else if (a <= 8.4e+183) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else if (a <= 1.2e+203) {
		tmp = (a * -0.25) * (y_45_scale_m * (Math.sin(((0.005555555555555556 * angle) * Math.PI)) * t_0));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * t_0));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.sqrt(8.0) * math.sqrt(2.0)
	tmp = 0
	if a <= 2.7e+105:
		tmp = (0.25 * ((y_45_scale_m * math.sqrt(8.0)) * (x_45_scale_m * b))) * (math.sqrt(2.0) / x_45_scale_m)
	elif a <= 8.4e+183:
		tmp = (0.25 * a) * (x_45_scale_m * 4.0)
	elif a <= 1.2e+203:
		tmp = (a * -0.25) * (y_45_scale_m * (math.sin(((0.005555555555555556 * angle) * math.pi)) * t_0))
	else:
		tmp = 0.25 * (a * (x_45_scale_m * t_0))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(sqrt(8.0) * sqrt(2.0))
	tmp = 0.0
	if (a <= 2.7e+105)
		tmp = Float64(Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(x_45_scale_m * b))) * Float64(sqrt(2.0) / x_45_scale_m));
	elseif (a <= 8.4e+183)
		tmp = Float64(Float64(0.25 * a) * Float64(x_45_scale_m * 4.0));
	elseif (a <= 1.2e+203)
		tmp = Float64(Float64(a * -0.25) * Float64(y_45_scale_m * Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * t_0)));
	else
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * t_0)));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = sqrt(8.0) * sqrt(2.0);
	tmp = 0.0;
	if (a <= 2.7e+105)
		tmp = (0.25 * ((y_45_scale_m * sqrt(8.0)) * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	elseif (a <= 8.4e+183)
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	elseif (a <= 1.2e+203)
		tmp = (a * -0.25) * (y_45_scale_m * (sin(((0.005555555555555556 * angle) * pi)) * t_0));
	else
		tmp = 0.25 * (a * (x_45_scale_m * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 2.7e+105], N[(N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 8.4e+183], N[(N[(0.25 * a), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.2e+203], N[(N[(a * -0.25), $MachinePrecision] * N[(y$45$scale$95$m * N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(a * N[(x$45$scale$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < 2.70000000000000016e105

   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 3.3%

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

   5. Taylor expanded in b around inf 9.6%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in angle around 0 21.6%

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

   if 2.70000000000000016e105 < a < 8.4e183

   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 16.7%

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

   5. Taylor expanded in x-scale around inf 17.1%

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

   7. Simplified 17.1%

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

   8. Taylor expanded in a around inf 36.9%

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

      1. associate-*r* 36.9%

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

      2. associate-*r* 36.9%

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

      3. *-commutative 36.9%

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

      4. associate-*r* 36.4%

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

      5. *-commutative 36.4%

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

   10. Simplified 36.4%

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

       1. sqrt-unprod 36.4%

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

       2. metadata-eval 36.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \sqrt{\color{blue}{16}}\right) \]

       3. metadata-eval 36.4%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   12. Applied egg-rr 36.4%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   13. Taylor expanded in angle around 0 68.2%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\color{blue}{1} \cdot x-scale\right) \cdot 4\right) \]

   if 8.4e183 < a < 1.2000000000000001e203

   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 y-scale around inf 21.1%

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

   7. Simplified 21.1%

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

   8. Taylor expanded in a around -inf 8.3%

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

      1. associate-*r* 8.3%

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

      2. associate-*r* 8.3%

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

      3. *-commutative 8.3%

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

   10. Simplified 8.3%

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

   if 1.2000000000000001e203 < a

   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 4.5%

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

   5. Taylor expanded in x-scale around inf 14.2%

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

   7. Simplified 18.2%

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

   8. Taylor expanded in angle around 0 18.1%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.7 \cdot 10^{+105}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale}\\ \mathbf{elif}\;a \leq 8.4 \cdot 10^{+183}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+203}:\\ \;\;\;\;\left(a \cdot -0.25\right) \cdot \left(y-scale \cdot \left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.0% accurate, 8.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.12 \cdot 10^{-41}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale\_m \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 1.12e-41)
   (* (* 0.25 a) (* x-scale_m 4.0))
   (*
    0.25
    (*
     (* y-scale_m (sqrt 8.0))
     (* (sqrt 2.0) (hypot (* a (* (* 0.005555555555555556 angle) PI)) b))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.12e-41) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else {
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * ((0.005555555555555556 * angle) * ((double) M_PI))), b)));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 1.12e-41) {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	} else {
		tmp = 0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * Math.hypot((a * ((0.005555555555555556 * angle) * Math.PI)), b)));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 1.12e-41:
		tmp = (0.25 * a) * (x_45_scale_m * 4.0)
	else:
		tmp = 0.25 * ((y_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * math.hypot((a * ((0.005555555555555556 * angle) * math.pi)), b)))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 1.12e-41)
		tmp = Float64(Float64(0.25 * a) * Float64(x_45_scale_m * 4.0));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * hypot(Float64(a * Float64(Float64(0.005555555555555556 * angle) * pi)), b))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 1.12e-41)
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	else
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * ((0.005555555555555556 * angle) * pi)), b)));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 1.12e-41], N[(N[(0.25 * a), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.11999999999999999e-41

