a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 60.0%

Time: 47.5s

Alternatives: 13

Speedup: 919.0×

• 31.4% 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: 60.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ \mathbf{if}\;x-scale\_m \leq 8.4 \cdot 10^{+58}:\\ \;\;\;\;0.25 \cdot {\left(\sqrt{y-scale\_m \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(t\_2 \cdot a, b \cdot t\_1\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_1, t\_2 \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (sin t_0)))
   (if (<= x-scale_m 8.4e+58)
     (*
      0.25
      (pow
       (sqrt
        (*
         y-scale_m
         (* (sqrt 8.0) (* (sqrt 2.0) (hypot (* t_2 a) (* b t_1))))))
       2.0))
     (*
      0.25
      (*
       (* x-scale_m (sqrt 8.0))
       (* (sqrt 2.0) (hypot (* a t_1) (* t_2 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 (x_45_scale_m <= 8.4e+58) {
		tmp = 0.25 * pow(sqrt((y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((t_2 * a), (b * t_1)))))), 2.0);
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * t_1), (t_2 * 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 (x_45_scale_m <= 8.4e+58) {
		tmp = 0.25 * Math.pow(Math.sqrt((y_45_scale_m * (Math.sqrt(8.0) * (Math.sqrt(2.0) * Math.hypot((t_2 * a), (b * t_1)))))), 2.0);
	} else {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * Math.hypot((a * t_1), (t_2 * 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 x_45_scale_m <= 8.4e+58:
		tmp = 0.25 * math.pow(math.sqrt((y_45_scale_m * (math.sqrt(8.0) * (math.sqrt(2.0) * math.hypot((t_2 * a), (b * t_1)))))), 2.0)
	else:
		tmp = 0.25 * ((x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * math.hypot((a * t_1), (t_2 * 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 (x_45_scale_m <= 8.4e+58)
		tmp = Float64(0.25 * (sqrt(Float64(y_45_scale_m * Float64(sqrt(8.0) * Float64(sqrt(2.0) * hypot(Float64(t_2 * a), Float64(b * t_1)))))) ^ 2.0));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * hypot(Float64(a * t_1), Float64(t_2 * 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 (x_45_scale_m <= 8.4e+58)
		tmp = 0.25 * (sqrt((y_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((t_2 * a), (b * t_1)))))) ^ 2.0);
	else
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * t_1), (t_2 * 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[x$45$scale$95$m, 8.4e+58], N[(0.25 * N[Power[N[Sqrt[N[(y$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t$95$2 * a), $MachinePrecision] ^ 2 + N[(b * t$95$1), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$1), $MachinePrecision] ^ 2 + N[(t$95$2 * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.40000000000000048e58

