a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 59.0%

Time: 1.9min

Alternatives: 12

Speedup: 393.9×

• 29.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 59.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ \mathbf{if}\;y-scale\_m \leq 6.4 \cdot 10^{+86}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot t\_1, b \cdot t\_2\right) \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(\sqrt{8} \cdot {\left(\sqrt{\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, t\_1 \cdot b\right)}\right)}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0)))
   (if (<= y-scale_m 6.4e+86)
     (*
      (* 0.25 (* x-scale_m (sqrt 8.0)))
      (* (hypot (* a t_1) (* b t_2)) (* (pow 2.0 0.25) (pow 2.0 0.25))))
     (*
      0.25
      (*
       y-scale_m
       (*
        (sqrt 8.0)
        (pow (sqrt (* (sqrt 2.0) (hypot (* a t_2) (* t_1 b)))) 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 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double tmp;
	if (y_45_scale_m <= 6.4e+86) {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (hypot((a * t_1), (b * t_2)) * (pow(2.0, 0.25) * pow(2.0, 0.25)));
	} else {
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * pow(sqrt((sqrt(2.0) * hypot((a * t_2), (t_1 * b)))), 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 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double t_2 = Math.sin(t_0);
	double tmp;
	if (y_45_scale_m <= 6.4e+86) {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.hypot((a * t_1), (b * t_2)) * (Math.pow(2.0, 0.25) * Math.pow(2.0, 0.25)));
	} else {
		tmp = 0.25 * (y_45_scale_m * (Math.sqrt(8.0) * Math.pow(Math.sqrt((Math.sqrt(2.0) * Math.hypot((a * t_2), (t_1 * b)))), 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 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.cos(t_0)
	t_2 = math.sin(t_0)
	tmp = 0
	if y_45_scale_m <= 6.4e+86:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.hypot((a * t_1), (b * t_2)) * (math.pow(2.0, 0.25) * math.pow(2.0, 0.25)))
	else:
		tmp = 0.25 * (y_45_scale_m * (math.sqrt(8.0) * math.pow(math.sqrt((math.sqrt(2.0) * math.hypot((a * t_2), (t_1 * b)))), 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(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	tmp = 0.0
	if (y_45_scale_m <= 6.4e+86)
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(hypot(Float64(a * t_1), Float64(b * t_2)) * Float64((2.0 ^ 0.25) * (2.0 ^ 0.25))));
	else
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(sqrt(8.0) * (sqrt(Float64(sqrt(2.0) * hypot(Float64(a * t_2), Float64(t_1 * b)))) ^ 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 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0);
	t_2 = sin(t_0);
	tmp = 0.0;
	if (y_45_scale_m <= 6.4e+86)
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (hypot((a * t_1), (b * t_2)) * ((2.0 ^ 0.25) * (2.0 ^ 0.25)));
	else
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * (sqrt((sqrt(2.0) * hypot((a * t_2), (t_1 * b)))) ^ 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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 6.4e+86], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + N[(b * t$95$2), $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(y$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + N[(t$95$1 * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 6.4000000000000001e86

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in y-scale around 0 22.8%

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

      1. associate-*r* 22.8%

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

      2. distribute-lft-out 22.8%

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

   7. Simplified 22.9%

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

      1. distribute-lft-in 22.9%

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

      2. unpow-prod-down 22.9%

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

      3. unpow-prod-down 22.8%

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

      4. add-cbrt-cube 21.1%

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

   9. Applied egg-rr 21.3%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left({\left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   11. Applied egg-rr 23.5%

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

       1. associate-*l* 23.5%

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

       2. associate-*r* 23.6%

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

       3. associate-*r* 23.6%

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

   13. Simplified 23.6%

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

   if 6.4000000000000001e86 < y-scale

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in x-scale around 0 55.9%

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

   7. Simplified 58.5%

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

      1. add-sqr-sqrt 58.5%

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

      2. pow2 58.5%

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

   9. Applied egg-rr 57.0%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 6.4 \cdot 10^{+86}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot {\left(\sqrt{\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}\right)}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ \mathbf{if}\;y-scale\_m \leq 9.4 \cdot 10^{+86}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot t\_1, b \cdot t\_2\right) \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, t\_1 \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0)))
   (if (<= y-scale_m 9.4e+86)
     (*
      (* 0.25 (* x-scale_m (sqrt 8.0)))
      (* (hypot (* a t_1) (* b t_2)) (* (pow 2.0 0.25) (pow 2.0 0.25))))
     (*
      0.25
      (*
       y-scale_m
       (* (sqrt 8.0) (* (sqrt 2.0) (hypot (* a t_2) (* t_1 b)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double tmp;
	if (y_45_scale_m <= 9.4e+86) {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (hypot((a * t_1), (b * t_2)) * (pow(2.0, 0.25) * pow(2.0, 0.25)));
	} else {
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * t_2), (t_1 * b)))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double t_2 = Math.sin(t_0);
	double tmp;
	if (y_45_scale_m <= 9.4e+86) {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.hypot((a * t_1), (b * t_2)) * (Math.pow(2.0, 0.25) * Math.pow(2.0, 0.25)));
	} else {
		tmp = 0.25 * (y_45_scale_m * (Math.sqrt(8.0) * (Math.sqrt(2.0) * Math.hypot((a * t_2), (t_1 * b)))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.cos(t_0)
	t_2 = math.sin(t_0)
	tmp = 0
	if y_45_scale_m <= 9.4e+86:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.hypot((a * t_1), (b * t_2)) * (math.pow(2.0, 0.25) * math.pow(2.0, 0.25)))
	else:
		tmp = 0.25 * (y_45_scale_m * (math.sqrt(8.0) * (math.sqrt(2.0) * math.hypot((a * t_2), (t_1 * b)))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	tmp = 0.0
	if (y_45_scale_m <= 9.4e+86)
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(hypot(Float64(a * t_1), Float64(b * t_2)) * Float64((2.0 ^ 0.25) * (2.0 ^ 0.25))));
	else
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(sqrt(8.0) * Float64(sqrt(2.0) * hypot(Float64(a * t_2), Float64(t_1 * b))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0);
	t_2 = sin(t_0);
	tmp = 0.0;
	if (y_45_scale_m <= 9.4e+86)
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (hypot((a * t_1), (b * t_2)) * ((2.0 ^ 0.25) * (2.0 ^ 0.25)));
	else
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * t_2), (t_1 * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 9.4e+86], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + N[(b * t$95$2), $MachinePrecision] ^ 2], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(y$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + N[(t$95$1 * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 9.4000000000000004e86