   1. Initial program 2.6%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x-scale around inf 11.7%

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

   7. Simplified 13.4%

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

   8. Taylor expanded in a around inf 14.0%

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

      1. associate-*r* 14.0%

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

      2. associate-*r* 14.0%

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

      3. *-commutative 14.0%

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

      4. associate-*r* 14.0%

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

      5. *-commutative 14.0%

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

   10. Simplified 14.0%

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

       1. sqrt-unprod 14.1%

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

       2. metadata-eval 14.1%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \sqrt{\color{blue}{16}}\right) \]

       3. metadata-eval 14.1%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   12. Applied egg-rr 14.1%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   13. Taylor expanded in angle around 0 16.2%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\color{blue}{1} \cdot x-scale\right) \cdot 4\right) \]

   if 1.11999999999999999e-41 < y-scale

   1. Initial program 4.9%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in x-scale around 0 50.9%

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

      1. distribute-lft-out 50.9%

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

   7. Simplified 56.6%

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

      1. pow1/2 56.6%

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

      2. *-commutative 56.6%

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

      3. unpow-prod-down 56.5%

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

   9. Applied egg-rr 64.7%

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

   10. Taylor expanded in angle around 0 64.6%

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

   11. Taylor expanded in angle around 0 62.0%

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

       1. associate-*r* 62.1%

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

       2. *-commutative 62.1%

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

       3. *-commutative 62.1%

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

   13. Simplified 62.1%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.12 \cdot 10^{-41}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right), b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 24.8% accurate, 12.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \left(x-scale\_m \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 2.15e+105)
   (*
    (* 0.25 (* (* y-scale_m (sqrt 8.0)) (* x-scale_m b)))
    (/ (sqrt 2.0) x-scale_m))
   (* 0.25 (* a (* x-scale_m (* (sqrt 8.0) (sqrt 2.0)))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 2.15e+105) {
		tmp = (0.25 * ((y_45_scale_m * sqrt(8.0)) * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 2.15d+105) then
        tmp = (0.25d0 * ((y_45scale_m * sqrt(8.0d0)) * (x_45scale_m * b))) * (sqrt(2.0d0) / x_45scale_m)
    else
        tmp = 0.25d0 * (a * (x_45scale_m * (sqrt(8.0d0) * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 2.15e+105) {
		tmp = (0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * (x_45_scale_m * b))) * (Math.sqrt(2.0) / x_45_scale_m);
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (Math.sqrt(8.0) * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 2.15e+105:
		tmp = (0.25 * ((y_45_scale_m * math.sqrt(8.0)) * (x_45_scale_m * b))) * (math.sqrt(2.0) / x_45_scale_m)
	else:
		tmp = 0.25 * (a * (x_45_scale_m * (math.sqrt(8.0) * math.sqrt(2.0))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 2.15e+105)
		tmp = Float64(Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * Float64(x_45_scale_m * b))) * Float64(sqrt(2.0) / x_45_scale_m));
	else
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * Float64(sqrt(8.0) * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 2.15e+105)
		tmp = (0.25 * ((y_45_scale_m * sqrt(8.0)) * (x_45_scale_m * b))) * (sqrt(2.0) / x_45_scale_m);
	else
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 2.15e+105], N[(N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(a * N[(x$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.1500000000000001e105

   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 3.3%

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

   5. Taylor expanded in b around inf 9.6%

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

   7. Simplified 10.1%

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

   8. Taylor expanded in angle around 0 21.6%

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

   if 2.1500000000000001e105 < a

   1. Initial program 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x-scale around inf 13.0%

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

   7. Simplified 15.8%

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

   8. Taylor expanded in angle around 0 24.5%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{+105}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \left(x-scale \cdot b\right)\right)\right) \cdot \frac{\sqrt{2}}{x-scale}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 23.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+105}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 2.3e+105)
   (* 0.25 (* b (* y-scale_m 4.0)))
   (* 0.25 (* a (* x-scale_m (* (sqrt 8.0) (sqrt 2.0)))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 2.3e+105) {
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 2.3d+105) then
        tmp = 0.25d0 * (b * (y_45scale_m * 4.0d0))
    else
        tmp = 0.25d0 * (a * (x_45scale_m * (sqrt(8.0d0) * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 2.3e+105) {
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (Math.sqrt(8.0) * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 2.3e+105:
		tmp = 0.25 * (b * (y_45_scale_m * 4.0))
	else:
		tmp = 0.25 * (a * (x_45_scale_m * (math.sqrt(8.0) * math.sqrt(2.0))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 2.3e+105)
		tmp = Float64(0.25 * Float64(b * Float64(y_45_scale_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * Float64(sqrt(8.0) * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 2.3e+105)
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	else
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 2.3e+105], N[(0.25 * N[(b * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(a * N[(x$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.2999999999999998e105