   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 2.9%

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

   5. Taylor expanded in x-scale around 0 19.1%

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

      1. *-un-lft-identity 19.1%

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

      2. distribute-lft-out 19.1%

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

      3. *-commutative 19.1%

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

      4. pow-prod-down 19.7%

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

      5. pow-prod-down 19.7%

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

   7. Applied egg-rr 19.7%

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

      1. *-lft-identity 19.7%

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

      2. *-commutative 19.7%

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

      3. associate-*r* 19.7%

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

      4. *-commutative 19.7%

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

      5. associate-*r* 19.6%

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

   9. Simplified 19.6%

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

       1. add-sqr-sqrt 19.3%

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

       2. pow2 19.3%

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

   11. Applied egg-rr 19.6%

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

   if 8.40000000000000048e58 < x-scale

   1. Initial program 4.0%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in y-scale around 0 51.1%

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

      1. pow1/2 51.1%

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

      2. distribute-lft-out 51.1%

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

      3. unpow-prod-down 51.0%

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

      4. pow1/2 51.0%

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

      5. pow-prod-down 51.0%

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

      6. *-commutative 51.0%

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

      7. pow-prod-down 58.2%

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

   7. Applied egg-rr 58.2%

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

      1. unpow1/2 58.2%

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

      2. unpow2 58.2%

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

      3. *-commutative 58.2%

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

      4. unpow2 58.2%

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

      5. hypot-define 60.8%

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

   9. Simplified 60.8%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 8.4 \cdot 10^{+58}:\\ \;\;\;\;0.25 \cdot {\left(\sqrt{y-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;x-scale\_m \leq 1.45 \cdot 10^{+59}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{2} \cdot \mathsf{hypot}\left(t\_1 \cdot a, b \cdot t\_2\right)\right) \cdot \left(y-scale\_m \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, t\_1 \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (if (<= x-scale_m 1.45e+59)
     (*
      0.25
      (* (* (sqrt 2.0) (hypot (* t_1 a) (* b t_2))) (* y-scale_m (sqrt 8.0))))
     (*
      0.25
      (*
       (* x-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 = sin(t_0);
	double t_2 = cos(t_0);
	double tmp;
	if (x_45_scale_m <= 1.45e+59) {
		tmp = 0.25 * ((sqrt(2.0) * hypot((t_1 * a), (b * t_2))) * (y_45_scale_m * sqrt(8.0)));
	} else {
		tmp = 0.25 * ((x_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.sin(t_0);
	double t_2 = Math.cos(t_0);
	double tmp;
	if (x_45_scale_m <= 1.45e+59) {
		tmp = 0.25 * ((Math.sqrt(2.0) * Math.hypot((t_1 * a), (b * t_2))) * (y_45_scale_m * Math.sqrt(8.0)));
	} else {
		tmp = 0.25 * ((x_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.sin(t_0)
	t_2 = math.cos(t_0)
	tmp = 0
	if x_45_scale_m <= 1.45e+59:
		tmp = 0.25 * ((math.sqrt(2.0) * math.hypot((t_1 * a), (b * t_2))) * (y_45_scale_m * math.sqrt(8.0)))
	else:
		tmp = 0.25 * ((x_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 = sin(t_0)
	t_2 = cos(t_0)
	tmp = 0.0
	if (x_45_scale_m <= 1.45e+59)
		tmp = Float64(0.25 * Float64(Float64(sqrt(2.0) * hypot(Float64(t_1 * a), Float64(b * t_2))) * Float64(y_45_scale_m * sqrt(8.0))));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * 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 = sin(t_0);
	t_2 = cos(t_0);
	tmp = 0.0;
	if (x_45_scale_m <= 1.45e+59)
		tmp = 0.25 * ((sqrt(2.0) * hypot((t_1 * a), (b * t_2))) * (y_45_scale_m * sqrt(8.0)));
	else
		tmp = 0.25 * ((x_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[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.45e+59], N[(0.25 * N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(t$95$1 * a), $MachinePrecision] ^ 2 + N[(b * t$95$2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + N[(t$95$1 * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.44999999999999995e59

   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 2.9%

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

   5. Taylor expanded in x-scale around 0 19.1%

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

      1. pow1/2 19.1%

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

      2. distribute-lft-out 19.1%

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

      3. unpow-prod-down 19.1%

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

      4. pow1/2 19.1%

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

      5. *-commutative 19.1%

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

      6. pow-prod-down 19.6%

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

      7. pow-prod-down 19.6%

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

   7. Applied egg-rr 19.6%

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

      1. unpow1/2 19.6%

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

      2. *-commutative 19.6%

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

      3. unpow2 19.6%

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

      4. unpow2 19.6%

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

      5. hypot-define 20.0%

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

      6. *-commutative 20.0%

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

   9. Simplified 20.0%

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

   if 1.44999999999999995e59 < x-scale

   1. Initial program 4.0%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in y-scale around 0 51.1%

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

      1. pow1/2 51.1%

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

      2. distribute-lft-out 51.1%

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

      3. unpow-prod-down 51.0%

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

      4. pow1/2 51.0%

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

      5. pow-prod-down 51.0%

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

      6. *-commutative 51.0%

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

      7. pow-prod-down 58.2%

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

   7. Applied egg-rr 58.2%

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

      1. unpow1/2 58.2%

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

      2. unpow2 58.2%

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

      3. *-commutative 58.2%

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

      4. unpow2 58.2%

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

      5. hypot-define 60.8%

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

   9. Simplified 60.8%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.45 \cdot 10^{+59}:\\ \;\;\;\;0.25 \cdot \left(\left(\sqrt{2} \cdot \mathsf{hypot}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \left(y-scale \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;x-scale\_m \leq 8.4 \cdot 10^{+58}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(t\_1 \cdot a, b \cdot t\_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, t\_1 \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (if (<= x-scale_m 8.4e+58)
     (*
      0.25
      (* y-scale_m (* (* (sqrt 8.0) (sqrt 2.0)) (hypot (* t_1 a) (* b t_2)))))
     (*
      0.25
      (*
       (* x-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 = sin(t_0);
	double t_2 = cos(t_0);
	double tmp;
	if (x_45_scale_m <= 8.4e+58) {
		tmp = 0.25 * (y_45_scale_m * ((sqrt(8.0) * sqrt(2.0)) * hypot((t_1 * a), (b * t_2))));
	} else {
		tmp = 0.25 * ((x_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.sin(t_0);
	double t_2 = Math.cos(t_0);
	double tmp;
	if (x_45_scale_m <= 8.4e+58) {
		tmp = 0.25 * (y_45_scale_m * ((Math.sqrt(8.0) * Math.sqrt(2.0)) * Math.hypot((t_1 * a), (b * t_2))));
	} else {
		tmp = 0.25 * ((x_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.sin(t_0)
	t_2 = math.cos(t_0)
	tmp = 0
	if x_45_scale_m <= 8.4e+58:
		tmp = 0.25 * (y_45_scale_m * ((math.sqrt(8.0) * math.sqrt(2.0)) * math.hypot((t_1 * a), (b * t_2))))
	else:
		tmp = 0.25 * ((x_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 = sin(t_0)
	t_2 = cos(t_0)
	tmp = 0.0
	if (x_45_scale_m <= 8.4e+58)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(Float64(sqrt(8.0) * sqrt(2.0)) * hypot(Float64(t_1 * a), Float64(b * t_2)))));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * 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 = sin(t_0);
	t_2 = cos(t_0);
	tmp = 0.0;
	if (x_45_scale_m <= 8.4e+58)
		tmp = 0.25 * (y_45_scale_m * ((sqrt(8.0) * sqrt(2.0)) * hypot((t_1 * a), (b * t_2))));
	else
		tmp = 0.25 * ((x_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[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 8.4e+58], N[(0.25 * N[(y$45$scale$95$m * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 * a), $MachinePrecision] ^ 2 + N[(b * t$95$2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + N[(t$95$1 * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.40000000000000048e58