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in y-scale around 0 22.8%

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

      1. associate-*r* 22.8%

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

      2. distribute-lft-out 22.8%

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

   7. Simplified 22.9%

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

      1. distribute-lft-in 22.9%

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

      2. unpow-prod-down 22.9%

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

      3. unpow-prod-down 22.8%

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

      4. add-cbrt-cube 21.1%

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

   9. Applied egg-rr 21.3%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left({\left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   11. Applied egg-rr 23.5%

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

       1. associate-*l* 23.5%

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

       2. associate-*r* 23.6%

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

       3. associate-*r* 23.6%

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

   13. Simplified 23.6%

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

   if 9.4000000000000004e86 < y-scale

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in x-scale around 0 55.9%

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

   7. Simplified 58.5%

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

      1. pow1/2 58.5%

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

      2. *-commutative 58.5%

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

      3. unpow-prod-down 58.4%

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

      4. pow1/2 58.4%

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

      5. unpow2 58.4%

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

      6. unpow2 58.4%

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

      7. hypot-define 56.9%

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

      8. pow1/2 56.9%

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

   9. Applied egg-rr 56.9%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 9.4 \cdot 10^{+86}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left({2}^{0.25} \cdot {2}^{0.25}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 59.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;y-scale\_m \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \mathsf{hypot}\left(a \cdot \cos t\_0, b \cdot \sin t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_1, \cos t\_1 \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* PI (* 0.005555555555555556 angle)))
        (t_1 (* 0.005555555555555556 (* angle PI))))
   (if (<= y-scale_m 6.8e+86)
     (*
      (* 0.25 (* x-scale_m (sqrt 8.0)))
      (*
       (pow 2.0 0.25)
       (* (pow 2.0 0.25) (hypot (* a (cos t_0)) (* b (sin t_0))))))
     (*
      0.25
      (*
       y-scale_m
       (*
        (sqrt 8.0)
        (* (sqrt 2.0) (hypot (* a (sin t_1)) (* (cos t_1) b)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_1 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (y_45_scale_m <= 6.8e+86) {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (pow(2.0, 0.25) * (pow(2.0, 0.25) * hypot((a * cos(t_0)), (b * sin(t_0)))));
	} else {
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * sin(t_1)), (cos(t_1) * b)))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double t_1 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (y_45_scale_m <= 6.8e+86) {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.pow(2.0, 0.25) * (Math.pow(2.0, 0.25) * Math.hypot((a * Math.cos(t_0)), (b * Math.sin(t_0)))));
	} else {
		tmp = 0.25 * (y_45_scale_m * (Math.sqrt(8.0) * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_1)), (Math.cos(t_1) * b)))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.pi * (0.005555555555555556 * angle)
	t_1 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if y_45_scale_m <= 6.8e+86:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.pow(2.0, 0.25) * (math.pow(2.0, 0.25) * math.hypot((a * math.cos(t_0)), (b * math.sin(t_0)))))
	else:
		tmp = 0.25 * (y_45_scale_m * (math.sqrt(8.0) * (math.sqrt(2.0) * math.hypot((a * math.sin(t_1)), (math.cos(t_1) * b)))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_1 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (y_45_scale_m <= 6.8e+86)
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64((2.0 ^ 0.25) * Float64((2.0 ^ 0.25) * hypot(Float64(a * cos(t_0)), Float64(b * sin(t_0))))));
	else
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(sqrt(8.0) * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_1)), Float64(cos(t_1) * b))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = pi * (0.005555555555555556 * angle);
	t_1 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (y_45_scale_m <= 6.8e+86)
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * ((2.0 ^ 0.25) * ((2.0 ^ 0.25) * hypot((a * cos(t_0)), (b * sin(t_0)))));
	else
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * sin(t_1)), (cos(t_1) * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 6.8e+86], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(N[Power[2.0, 0.25], $MachinePrecision] * N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(y$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Cos[t$95$1], $MachinePrecision] * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 6.7999999999999995e86

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in y-scale around 0 22.8%

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

      1. associate-*r* 22.8%

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

      2. distribute-lft-out 22.8%

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

   7. Simplified 22.9%

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

      1. distribute-lft-in 22.9%

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

      2. unpow-prod-down 22.9%

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

      3. unpow-prod-down 22.8%

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

      4. add-cbrt-cube 21.1%

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

   9. Applied egg-rr 21.3%

      \[\leadsto \left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \color{blue}{{\left({\left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   11. Applied egg-rr 23.5%