   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 3.3%

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

   5. Taylor expanded in angle around 0 19.6%

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

      1. *-commutative 19.6%

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

   7. Simplified 19.6%

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

      1. sqrt-unprod 19.8%

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

      2. metadata-eval 19.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 19.8%

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

   9. Applied egg-rr 19.8%

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

   if 2.2999999999999998e105 < a

   1. Initial program 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x-scale around inf 13.0%

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

   7. Simplified 15.8%

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

   8. Taylor expanded in angle around 0 24.5%

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

4. Final simplification 20.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.3 \cdot 10^{+105}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 24.0% accurate, 229.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{+105}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale\_m \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 2.5e+105)
   (* 0.25 (* b (* y-scale_m 4.0)))
   (* (* 0.25 a) (* x-scale_m 4.0))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 2.5e+105) {
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	} else {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 2.5d+105) then
        tmp = 0.25d0 * (b * (y_45scale_m * 4.0d0))
    else
        tmp = (0.25d0 * a) * (x_45scale_m * 4.0d0)
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 2.5e+105) {
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	} else {
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 2.5e+105:
		tmp = 0.25 * (b * (y_45_scale_m * 4.0))
	else:
		tmp = (0.25 * a) * (x_45_scale_m * 4.0)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 2.5e+105)
		tmp = Float64(0.25 * Float64(b * Float64(y_45_scale_m * 4.0)));
	else
		tmp = Float64(Float64(0.25 * a) * Float64(x_45_scale_m * 4.0));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 2.5e+105)
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	else
		tmp = (0.25 * a) * (x_45_scale_m * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 2.5e+105], N[(0.25 * N[(b * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * a), $MachinePrecision] * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.50000000000000023e105

   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 3.3%

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

   5. Taylor expanded in angle around 0 19.6%

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

      1. *-commutative 19.6%

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

   7. Simplified 19.6%

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

      1. sqrt-unprod 19.8%

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

      2. metadata-eval 19.8%

         \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

      3. metadata-eval 19.8%

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

   9. Applied egg-rr 19.8%

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

   if 2.50000000000000023e105 < a

   1. Initial program 5.9%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in x-scale around inf 13.0%

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

   7. Simplified 15.8%

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

   8. Taylor expanded in a around inf 16.8%

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

      1. associate-*r* 16.8%

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

      2. associate-*r* 16.8%

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

      3. *-commutative 16.8%

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

      4. associate-*r* 16.6%

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

      5. *-commutative 16.6%

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

   10. Simplified 16.6%

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

       1. sqrt-unprod 16.6%

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

       2. metadata-eval 16.6%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \sqrt{\color{blue}{16}}\right) \]

       3. metadata-eval 16.6%

          \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   12. Applied egg-rr 16.6%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot x-scale\right) \cdot \color{blue}{4}\right) \]

   13. Taylor expanded in angle around 0 24.6%

       \[\leadsto \left(0.25 \cdot a\right) \cdot \left(\left(\color{blue}{1} \cdot x-scale\right) \cdot 4\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.5 \cdot 10^{+105}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(x-scale \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 17.6% accurate, 393.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ 0.25 \cdot \left(b \cdot \left(y-scale\_m \cdot 4\right)\right) \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (* 0.25 (* b (* y-scale_m 4.0))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.25 * (b * (y_45_scale_m * 4.0));
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = 0.25d0 * (b * (y_45scale_m * 4.0d0))
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	return 0.25 * (b * (y_45_scale_m * 4.0));
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	return 0.25 * (b * (y_45_scale_m * 4.0))

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	return Float64(0.25 * Float64(b * Float64(y_45_scale_m * 4.0)))
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.25 * (b * (y_45_scale_m * 4.0));
end

Wolfram

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

Derivation

1. Initial program 3.2%

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

3. Simplified 3.6%

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

5. Taylor expanded in angle around 0 17.6%

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

   1. *-commutative 17.6%

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

7. Simplified 17.6%

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

   1. sqrt-unprod 17.7%

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

   2. metadata-eval 17.7%

      \[\leadsto 0.25 \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \]

   3. metadata-eval 17.7%

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

9. Applied egg-rr 17.7%

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

Reproduce

herbie shell --seed 2024092 
(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))))