   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 2.9%

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

   5. Taylor expanded in x-scale around 0 19.1%

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

      1. *-un-lft-identity 19.1%

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

      2. distribute-lft-out 19.1%

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

      3. *-commutative 19.1%

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

      4. pow-prod-down 19.7%

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

      5. pow-prod-down 19.7%

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

   7. Applied egg-rr 19.7%

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

      1. *-lft-identity 19.7%

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

      2. *-commutative 19.7%

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

      3. associate-*r* 19.7%

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

      4. *-commutative 19.7%

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

      5. associate-*r* 19.6%

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

   9. Simplified 19.6%

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

   10. Taylor expanded in y-scale around 0 19.0%

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

       1. associate-*l* 19.0%

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

       2. *-commutative 19.0%

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

       3. *-commutative 19.0%

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

       4. unpow2 19.0%

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

       5. unpow2 19.0%

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

       6. swap-sqr 19.6%

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

       7. unpow2 19.6%

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

   12. Simplified 20.0%

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

   if 8.40000000000000048e58 < x-scale

   1. Initial program 4.0%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in y-scale around 0 51.1%

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

      1. pow1/2 51.1%

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

      2. distribute-lft-out 51.1%

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

      3. unpow-prod-down 51.0%

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

      4. pow1/2 51.0%

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

      5. pow-prod-down 51.0%

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

      6. *-commutative 51.0%

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

      7. pow-prod-down 58.2%

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

   7. Applied egg-rr 58.2%

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

      1. unpow1/2 58.2%

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

      2. unpow2 58.2%

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

      3. *-commutative 58.2%

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

      4. unpow2 58.2%

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

      5. hypot-define 60.8%

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

   9. Simplified 60.8%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 8.4 \cdot 10^{+58}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;x-scale\_m \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(t\_1 \cdot a, b \cdot t\_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale\_m \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot t\_2, t\_1 \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

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 (sin t_0))
        (t_2 (cos t_0)))
   (if (<= x-scale_m 8.5e+58)
     (*
      0.25
      (* y-scale_m (* (* (sqrt 8.0) (sqrt 2.0)) (hypot (* t_1 a) (* b t_2)))))
     (*
      0.25
      (*
       x-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 = sin(t_0);
	double t_2 = cos(t_0);
	double tmp;
	if (x_45_scale_m <= 8.5e+58) {
		tmp = 0.25 * (y_45_scale_m * ((sqrt(8.0) * sqrt(2.0)) * hypot((t_1 * a), (b * t_2))));
	} else {
		tmp = 0.25 * (x_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.sin(t_0);
	double t_2 = Math.cos(t_0);
	double tmp;
	if (x_45_scale_m <= 8.5e+58) {
		tmp = 0.25 * (y_45_scale_m * ((Math.sqrt(8.0) * Math.sqrt(2.0)) * Math.hypot((t_1 * a), (b * t_2))));
	} else {
		tmp = 0.25 * (x_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.sin(t_0)
	t_2 = math.cos(t_0)
	tmp = 0
	if x_45_scale_m <= 8.5e+58:
		tmp = 0.25 * (y_45_scale_m * ((math.sqrt(8.0) * math.sqrt(2.0)) * math.hypot((t_1 * a), (b * t_2))))
	else:
		tmp = 0.25 * (x_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 = sin(t_0)
	t_2 = cos(t_0)
	tmp = 0.0
	if (x_45_scale_m <= 8.5e+58)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(Float64(sqrt(8.0) * sqrt(2.0)) * hypot(Float64(t_1 * a), Float64(b * t_2)))));
	else
		tmp = Float64(0.25 * Float64(x_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 = sin(t_0);
	t_2 = cos(t_0);
	tmp = 0.0;
	if (x_45_scale_m <= 8.5e+58)
		tmp = 0.25 * (y_45_scale_m * ((sqrt(8.0) * sqrt(2.0)) * hypot((t_1 * a), (b * t_2))));
	else
		tmp = 0.25 * (x_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[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 8.5e+58], N[(0.25 * N[(y$45$scale$95$m * N[(N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 * a), $MachinePrecision] ^ 2 + N[(b * t$95$2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(x$45$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 x-scale < 8.50000000000000015e58