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

   if 6.7999999999999995e86 < y-scale

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in x-scale around 0 55.9%

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

   7. Simplified 58.5%

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

      1. pow1/2 58.5%

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

      2. *-commutative 58.5%

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

      3. unpow-prod-down 58.4%

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

      4. pow1/2 58.4%

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

      5. unpow2 58.4%

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

      6. unpow2 58.4%

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

      7. hypot-define 56.9%

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

      8. pow1/2 56.9%

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

   9. Applied egg-rr 56.9%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 6.8 \cdot 10^{+86}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left({2}^{0.25} \cdot \left({2}^{0.25} \cdot \mathsf{hypot}\left(a \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right), b \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ \mathbf{if}\;y-scale\_m \leq 1.25 \cdot 10^{+87}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot t\_1, b \cdot t\_2\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, t\_1 \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0)))
   (if (<= y-scale_m 1.25e+87)
     (*
      (* 0.25 (* x-scale_m (sqrt 8.0)))
      (* (hypot (* a t_1) (* b t_2)) (sqrt 2.0)))
     (*
      0.25
      (*
       y-scale_m
       (* (sqrt 8.0) (* (sqrt 2.0) (hypot (* a t_2) (* t_1 b)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double tmp;
	if (y_45_scale_m <= 1.25e+87) {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (hypot((a * t_1), (b * t_2)) * sqrt(2.0));
	} else {
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * t_2), (t_1 * b)))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double t_2 = Math.sin(t_0);
	double tmp;
	if (y_45_scale_m <= 1.25e+87) {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * (Math.hypot((a * t_1), (b * t_2)) * Math.sqrt(2.0));
	} else {
		tmp = 0.25 * (y_45_scale_m * (Math.sqrt(8.0) * (Math.sqrt(2.0) * Math.hypot((a * t_2), (t_1 * b)))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.cos(t_0)
	t_2 = math.sin(t_0)
	tmp = 0
	if y_45_scale_m <= 1.25e+87:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * (math.hypot((a * t_1), (b * t_2)) * math.sqrt(2.0))
	else:
		tmp = 0.25 * (y_45_scale_m * (math.sqrt(8.0) * (math.sqrt(2.0) * math.hypot((a * t_2), (t_1 * b)))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	tmp = 0.0
	if (y_45_scale_m <= 1.25e+87)
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(hypot(Float64(a * t_1), Float64(b * t_2)) * sqrt(2.0)));
	else
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(sqrt(8.0) * Float64(sqrt(2.0) * hypot(Float64(a * t_2), Float64(t_1 * b))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0);
	t_2 = sin(t_0);
	tmp = 0.0;
	if (y_45_scale_m <= 1.25e+87)
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * (hypot((a * t_1), (b * t_2)) * sqrt(2.0));
	else
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * t_2), (t_1 * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 1.25e+87], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + N[(b * t$95$2), $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(y$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + N[(t$95$1 * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.24999999999999995e87

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in y-scale around 0 22.8%

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

      1. associate-*r* 22.8%

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

      2. distribute-lft-out 22.8%

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

   7. Simplified 22.9%

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

      1. pow1/2 22.9%

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

      2. *-commutative 22.9%

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

      3. unpow-prod-down 22.9%

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

   9. Applied egg-rr 23.5%

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

   if 1.24999999999999995e87 < y-scale

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in x-scale around 0 55.9%

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

   7. Simplified 58.5%

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

      1. pow1/2 58.5%

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

      2. *-commutative 58.5%

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

      3. unpow-prod-down 58.4%

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

      4. pow1/2 58.4%

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

      5. unpow2 58.4%

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

      6. unpow2 58.4%

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

      7. hypot-define 56.9%

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

      8. pow1/2 56.9%

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

   9. Applied egg-rr 56.9%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.25 \cdot 10^{+87}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.2% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ \mathbf{if}\;y-scale\_m \leq 7.8 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a \cdot t\_1, \pi \cdot \left(b \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin t\_0, t\_1 \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))) (t_1 (cos t_0)))
   (if (<= y-scale_m 7.8e+86)
     (*
      (sqrt 2.0)
      (*
       (* 0.25 (* x-scale_m (sqrt 8.0)))
       (hypot (* a t_1) (* PI (* b (* 0.005555555555555556 angle))))))
     (*
      0.25
      (*
       y-scale_m
       (* (sqrt 8.0) (* (sqrt 2.0) (hypot (* a (sin t_0)) (* t_1 b)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double tmp;
	if (y_45_scale_m <= 7.8e+86) {
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((a * t_1), (((double) M_PI) * (b * (0.005555555555555556 * angle)))));
	} else {
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * sin(t_0)), (t_1 * b)))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double tmp;
	if (y_45_scale_m <= 7.8e+86) {
		tmp = Math.sqrt(2.0) * ((0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.hypot((a * t_1), (Math.PI * (b * (0.005555555555555556 * angle)))));
	} else {
		tmp = 0.25 * (y_45_scale_m * (Math.sqrt(8.0) * (Math.sqrt(2.0) * Math.hypot((a * Math.sin(t_0)), (t_1 * b)))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.cos(t_0)
	tmp = 0
	if y_45_scale_m <= 7.8e+86:
		tmp = math.sqrt(2.0) * ((0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.hypot((a * t_1), (math.pi * (b * (0.005555555555555556 * angle)))))
	else:
		tmp = 0.25 * (y_45_scale_m * (math.sqrt(8.0) * (math.sqrt(2.0) * math.hypot((a * math.sin(t_0)), (t_1 * b)))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	tmp = 0.0
	if (y_45_scale_m <= 7.8e+86)
		tmp = Float64(sqrt(2.0) * Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * hypot(Float64(a * t_1), Float64(pi * Float64(b * Float64(0.005555555555555556 * angle))))));
	else
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(sqrt(8.0) * Float64(sqrt(2.0) * hypot(Float64(a * sin(t_0)), Float64(t_1 * b))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = cos(t_0);
	tmp = 0.0;
	if (y_45_scale_m <= 7.8e+86)
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((a * t_1), (pi * (b * (0.005555555555555556 * angle)))));
	else
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * sin(t_0)), (t_1 * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 7.8e+86], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + N[(Pi * N[(b * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(y$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(t$95$1 * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.8000000000000004e86