   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 2.9%

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

   5. Taylor expanded in x-scale around 0 19.1%

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

      1. *-un-lft-identity 19.1%

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

      2. distribute-lft-out 19.1%

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

      3. *-commutative 19.1%

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

      4. pow-prod-down 19.7%

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

      5. pow-prod-down 19.7%

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

   7. Applied egg-rr 19.7%

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

      1. *-lft-identity 19.7%

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

      2. *-commutative 19.7%

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

      3. associate-*r* 19.7%

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

      4. *-commutative 19.7%

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

      5. associate-*r* 19.6%

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

   9. Simplified 19.6%

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

   10. Taylor expanded in y-scale around 0 19.0%

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

       1. associate-*l* 19.0%

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

       2. *-commutative 19.0%

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

       3. *-commutative 19.0%

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

       4. unpow2 19.0%

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

       5. unpow2 19.0%

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

       6. swap-sqr 19.6%

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

       7. unpow2 19.6%

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

   12. Simplified 20.0%

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

   if 8.50000000000000015e58 < x-scale

   1. Initial program 4.0%

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

   3. Simplified 4.1%

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

   5. Taylor expanded in y-scale around 0 51.1%

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

      1. add-cube-cbrt 51.0%

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

      2. pow3 51.0%

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

   7. Applied egg-rr 58.2%

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

   8. Taylor expanded in x-scale around 0 51.0%

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

      1. associate-*l* 51.0%

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

      2. *-commutative 51.0%

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

      3. unpow2 51.0%

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

      4. unpow2 51.0%

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

      5. swap-sqr 51.0%

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

      6. *-commutative 51.0%

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

      7. unpow2 51.0%

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

      8. unpow2 51.0%

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

      9. swap-sqr 58.2%

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

   10. Simplified 60.8%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 8.5 \cdot 10^{+58}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(\left(\sqrt{8} \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a, b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.9% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;y-scale\_m \leq 2.6 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(x-scale\_m \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos t\_0, \sin t\_0 \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {b}^{2}\right)}\right)\\ \end{array} \end{array} \]

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))))
   (if (<= y-scale_m 2.6e-27)
     (*
      0.25
      (*
       x-scale_m
       (* (sqrt 8.0) (* (sqrt 2.0) (hypot (* a (cos t_0)) (* (sin t_0) b))))))
     (*
      0.25
      (*
       (* y-scale_m (sqrt 8.0))
       (sqrt
        (*
         2.0
         (+
          (pow (* a (sin (* PI (* 0.005555555555555556 angle)))) 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 <= 2.6e-27) {
		tmp = 0.25 * (x_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * cos(t_0)), (sin(t_0) * b)))));
	} else {
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * sqrt((2.0 * (pow((a * sin((((double) M_PI) * (0.005555555555555556 * angle)))), 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 <= 2.6e-27) {
		tmp = 0.25 * (x_45_scale_m * (Math.sqrt(8.0) * (Math.sqrt(2.0) * Math.hypot((a * Math.cos(t_0)), (Math.sin(t_0) * b)))));
	} else {
		tmp = 0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * Math.sqrt((2.0 * (Math.pow((a * Math.sin((Math.PI * (0.005555555555555556 * angle)))), 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 <= 2.6e-27:
		tmp = 0.25 * (x_45_scale_m * (math.sqrt(8.0) * (math.sqrt(2.0) * math.hypot((a * math.cos(t_0)), (math.sin(t_0) * b)))))
	else:
		tmp = 0.25 * ((y_45_scale_m * math.sqrt(8.0)) * math.sqrt((2.0 * (math.pow((a * math.sin((math.pi * (0.005555555555555556 * angle)))), 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 <= 2.6e-27)
		tmp = Float64(0.25 * Float64(x_45_scale_m * Float64(sqrt(8.0) * Float64(sqrt(2.0) * hypot(Float64(a * cos(t_0)), Float64(sin(t_0) * b))))));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * sqrt(Float64(2.0 * Float64((Float64(a * sin(Float64(pi * Float64(0.005555555555555556 * angle)))) ^ 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 <= 2.6e-27)
		tmp = 0.25 * (x_45_scale_m * (sqrt(8.0) * (sqrt(2.0) * hypot((a * cos(t_0)), (sin(t_0) * b)))));
	else
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * sqrt((2.0 * (((a * sin((pi * (0.005555555555555556 * angle)))) ^ 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, 2.6e-27], N[(0.25 * N[(x$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[Sin[t$95$0], $MachinePrecision] * b), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[Power[N[(a * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.60000000000000017e-27

   1. Initial program 3.5%

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

   3. Simplified 3.5%

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

   5. Taylor expanded in y-scale around 0 17.2%

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

      1. add-cube-cbrt 17.2%

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

      2. pow3 17.2%

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

   7. Applied egg-rr 18.8%

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

   8. Taylor expanded in x-scale around 0 17.2%

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

      1. associate-*l* 17.2%

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

      2. *-commutative 17.2%

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

      3. unpow2 17.2%

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

      4. unpow2 17.2%

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

      5. swap-sqr 17.2%

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

      6. *-commutative 17.2%

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

      7. unpow2 17.2%

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

      8. unpow2 17.2%

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

      9. swap-sqr 18.8%

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

   10. Simplified 19.4%

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

   if 2.60000000000000017e-27 < y-scale

   1. Initial program 0.1%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in x-scale around 0 55.6%

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

      1. *-un-lft-identity 55.6%

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

      2. distribute-lft-out 55.6%

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

      3. *-commutative 55.6%

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

      4. pow-prod-down 54.4%

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

      5. pow-prod-down 54.4%

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

   7. Applied egg-rr 54.4%

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

      1. *-lft-identity 54.4%

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

      2. *-commutative 54.4%

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

      3. associate-*r* 54.4%

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

      4. *-commutative 54.4%

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

      5. associate-*r* 54.5%

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

   9. Simplified 54.5%

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

   10. Taylor expanded in angle around 0 55.1%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.6 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {b}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.2 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot e^{\log 8 \cdot 0.5}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {b}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 2.2e-27)
   (* 0.25 (* (* x-scale_m (exp (* (log 8.0) 0.5))) (* (sqrt 2.0) a)))
   (*
    0.25
    (*
     (* y-scale_m (sqrt 8.0))
     (sqrt
      (*
       2.0
       (+
        (pow (* a (sin (* PI (* 0.005555555555555556 angle)))) 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 tmp;
	if (y_45_scale_m <= 2.2e-27) {
		tmp = 0.25 * ((x_45_scale_m * exp((log(8.0) * 0.5))) * (sqrt(2.0) * a));
	} else {
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * sqrt((2.0 * (pow((a * sin((((double) M_PI) * (0.005555555555555556 * angle)))), 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 tmp;
	if (y_45_scale_m <= 2.2e-27) {
		tmp = 0.25 * ((x_45_scale_m * Math.exp((Math.log(8.0) * 0.5))) * (Math.sqrt(2.0) * a));
	} else {
		tmp = 0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * Math.sqrt((2.0 * (Math.pow((a * Math.sin((Math.PI * (0.005555555555555556 * angle)))), 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):
	tmp = 0
	if y_45_scale_m <= 2.2e-27:
		tmp = 0.25 * ((x_45_scale_m * math.exp((math.log(8.0) * 0.5))) * (math.sqrt(2.0) * a))
	else:
		tmp = 0.25 * ((y_45_scale_m * math.sqrt(8.0)) * math.sqrt((2.0 * (math.pow((a * math.sin((math.pi * (0.005555555555555556 * angle)))), 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)
	tmp = 0.0
	if (y_45_scale_m <= 2.2e-27)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * exp(Float64(log(8.0) * 0.5))) * Float64(sqrt(2.0) * a)));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * sqrt(Float64(2.0 * Float64((Float64(a * sin(Float64(pi * Float64(0.005555555555555556 * angle)))) ^ 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)
	tmp = 0.0;
	if (y_45_scale_m <= 2.2e-27)
		tmp = 0.25 * ((x_45_scale_m * exp((log(8.0) * 0.5))) * (sqrt(2.0) * a));
	else
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * sqrt((2.0 * (((a * sin((pi * (0.005555555555555556 * angle)))) ^ 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_] := If[LessEqual[y$45$scale$95$m, 2.2e-27], N[(0.25 * N[(N[(x$45$scale$95$m * N[Exp[N[(N[Log[8.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[Power[N[(a * N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.19999999999999987e-27

   1. Initial program 3.5%

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

   3. Simplified 3.5%

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

   5. Taylor expanded in y-scale around 0 17.2%

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

      1. pow1/2 17.2%

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

      2. pow-to-exp 17.2%

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

   7. Applied egg-rr 17.2%

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

   8. Taylor expanded in angle around 0 18.0%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot e^{\log 8 \cdot 0.5}\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)}\right) \]