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in y-scale around 0 22.8%

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

      1. associate-*r* 22.8%

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

      2. distribute-lft-out 22.8%

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

   7. Simplified 22.9%

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

   8. Taylor expanded in angle around 0 22.5%

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

   10. Applied egg-rr 23.0%

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

       1. unpow1 23.0%

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

       2. associate-*l* 22.9%

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

       3. associate-*r* 23.0%

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

       4. *-commutative 23.0%

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

       5. associate-*l* 23.0%

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

       6. *-commutative 23.0%

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

   12. Simplified 23.0%

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

   if 7.8000000000000004e86 < y-scale

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in x-scale around 0 55.9%

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

   7. Simplified 58.5%

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

      1. pow1/2 58.5%

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

      2. *-commutative 58.5%

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

      3. unpow-prod-down 58.4%

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

      4. pow1/2 58.4%

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

      5. unpow2 58.4%

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

      6. unpow2 58.4%

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

      7. hypot-define 56.9%

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

      8. pow1/2 56.9%

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

   9. Applied egg-rr 56.9%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 7.8 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \pi \cdot \left(b \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;y-scale\_m \leq 4.7 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a \cdot \cos t\_0, \pi \cdot \left(b \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin t\_0\right)}^{2} + {b}^{2}\right)}\right)\right)\\ \end{array} \end{array} \]

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 (* 0.005555555555555556 (* angle PI))))
   (if (<= y-scale_m 4.7e+87)
     (*
      (sqrt 2.0)
      (*
       (* 0.25 (* x-scale_m (sqrt 8.0)))
       (hypot (* a (cos t_0)) (* PI (* b (* 0.005555555555555556 angle))))))
     (*
      0.25
      (*
       y-scale_m
       (*
        (sqrt 8.0)
        (sqrt (* 2.0 (+ (pow (* a (sin t_0)) 2.0) (pow b 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 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (y_45_scale_m <= 4.7e+87) {
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((a * cos(t_0)), (((double) M_PI) * (b * (0.005555555555555556 * angle)))));
	} else {
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * sqrt((2.0 * (pow((a * sin(t_0)), 2.0) + pow(b, 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 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (y_45_scale_m <= 4.7e+87) {
		tmp = Math.sqrt(2.0) * ((0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.hypot((a * Math.cos(t_0)), (Math.PI * (b * (0.005555555555555556 * angle)))));
	} else {
		tmp = 0.25 * (y_45_scale_m * (Math.sqrt(8.0) * Math.sqrt((2.0 * (Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow(b, 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 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if y_45_scale_m <= 4.7e+87:
		tmp = math.sqrt(2.0) * ((0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.hypot((a * math.cos(t_0)), (math.pi * (b * (0.005555555555555556 * angle)))))
	else:
		tmp = 0.25 * (y_45_scale_m * (math.sqrt(8.0) * math.sqrt((2.0 * (math.pow((a * math.sin(t_0)), 2.0) + math.pow(b, 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(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (y_45_scale_m <= 4.7e+87)
		tmp = Float64(sqrt(2.0) * Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * hypot(Float64(a * cos(t_0)), Float64(pi * Float64(b * Float64(0.005555555555555556 * angle))))));
	else
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(sqrt(8.0) * sqrt(Float64(2.0 * Float64((Float64(a * sin(t_0)) ^ 2.0) + (b ^ 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 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (y_45_scale_m <= 4.7e+87)
		tmp = sqrt(2.0) * ((0.25 * (x_45_scale_m * sqrt(8.0))) * hypot((a * cos(t_0)), (pi * (b * (0.005555555555555556 * angle)))));
	else
		tmp = 0.25 * (y_45_scale_m * (sqrt(8.0) * sqrt((2.0 * (((a * sin(t_0)) ^ 2.0) + (b ^ 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[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 4.7e+87], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(Pi * N[(b * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(y$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.7000000000000004e87