   if 2.19999999999999987e-27 < y-scale

   1. Initial program 0.1%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in x-scale around 0 55.6%

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

      1. *-un-lft-identity 55.6%

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

      2. distribute-lft-out 55.6%

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

      3. *-commutative 55.6%

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

      4. pow-prod-down 54.4%

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

      5. pow-prod-down 54.4%

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

   7. Applied egg-rr 54.4%

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

      1. *-lft-identity 54.4%

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

      2. *-commutative 54.4%

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

      3. associate-*r* 54.4%

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

      4. *-commutative 54.4%

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

      5. associate-*r* 54.5%

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

   9. Simplified 54.5%

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

   10. Taylor expanded in angle around 0 55.1%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.2 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot e^{\log 8 \cdot 0.5}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(a \cdot \sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {b}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.8% accurate, 5.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 1.8 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot e^{\log 8 \cdot 0.5}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale\_m \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)}^{2} + {b}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 1.8e-27)
   (* 0.25 (* (* x-scale_m (exp (* (log 8.0) 0.5))) (* (sqrt 2.0) a)))
   (*
    0.25
    (*
     (* y-scale_m (sqrt 8.0))
     (sqrt
      (*
       2.0
       (+
        (pow (* (sin (* 0.005555555555555556 (* angle PI))) a) 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 tmp;
	if (y_45_scale_m <= 1.8e-27) {
		tmp = 0.25 * ((x_45_scale_m * exp((log(8.0) * 0.5))) * (sqrt(2.0) * a));
	} else {
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * sqrt((2.0 * (pow((sin((0.005555555555555556 * (angle * ((double) M_PI)))) * a), 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 tmp;
	if (y_45_scale_m <= 1.8e-27) {
		tmp = 0.25 * ((x_45_scale_m * Math.exp((Math.log(8.0) * 0.5))) * (Math.sqrt(2.0) * a));
	} else {
		tmp = 0.25 * ((y_45_scale_m * Math.sqrt(8.0)) * Math.sqrt((2.0 * (Math.pow((Math.sin((0.005555555555555556 * (angle * Math.PI))) * a), 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):
	tmp = 0
	if y_45_scale_m <= 1.8e-27:
		tmp = 0.25 * ((x_45_scale_m * math.exp((math.log(8.0) * 0.5))) * (math.sqrt(2.0) * a))
	else:
		tmp = 0.25 * ((y_45_scale_m * math.sqrt(8.0)) * math.sqrt((2.0 * (math.pow((math.sin((0.005555555555555556 * (angle * math.pi))) * a), 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)
	tmp = 0.0
	if (y_45_scale_m <= 1.8e-27)
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * exp(Float64(log(8.0) * 0.5))) * Float64(sqrt(2.0) * a)));
	else
		tmp = Float64(0.25 * Float64(Float64(y_45_scale_m * sqrt(8.0)) * sqrt(Float64(2.0 * Float64((Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * a) ^ 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)
	tmp = 0.0;
	if (y_45_scale_m <= 1.8e-27)
		tmp = 0.25 * ((x_45_scale_m * exp((log(8.0) * 0.5))) * (sqrt(2.0) * a));
	else
		tmp = 0.25 * ((y_45_scale_m * sqrt(8.0)) * sqrt((2.0 * (((sin((0.005555555555555556 * (angle * pi))) * a) ^ 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_] := If[LessEqual[y$45$scale$95$m, 1.8e-27], N[(0.25 * N[(N[(x$45$scale$95$m * N[Exp[N[(N[Log[8.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(y$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[Power[N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.7999999999999999e-27

   1. Initial program 3.5%

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

   3. Simplified 3.5%

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

   5. Taylor expanded in y-scale around 0 17.2%

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

      1. pow1/2 17.2%

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

      2. pow-to-exp 17.2%

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

   7. Applied egg-rr 17.2%

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

   8. Taylor expanded in angle around 0 18.0%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot e^{\log 8 \cdot 0.5}\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)}\right) \]

   if 1.7999999999999999e-27 < y-scale

   1. Initial program 0.1%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in x-scale around 0 55.6%

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

      1. add-cbrt-cube 54.0%

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

      2. pow1/3 53.8%

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

      3. pow3 53.8%

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

      4. unpow-prod-down 53.8%

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

      5. metadata-eval 53.8%

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

      6. *-commutative 53.8%

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

      7. pow-prod-down 52.4%

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

   7. Applied egg-rr 52.4%

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

   8. Taylor expanded in angle around 0 53.2%

      \[\leadsto 0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{{\left(8 \cdot {\left({\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333} + 2 \cdot \left({b}^{2} \cdot {\color{blue}{1}}^{2}\right)}\right) \]

   9. Taylor expanded in y-scale around 0 56.4%

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

       1. distribute-lft-out 56.4%

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

       2. *-commutative 56.4%

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

       3. unpow2 56.4%

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

       4. unpow2 56.4%

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

       5. swap-sqr 55.2%

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

       6. unpow2 55.2%

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

       7. *-commutative 55.2%

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

   11. Simplified 55.2%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 1.8 \cdot 10^{-27}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot e^{\log 8 \cdot 0.5}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(y-scale \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left({\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot a\right)}^{2} + {b}^{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.2% accurate, 8.7× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+34}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot e^{\log 8 \cdot 0.5}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 9.5e+34)
   (* y-scale_m b)
   (* 0.25 (* (* x-scale_m (exp (* (log 8.0) 0.5))) (* (sqrt 2.0) a)))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 9.5e+34) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * ((x_45_scale_m * exp((log(8.0) * 0.5))) * (sqrt(2.0) * a));
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 9.5d+34) then
        tmp = y_45scale_m * b
    else
        tmp = 0.25d0 * ((x_45scale_m * exp((log(8.0d0) * 0.5d0))) * (sqrt(2.0d0) * a))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 9.5e+34) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * ((x_45_scale_m * Math.exp((Math.log(8.0) * 0.5))) * (Math.sqrt(2.0) * a));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 9.5e+34:
		tmp = y_45_scale_m * b
	else:
		tmp = 0.25 * ((x_45_scale_m * math.exp((math.log(8.0) * 0.5))) * (math.sqrt(2.0) * a))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 9.5e+34)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * exp(Float64(log(8.0) * 0.5))) * Float64(sqrt(2.0) * a)));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 9.5e+34)
		tmp = y_45_scale_m * b;
	else
		tmp = 0.25 * ((x_45_scale_m * exp((log(8.0) * 0.5))) * (sqrt(2.0) * a));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 9.5e+34], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Exp[N[(N[Log[8.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.4999999999999999e34