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in y-scale around 0 22.8%

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

      1. associate-*r* 22.8%

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

      2. distribute-lft-out 22.8%

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

   7. Simplified 22.9%

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

   8. Taylor expanded in angle around 0 22.5%

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

   10. Applied egg-rr 23.0%

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

       1. unpow1 23.0%

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

       2. associate-*l* 22.9%

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

       3. associate-*r* 23.0%

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

       4. *-commutative 23.0%

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

       5. associate-*l* 23.0%

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

       6. *-commutative 23.0%

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

   12. Simplified 23.0%

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

   if 4.7000000000000004e87 < y-scale

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in x-scale around 0 55.9%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in angle around 0 58.6%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 4.7 \cdot 10^{+87}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \pi \cdot \left(b \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {b}^{2}\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 43.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\\ t_1 := 0.25 \cdot \left(b \cdot \left(y-scale\_m \cdot 4\right)\right)\\ t_2 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale\_m \leq 1.8 \cdot 10^{-180}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x-scale\_m \leq 1.3 \cdot 10^{-151}:\\ \;\;\;\;t\_0 \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a, t\_2 \cdot b\right)\right)\\ \mathbf{elif}\;x-scale\_m \leq 1.36 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(t\_0 \cdot \mathsf{hypot}\left(a \cdot \cos t\_2, \pi \cdot \left(b \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* 0.25 (* x-scale_m (sqrt 8.0))))
        (t_1 (* 0.25 (* b (* y-scale_m 4.0))))
        (t_2 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale_m 1.8e-180)
     t_1
     (if (<= x-scale_m 1.3e-151)
       (* t_0 (* (sqrt 2.0) (hypot a (* t_2 b))))
       (if (<= x-scale_m 1.36e-49)
         t_1
         (*
          (sqrt 2.0)
          (*
           t_0
           (hypot
            (* a (cos t_2))
            (* PI (* b (* 0.005555555555555556 angle)))))))))))

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 = 0.25 * (x_45_scale_m * sqrt(8.0));
	double t_1 = 0.25 * (b * (y_45_scale_m * 4.0));
	double t_2 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale_m <= 1.8e-180) {
		tmp = t_1;
	} else if (x_45_scale_m <= 1.3e-151) {
		tmp = t_0 * (sqrt(2.0) * hypot(a, (t_2 * b)));
	} else if (x_45_scale_m <= 1.36e-49) {
		tmp = t_1;
	} else {
		tmp = sqrt(2.0) * (t_0 * hypot((a * cos(t_2)), (((double) M_PI) * (b * (0.005555555555555556 * angle)))));
	}
	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 = 0.25 * (x_45_scale_m * Math.sqrt(8.0));
	double t_1 = 0.25 * (b * (y_45_scale_m * 4.0));
	double t_2 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale_m <= 1.8e-180) {
		tmp = t_1;
	} else if (x_45_scale_m <= 1.3e-151) {
		tmp = t_0 * (Math.sqrt(2.0) * Math.hypot(a, (t_2 * b)));
	} else if (x_45_scale_m <= 1.36e-49) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(2.0) * (t_0 * Math.hypot((a * Math.cos(t_2)), (Math.PI * (b * (0.005555555555555556 * angle)))));
	}
	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 = 0.25 * (x_45_scale_m * math.sqrt(8.0))
	t_1 = 0.25 * (b * (y_45_scale_m * 4.0))
	t_2 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale_m <= 1.8e-180:
		tmp = t_1
	elif x_45_scale_m <= 1.3e-151:
		tmp = t_0 * (math.sqrt(2.0) * math.hypot(a, (t_2 * b)))
	elif x_45_scale_m <= 1.36e-49:
		tmp = t_1
	else:
		tmp = math.sqrt(2.0) * (t_0 * math.hypot((a * math.cos(t_2)), (math.pi * (b * (0.005555555555555556 * angle)))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0)))
	t_1 = Float64(0.25 * Float64(b * Float64(y_45_scale_m * 4.0)))
	t_2 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale_m <= 1.8e-180)
		tmp = t_1;
	elseif (x_45_scale_m <= 1.3e-151)
		tmp = Float64(t_0 * Float64(sqrt(2.0) * hypot(a, Float64(t_2 * b))));
	elseif (x_45_scale_m <= 1.36e-49)
		tmp = t_1;
	else
		tmp = Float64(sqrt(2.0) * Float64(t_0 * hypot(Float64(a * cos(t_2)), Float64(pi * Float64(b * Float64(0.005555555555555556 * angle))))));
	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 = 0.25 * (x_45_scale_m * sqrt(8.0));
	t_1 = 0.25 * (b * (y_45_scale_m * 4.0));
	t_2 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale_m <= 1.8e-180)
		tmp = t_1;
	elseif (x_45_scale_m <= 1.3e-151)
		tmp = t_0 * (sqrt(2.0) * hypot(a, (t_2 * b)));
	elseif (x_45_scale_m <= 1.36e-49)
		tmp = t_1;
	else
		tmp = sqrt(2.0) * (t_0 * hypot((a * cos(t_2)), (pi * (b * (0.005555555555555556 * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.25 * N[(b * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.8e-180], t$95$1, If[LessEqual[x$45$scale$95$m, 1.3e-151], N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[a ^ 2 + N[(t$95$2 * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 1.36e-49], t$95$1, N[(N[Sqrt[2.0], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(a * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(Pi * N[(b * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 1.8e-180 or 1.3e-151 < x-scale < 1.36000000000000006e-49

   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 2.6%

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

   5. Taylor expanded in angle around 0 19.1%

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

      1. *-commutative 19.1%

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

   7. Simplified 19.1%

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

      1. sqrt-unprod 19.2%

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

      2. metadata-eval 19.2%

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

      3. metadata-eval 19.2%

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

   9. Applied egg-rr 19.2%

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

   if 1.8e-180 < x-scale < 1.3e-151

   1. Initial program 1.4%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in y-scale around 0 51.3%

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

      1. associate-*r* 51.3%

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

      2. distribute-lft-out 51.3%

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

   7. Simplified 51.3%

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

   8. Taylor expanded in angle around 0 51.8%

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

   9. Taylor expanded in angle around 0 51.8%

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

       1. pow1/2 51.8%

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

       2. *-commutative 51.8%

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

       3. unpow-prod-down 51.8%

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

       4. pow1/2 51.8%

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

       5. *-rgt-identity 51.8%

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

       6. pow2 51.8%

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

       7. unpow2 51.8%

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

       8. hypot-define 53.9%

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

       9. pow1/2 53.9%

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

   11. Applied egg-rr 53.9%

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

   if 1.36000000000000006e-49 < x-scale

   1. Initial program 1.6%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in y-scale around 0 52.1%

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

      1. associate-*r* 52.1%

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

      2. distribute-lft-out 52.1%

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

   7. Simplified 54.9%

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

   8. Taylor expanded in angle around 0 52.9%

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

   10. Applied egg-rr 56.6%

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

       1. unpow1 56.6%

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

       2. associate-*l* 56.6%

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

       3. associate-*r* 56.7%

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

       4. *-commutative 56.7%

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

       5. associate-*l* 56.7%

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

       6. *-commutative 56.7%

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

   12. Simplified 56.7%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.8 \cdot 10^{-180}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.3 \cdot 10^{-151}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a, \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\\ \mathbf{elif}\;x-scale \leq 1.36 \cdot 10^{-49}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \pi \cdot \left(b \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 1.8 \cdot 10^{-180} \lor \neg \left(x-scale\_m \leq 10^{-146}\right) \land x-scale\_m \leq 1.4 \cdot 10^{-51}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a, \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (or (<= x-scale_m 1.8e-180)
         (and (not (<= x-scale_m 1e-146)) (<= x-scale_m 1.4e-51)))
   (* 0.25 (* b (* y-scale_m 4.0)))
   (*
    (* 0.25 (* x-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 ((x_45_scale_m <= 1.8e-180) || (!(x_45_scale_m <= 1e-146) && (x_45_scale_m <= 1.4e-51))) {
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	} else {
		tmp = (0.25 * (x_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 ((x_45_scale_m <= 1.8e-180) || (!(x_45_scale_m <= 1e-146) && (x_45_scale_m <= 1.4e-51))) {
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	} else {
		tmp = (0.25 * (x_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 (x_45_scale_m <= 1.8e-180) or (not (x_45_scale_m <= 1e-146) and (x_45_scale_m <= 1.4e-51)):
		tmp = 0.25 * (b * (y_45_scale_m * 4.0))
	else:
		tmp = (0.25 * (x_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 ((x_45_scale_m <= 1.8e-180) || (!(x_45_scale_m <= 1e-146) && (x_45_scale_m <= 1.4e-51)))
		tmp = Float64(0.25 * Float64(b * Float64(y_45_scale_m * 4.0)));
	else
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * Float64(sqrt(2.0) * hypot(a, Float64(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 ((x_45_scale_m <= 1.8e-180) || (~((x_45_scale_m <= 1e-146)) && (x_45_scale_m <= 1.4e-51)))
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	else
		tmp = (0.25 * (x_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[Or[LessEqual[x$45$scale$95$m, 1.8e-180], And[N[Not[LessEqual[x$45$scale$95$m, 1e-146]], $MachinePrecision], LessEqual[x$45$scale$95$m, 1.4e-51]]], N[(0.25 * N[(b * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[a ^ 2 + N[(N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.8e-180 or 1.00000000000000003e-146 < x-scale < 1.4e-51