   1. Initial program 3.3%

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

   3. Simplified 3.3%

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

   5. Taylor expanded in angle around 0 20.1%

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

      1. pow1 20.1%

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

      2. associate-*r* 20.1%

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

      3. sqrt-unprod 20.3%

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

      4. metadata-eval 20.3%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)}^{1} \]

      5. metadata-eval 20.3%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)}^{1} \]

   7. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{{\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot 4\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 20.3%

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

      2. *-commutative 20.3%

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

   9. Simplified 20.3%

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

   10. Taylor expanded in b around 0 20.3%

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

       1. *-commutative 20.3%

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

   12. Simplified 20.3%

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

   if 9.4999999999999999e34 < a

   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 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 30.9%

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

      1. pow1/2 30.9%

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

      2. pow-to-exp 30.9%

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

   7. Applied egg-rr 30.9%

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

   8. Taylor expanded in angle around 0 26.1%

      \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot e^{\log 8 \cdot 0.5}\right) \cdot \color{blue}{\left(a \cdot \sqrt{2}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.5 \cdot 10^{+34}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot e^{\log 8 \cdot 0.5}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 26.3% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 6.1 \cdot 10^{+32}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 6.1e+32)
   (* y-scale_m b)
   (* 0.25 (* (* x-scale_m (sqrt 8.0)) (* (sqrt 2.0) a)))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 6.1e+32) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 6.1d+32) then
        tmp = y_45scale_m * b
    else
        tmp = 0.25d0 * ((x_45scale_m * sqrt(8.0d0)) * (sqrt(2.0d0) * a))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 6.1e+32) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * a));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 6.1e+32:
		tmp = y_45_scale_m * b
	else:
		tmp = 0.25 * ((x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * a))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 6.1e+32)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * a)));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 6.1e+32)
		tmp = y_45_scale_m * b;
	else
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 6.1e+32], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 6.10000000000000027e32

   1. Initial program 3.3%

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

   3. Simplified 3.3%

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

   5. Taylor expanded in angle around 0 20.1%

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

      1. pow1 20.1%

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

      2. associate-*r* 20.1%

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

      3. sqrt-unprod 20.3%

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

      4. metadata-eval 20.3%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)}^{1} \]

      5. metadata-eval 20.3%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)}^{1} \]

   7. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{{\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot 4\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 20.3%

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

      2. *-commutative 20.3%

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

   9. Simplified 20.3%

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

   10. Taylor expanded in b around 0 20.3%

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

       1. *-commutative 20.3%

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

   12. Simplified 20.3%

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

   if 6.10000000000000027e32 < a

   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 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 30.9%

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

   6. Taylor expanded in angle around 0 26.0%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6.1 \cdot 10^{+32}:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.2% accurate, 12.9× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{+36}:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 1.8e+36)
   (* y-scale_m b)
   (* 0.25 (* a (* x-scale_m (* (sqrt 8.0) (sqrt 2.0)))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.8e+36) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (a <= 1.8d+36) then
        tmp = y_45scale_m * b
    else
        tmp = 0.25d0 * (a * (x_45scale_m * (sqrt(8.0d0) * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.8e+36) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (Math.sqrt(8.0) * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 1.8e+36:
		tmp = y_45_scale_m * b
	else:
		tmp = 0.25 * (a * (x_45_scale_m * (math.sqrt(8.0) * math.sqrt(2.0))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 1.8e+36)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * Float64(sqrt(8.0) * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (a <= 1.8e+36)
		tmp = y_45_scale_m * b;
	else
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 1.8e+36], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[(a * N[(x$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.7999999999999999e36

   1. Initial program 3.3%

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

   3. Simplified 3.3%

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

   5. Taylor expanded in angle around 0 20.1%

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

      1. pow1 20.1%

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

      2. associate-*r* 20.1%

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

      3. sqrt-unprod 20.3%

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

      4. metadata-eval 20.3%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)}^{1} \]

      5. metadata-eval 20.3%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)}^{1} \]

   7. Applied egg-rr 20.3%

      \[\leadsto \color{blue}{{\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot 4\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 20.3%

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

      2. *-commutative 20.3%

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

   9. Simplified 20.3%

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

   10. Taylor expanded in b around 0 20.3%

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

       1. *-commutative 20.3%

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

   12. Simplified 20.3%

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

   if 1.7999999999999999e36 < a

   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 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 13.9%