   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 2.6%

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

   5. Taylor expanded in angle around 0 19.1%

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

      1. *-commutative 19.1%

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

   7. Simplified 19.1%

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

      1. sqrt-unprod 19.2%

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

      2. metadata-eval 19.2%

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

      3. metadata-eval 19.2%

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

   9. Applied egg-rr 19.2%

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

   if 1.8e-180 < x-scale < 1.00000000000000003e-146 or 1.4e-51 < x-scale

   1. Initial program 1.6%

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

   3. Simplified 1.7%

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

   5. Taylor expanded in y-scale around 0 52.0%

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

      1. associate-*r* 52.0%

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

      2. distribute-lft-out 52.0%

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

   7. Simplified 54.5%

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

   8. Taylor expanded in angle around 0 52.8%

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

   9. Taylor expanded in angle around 0 52.6%

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

       1. pow1/2 52.6%

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

       2. *-commutative 52.6%

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

       3. unpow-prod-down 52.6%

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

       4. pow1/2 52.6%

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

       5. *-rgt-identity 52.6%

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

       6. pow2 52.6%

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

       7. unpow2 52.6%

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

       8. hypot-define 56.3%

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

       9. pow1/2 56.3%

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

   11. Applied egg-rr 56.3%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.8 \cdot 10^{-180} \lor \neg \left(x-scale \leq 10^{-146}\right) \land x-scale \leq 1.4 \cdot 10^{-51}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a, \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 23.0% accurate, 8.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.6 \cdot 10^{+87}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot 4\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 4.3 \cdot 10^{+209}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{elif}\;y-scale\_m \leq 5.4 \cdot 10^{+253}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale\_m \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot {a}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(b \cdot \frac{\sqrt{2}}{x-scale\_m}\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= y-scale_m 2.6e+87)
   (* 0.25 (* a (* x-scale_m 4.0)))
   (if (<= y-scale_m 4.3e+209)
     (* 0.25 (* b (* y-scale_m 4.0)))
     (if (<= y-scale_m 5.4e+253)
       (* (* 0.25 (* x-scale_m (sqrt 8.0))) (sqrt (* 2.0 (pow a 2.0))))
       (*
        0.25
        (*
         (* x-scale_m (* y-scale_m (sqrt 8.0)))
         (* b (/ (sqrt 2.0) x-scale_m))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.6e+87) {
		tmp = 0.25 * (a * (x_45_scale_m * 4.0));
	} else if (y_45_scale_m <= 4.3e+209) {
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	} else if (y_45_scale_m <= 5.4e+253) {
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * pow(a, 2.0)));
	} else {
		tmp = 0.25 * ((x_45_scale_m * (y_45_scale_m * sqrt(8.0))) * (b * (sqrt(2.0) / x_45_scale_m)));
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (y_45scale_m <= 2.6d+87) then
        tmp = 0.25d0 * (a * (x_45scale_m * 4.0d0))
    else if (y_45scale_m <= 4.3d+209) then
        tmp = 0.25d0 * (b * (y_45scale_m * 4.0d0))
    else if (y_45scale_m <= 5.4d+253) then
        tmp = (0.25d0 * (x_45scale_m * sqrt(8.0d0))) * sqrt((2.0d0 * (a ** 2.0d0)))
    else
        tmp = 0.25d0 * ((x_45scale_m * (y_45scale_m * sqrt(8.0d0))) * (b * (sqrt(2.0d0) / x_45scale_m)))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.6e+87) {
		tmp = 0.25 * (a * (x_45_scale_m * 4.0));
	} else if (y_45_scale_m <= 4.3e+209) {
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	} else if (y_45_scale_m <= 5.4e+253) {
		tmp = (0.25 * (x_45_scale_m * Math.sqrt(8.0))) * Math.sqrt((2.0 * Math.pow(a, 2.0)));
	} else {
		tmp = 0.25 * ((x_45_scale_m * (y_45_scale_m * Math.sqrt(8.0))) * (b * (Math.sqrt(2.0) / x_45_scale_m)));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 2.6e+87:
		tmp = 0.25 * (a * (x_45_scale_m * 4.0))
	elif y_45_scale_m <= 4.3e+209:
		tmp = 0.25 * (b * (y_45_scale_m * 4.0))
	elif y_45_scale_m <= 5.4e+253:
		tmp = (0.25 * (x_45_scale_m * math.sqrt(8.0))) * math.sqrt((2.0 * math.pow(a, 2.0)))
	else:
		tmp = 0.25 * ((x_45_scale_m * (y_45_scale_m * math.sqrt(8.0))) * (b * (math.sqrt(2.0) / x_45_scale_m)))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 2.6e+87)
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * 4.0)));
	elseif (y_45_scale_m <= 4.3e+209)
		tmp = Float64(0.25 * Float64(b * Float64(y_45_scale_m * 4.0)));
	elseif (y_45_scale_m <= 5.4e+253)
		tmp = Float64(Float64(0.25 * Float64(x_45_scale_m * sqrt(8.0))) * sqrt(Float64(2.0 * (a ^ 2.0))));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * sqrt(8.0))) * Float64(b * Float64(sqrt(2.0) / x_45_scale_m))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 2.6e+87)
		tmp = 0.25 * (a * (x_45_scale_m * 4.0));
	elseif (y_45_scale_m <= 4.3e+209)
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	elseif (y_45_scale_m <= 5.4e+253)
		tmp = (0.25 * (x_45_scale_m * sqrt(8.0))) * sqrt((2.0 * (a ^ 2.0)));
	else
		tmp = 0.25 * ((x_45_scale_m * (y_45_scale_m * sqrt(8.0))) * (b * (sqrt(2.0) / x_45_scale_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 2.6e+87], N[(0.25 * N[(a * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 4.3e+209], N[(0.25 * N[(b * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$45$scale$95$m, 5.4e+253], N[(N[(0.25 * N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y-scale < 2.59999999999999998e87

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in y-scale around 0 10.1%

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

   6. Taylor expanded in angle around 0 20.3%

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

      1. sqrt-unprod 20.5%

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

      2. metadata-eval 20.5%

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

      3. metadata-eval 20.5%

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

   8. Applied egg-rr 20.5%

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

   if 2.59999999999999998e87 < y-scale < 4.29999999999999988e209

   1. Initial program 0.5%

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

   3. Simplified 0.5%

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

   5. Taylor expanded in angle around 0 34.3%

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

      1. *-commutative 34.3%

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

   7. Simplified 34.3%

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

      1. sqrt-unprod 34.6%

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

      2. metadata-eval 34.6%

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

      3. metadata-eval 34.6%

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

   9. Applied egg-rr 34.6%

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

   if 4.29999999999999988e209 < y-scale < 5.40000000000000005e253

   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 0 29.0%

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

      1. associate-*r* 29.0%

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

      2. distribute-lft-out 29.0%

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

   7. Simplified 29.0%

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

   8. Taylor expanded in a around -inf 2.7%

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

      1. mul-1-neg 2.7%

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

      2. associate-*r* 2.7%

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

      3. distribute-rgt-neg-in 2.7%

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

   10. Simplified 2.7%

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

   12. Applied egg-rr 29.0%

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

   13. Taylor expanded in angle around 0 29.0%

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

   if 5.40000000000000005e253 < y-scale

   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 a around 0 1.2%

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

   6. Taylor expanded in angle around 0 9.8%

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

      1. associate-/l* 9.8%

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

   8. Simplified 9.8%

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

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.6 \cdot 10^{+87}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale \cdot 4\right)\right)\\ \mathbf{elif}\;y-scale \leq 4.3 \cdot 10^{+209}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \mathbf{elif}\;y-scale \leq 5.4 \cdot 10^{+253}:\\ \;\;\;\;\left(0.25 \cdot \left(x-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2 \cdot {a}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \left(b \cdot \frac{\sqrt{2}}{x-scale}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 22.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.8 \cdot 10^{+87}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right) \cdot \left(b \cdot \frac{\sqrt{2}}{x-scale\_m}\right)\right)\\ \end{array} \end{array} \]

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
 (if (<= y-scale_m 2.8e+87)
   (* 0.25 (* a (* x-scale_m 4.0)))
   (*
    0.25
    (*
     (* x-scale_m (* y-scale_m (sqrt 8.0)))
     (* b (/ (sqrt 2.0) x-scale_m))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.8e+87) {
		tmp = 0.25 * (a * (x_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * (y_45_scale_m * sqrt(8.0))) * (b * (sqrt(2.0) / x_45_scale_m)));
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (y_45scale_m <= 2.8d+87) then
        tmp = 0.25d0 * (a * (x_45scale_m * 4.0d0))
    else
        tmp = 0.25d0 * ((x_45scale_m * (y_45scale_m * sqrt(8.0d0))) * (b * (sqrt(2.0d0) / x_45scale_m)))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.8e+87) {
		tmp = 0.25 * (a * (x_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * (y_45_scale_m * Math.sqrt(8.0))) * (b * (Math.sqrt(2.0) / x_45_scale_m)));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 2.8e+87:
		tmp = 0.25 * (a * (x_45_scale_m * 4.0))
	else:
		tmp = 0.25 * ((x_45_scale_m * (y_45_scale_m * math.sqrt(8.0))) * (b * (math.sqrt(2.0) / x_45_scale_m)))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 2.8e+87)
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * Float64(y_45_scale_m * sqrt(8.0))) * Float64(b * Float64(sqrt(2.0) / x_45_scale_m))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 2.8e+87)
		tmp = 0.25 * (a * (x_45_scale_m * 4.0));
	else
		tmp = 0.25 * ((x_45_scale_m * (y_45_scale_m * sqrt(8.0))) * (b * (sqrt(2.0) / x_45_scale_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 2.8e+87], N[(0.25 * N[(a * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * N[(N[Sqrt[2.0], $MachinePrecision] / x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.80000000000000015e87

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in y-scale around 0 10.1%

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

   6. Taylor expanded in angle around 0 20.3%

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

      1. sqrt-unprod 20.5%

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

      2. metadata-eval 20.5%

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

      3. metadata-eval 20.5%

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

   8. Applied egg-rr 20.5%

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

   if 2.80000000000000015e87 < y-scale

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in a around 0 3.4%

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

   6. Taylor expanded in angle around 0 27.1%

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

      1. associate-/l* 27.1%

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

   8. Simplified 27.1%

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

Alternative 11: 23.0% accurate, 229.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.45 \cdot 10^{+87}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 1.45e+87)
   (* 0.25 (* a (* x-scale_m 4.0)))
   (* 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) {
	double tmp;
	if (y_45_scale_m <= 1.45e+87) {
		tmp = 0.25 * (a * (x_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * (b * (y_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 (y_45scale_m <= 1.45d+87) then
        tmp = 0.25d0 * (a * (x_45scale_m * 4.0d0))
    else
        tmp = 0.25d0 * (b * (y_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 (y_45_scale_m <= 1.45e+87) {
		tmp = 0.25 * (a * (x_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 1.45e+87:
		tmp = 0.25 * (a * (x_45_scale_m * 4.0))
	else:
		tmp = 0.25 * (b * (y_45_scale_m * 4.0))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 1.45e+87)
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(b * Float64(y_45_scale_m * 4.0)));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 1.45e+87)
		tmp = 0.25 * (a * (x_45_scale_m * 4.0));
	else
		tmp = 0.25 * (b * (y_45_scale_m * 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 1.45e+87], N[(0.25 * N[(a * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(b * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.4499999999999999e87

   1. Initial program 2.7%

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

   3. Simplified 2.7%

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

   5. Taylor expanded in y-scale around 0 10.1%

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

   6. Taylor expanded in angle around 0 20.3%

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

      1. sqrt-unprod 20.5%

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

      2. metadata-eval 20.5%

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

      3. metadata-eval 20.5%

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

   8. Applied egg-rr 20.5%

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

   if 1.4499999999999999e87 < y-scale

   1. Initial program 0.3%

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

   3. Simplified 0.3%

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

   5. Taylor expanded in angle around 0 25.6%

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

      1. *-commutative 25.6%

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

   7. Simplified 25.6%

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

      1. sqrt-unprod 25.8%

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

      2. metadata-eval 25.8%

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

      3. metadata-eval 25.8%

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

   9. Applied egg-rr 25.8%

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

Alternative 12: 17.8% accurate, 393.9× speedup

Math

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

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 (* 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) {
	return 0.25 * (a * (x_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 * (a * (x_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 * (a * (x_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 * (a * (x_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(a * Float64(x_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 * (a * (x_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[(a * N[(x$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.3%

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

3. Simplified 2.3%

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

5. Taylor expanded in y-scale around 0 8.9%

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

6. Taylor expanded in angle around 0 19.5%

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

   1. sqrt-unprod 19.7%

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

   2. metadata-eval 19.7%

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

   3. metadata-eval 19.7%

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

8. Applied egg-rr 19.7%

   \[\leadsto 0.25 \cdot \left(a \cdot \left(x-scale \cdot \color{blue}{4}\right)\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024103 
(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))))