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

   6. Taylor expanded in angle around 0 26.0%

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

      1. *-commutative 26.0%

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

   8. Simplified 26.0%

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

Alternative 11: 19.4% accurate, 13.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 960000000:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y-scale\_m \cdot 4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 960000000.0)
   (* y-scale_m b)
   (* 0.25 (* b (log1p (expm1 (* 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 (a <= 960000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (b * log1p(expm1((y_45_scale_m * 4.0))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 960000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = 0.25 * (b * Math.log1p(Math.expm1((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 a <= 960000000.0:
		tmp = y_45_scale_m * b
	else:
		tmp = 0.25 * (b * math.log1p(math.expm1((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 (a <= 960000000.0)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(0.25 * Float64(b * log1p(expm1(Float64(y_45_scale_m * 4.0)))));
	end
	return tmp
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 960000000.0], N[(y$45$scale$95$m * b), $MachinePrecision], N[(0.25 * N[(b * N[Log[1 + N[(Exp[N[(y$45$scale$95$m * 4.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 9.6e8

   1. Initial program 2.8%

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

   3. Simplified 2.9%

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

   5. Taylor expanded in angle around 0 20.6%

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

      1. pow1 20.6%

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

      2. associate-*r* 20.6%

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

      3. sqrt-unprod 20.8%

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

      4. metadata-eval 20.8%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)}^{1} \]

      5. metadata-eval 20.8%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)}^{1} \]

   7. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{{\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot 4\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 20.8%

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

      2. *-commutative 20.8%

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

   9. Simplified 20.8%

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

   10. Taylor expanded in b around 0 20.8%

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

       1. *-commutative 20.8%

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

   12. Simplified 20.8%

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

   if 9.6e8 < a

   1. Initial program 2.0%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in angle around 0 13.4%

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

      1. log1p-expm1-u 25.1%

         \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)}\right) \]

      2. sqrt-unprod 25.1%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right)\right)\right) \]

      3. metadata-eval 25.1%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right) \]

      4. metadata-eval 25.1%

         \[\leadsto 0.25 \cdot \left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot \color{blue}{4}\right)\right)\right) \]

   7. Applied egg-rr 25.1%

      \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot 4\right)\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 19.0% accurate, 13.3× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3450000000:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y-scale\_m \cdot b\right)\right)\\ \end{array} \end{array} \]

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 (<= a 3450000000.0) (* y-scale_m b) (log1p (expm1 (* y-scale_m b)))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 3450000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = log1p(expm1((y_45_scale_m * b)));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 3450000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = Math.log1p(Math.expm1((y_45_scale_m * b)));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if a <= 3450000000.0:
		tmp = y_45_scale_m * b
	else:
		tmp = math.log1p(math.expm1((y_45_scale_m * b)))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 3450000000.0)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = log1p(expm1(Float64(y_45_scale_m * b)));
	end
	return tmp
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 3450000000.0], N[(y$45$scale$95$m * b), $MachinePrecision], N[Log[1 + N[(Exp[N[(y$45$scale$95$m * b), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.45e9

   1. Initial program 2.8%

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

   3. Simplified 2.9%

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

   5. Taylor expanded in angle around 0 20.6%

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

      1. pow1 20.6%

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

      2. associate-*r* 20.6%

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

      3. sqrt-unprod 20.8%

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

      4. metadata-eval 20.8%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)}^{1} \]

      5. metadata-eval 20.8%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)}^{1} \]

   7. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{{\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot 4\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 20.8%

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

      2. *-commutative 20.8%

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

   9. Simplified 20.8%

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

   10. Taylor expanded in b around 0 20.8%

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

       1. *-commutative 20.8%

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

   12. Simplified 20.8%

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

   if 3.45e9 < a

   1. Initial program 2.0%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in angle around 0 13.4%

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

      1. pow1 13.4%

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

      2. associate-*r* 13.4%

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

      3. sqrt-unprod 13.4%

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

      4. metadata-eval 13.4%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)}^{1} \]

      5. metadata-eval 13.4%

         \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)}^{1} \]

   7. Applied egg-rr 13.4%

      \[\leadsto \color{blue}{{\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot 4\right)\right)}^{1}} \]
   8. Step-by-step derivation

      1. unpow1 13.4%

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

      2. *-commutative 13.4%

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

   9. Simplified 13.4%

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

   10. Taylor expanded in b around 0 13.4%

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

       1. *-commutative 13.4%

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

   12. Simplified 13.4%

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

       1. log1p-expm1-u 25.2%

          \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot b\right)\right)} \]

   14. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y-scale \cdot b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 17.6% accurate, 919.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 2.7%

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

3. Simplified 3.1%

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

5. Taylor expanded in angle around 0 19.0%

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

   1. pow1 19.0%

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

   2. associate-*r* 19.0%

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

   3. sqrt-unprod 19.2%

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

   4. metadata-eval 19.2%

      \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)}^{1} \]

   5. metadata-eval 19.2%

      \[\leadsto {\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)}^{1} \]

7. Applied egg-rr 19.2%

   \[\leadsto \color{blue}{{\left(\left(0.25 \cdot b\right) \cdot \left(y-scale \cdot 4\right)\right)}^{1}} \]
8. Step-by-step derivation

   1. unpow1 19.2%

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

   2. *-commutative 19.2%

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

9. Simplified 19.2%

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

10. Taylor expanded in b around 0 19.2%

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

    1. *-commutative 19.2%

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

12. Simplified 19.2%

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

Reproduce

herbie shell --seed 2024182 
(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))))