a from scale-rotated-ellipse

Percentage Accurate: 2.9% → 49.4%

Time: 1.5min

Alternatives: 9

Speedup: N/A×

• 30.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 49.4% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)\\ \mathbf{if}\;x-scale\_m \leq 6.6 \cdot 10^{+15}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot {8}^{0.5}\right) \cdot {\left(\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \left(t\_1 \cdot t\_1\right), 2 \cdot \left(\left(b\_m \cdot b\_m\right) \cdot \left(t\_0 \cdot t\_0\right)\right)\right)\right)}^{0.5}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI))))
        (t_1 (sin (fma 0.005555555555555556 (* angle PI) (/ PI 2.0)))))
   (if (<= x-scale_m 6.6e+15)
     (* 0.25 (* b_m (* y-scale_m 4.0)))
     (*
      0.25
      (*
       (* x-scale_m (pow 8.0 0.5))
       (pow
        (fma 2.0 (* (* a a) (* t_1 t_1)) (* 2.0 (* (* b_m b_m) (* t_0 t_0))))
        0.5))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (((double) M_PI) / 2.0)));
	double tmp;
	if (x_45_scale_m <= 6.6e+15) {
		tmp = 0.25 * (b_m * (y_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * pow(8.0, 0.5)) * pow(fma(2.0, ((a * a) * (t_1 * t_1)), (2.0 * ((b_m * b_m) * (t_0 * t_0)))), 0.5));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(pi / 2.0)))
	tmp = 0.0
	if (x_45_scale_m <= 6.6e+15)
		tmp = Float64(0.25 * Float64(b_m * Float64(y_45_scale_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * (8.0 ^ 0.5)) * (fma(2.0, Float64(Float64(a * a) * Float64(t_1 * t_1)), Float64(2.0 * Float64(Float64(b_m * b_m) * Float64(t_0 * t_0)))) ^ 0.5)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 6.6e+15], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Power[8.0, 0.5], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * N[(N[(a * a), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(b$95$m * b$95$m), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 6.6e15

   1. Initial program 2.4%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 23.3%

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

   if 6.6e15 < x-scale

   1. Initial program 2.2%

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

   3. Taylor expanded in y-scale around 0

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

   5. Applied rewrites 64.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(x-scale \cdot {8}^{0.5}\right) \cdot {\left(\mathsf{fma}\left(2, \left(a \cdot a\right) \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)\right), 2 \cdot \left(\left(b \cdot b\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\right)}^{0.5}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 49.4% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale\_m \leq 6.6 \cdot 10^{+15}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \sqrt{\mathsf{fma}\left(2, {a}^{2} \cdot {\cos t\_0}^{2}, 2 \cdot \left({b\_m}^{2} \cdot {\sin t\_0}^{2}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale_m 6.6e+15)
     (* 0.25 (* b_m (* y-scale_m 4.0)))
     (*
      0.25
      (*
       (* x-scale_m (sqrt 8.0))
       (sqrt
        (fma
         2.0
         (* (pow a 2.0) (pow (cos t_0) 2.0))
         (* 2.0 (* (pow b_m 2.0) (pow (sin t_0) 2.0))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale_m <= 6.6e+15) {
		tmp = 0.25 * (b_m * (y_45_scale_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * sqrt(fma(2.0, (pow(a, 2.0) * pow(cos(t_0), 2.0)), (2.0 * (pow(b_m, 2.0) * pow(sin(t_0), 2.0))))));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale_m <= 6.6e+15)
		tmp = Float64(0.25 * Float64(b_m * Float64(y_45_scale_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * sqrt(fma(2.0, Float64((a ^ 2.0) * (cos(t_0) ^ 2.0)), Float64(2.0 * Float64((b_m ^ 2.0) * (sin(t_0) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 6.6e+15], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[Power[b$95$m, 2.0], $MachinePrecision] * N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 6.6e15

   1. Initial program 2.4%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 23.3%

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

   if 6.6e15 < x-scale

   1. Initial program 2.2%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 21.0%

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

   6. Taylor expanded in y-scale around 0

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

   8. Applied rewrites 64.8%

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

Alternative 3: 36.2% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := t\_1 \cdot t\_1\\ t_3 := \cos t\_0\\ \mathbf{if}\;b\_m \leq 4.4 \cdot 10^{+24}:\\ \;\;\;\;0.25 \cdot \mathsf{fma}\left(b\_m, y-scale\_m \cdot \left(t\_3 \cdot 4\right), \frac{\left(a \cdot a\right) \cdot \left(y-scale\_m \cdot \left(t\_2 \cdot {8}^{0.5}\right)\right)}{b\_m \cdot \left(t\_3 \cdot {2}^{0.5}\right)}\right)\\ \mathbf{elif}\;b\_m \leq 1.1 \cdot 10^{+30}:\\ \;\;\;\;0.25 \cdot \left(\left(b\_m \cdot \left(x-scale\_m \cdot \left(y-scale\_m \cdot {8}^{0.5}\right)\right)\right) \cdot {\left(\frac{t\_2}{y-scale\_m \cdot y-scale\_m} + \left(\frac{{t\_1}^{2}}{{y-scale\_m}^{2}} + \frac{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale\_m}^{2}}\right)\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (* t_1 t_1))
        (t_3 (cos t_0)))
   (if (<= b_m 4.4e+24)
     (*
      0.25
      (fma
       b_m
       (* y-scale_m (* t_3 4.0))
       (/
        (* (* a a) (* y-scale_m (* t_2 (pow 8.0 0.5))))
        (* b_m (* t_3 (pow 2.0 0.5))))))
     (if (<= b_m 1.1e+30)
       (*
        0.25
        (*
         (* b_m (* x-scale_m (* y-scale_m (pow 8.0 0.5))))
         (pow
          (+
           (/ t_2 (* y-scale_m y-scale_m))
           (+
            (/ (pow t_1 2.0) (pow y-scale_m 2.0))
            (/
             (pow (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))) 2.0)
             (pow x-scale_m 2.0))))
          0.5)))
       (* 0.25 (* b_m (* y-scale_m 4.0)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = t_1 * t_1;
	double t_3 = cos(t_0);
	double tmp;
	if (b_m <= 4.4e+24) {
		tmp = 0.25 * fma(b_m, (y_45_scale_m * (t_3 * 4.0)), (((a * a) * (y_45_scale_m * (t_2 * pow(8.0, 0.5)))) / (b_m * (t_3 * pow(2.0, 0.5)))));
	} else if (b_m <= 1.1e+30) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * (y_45_scale_m * pow(8.0, 0.5)))) * pow(((t_2 / (y_45_scale_m * y_45_scale_m)) + ((pow(t_1, 2.0) / pow(y_45_scale_m, 2.0)) + (pow(sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI)))), 2.0) / pow(x_45_scale_m, 2.0)))), 0.5));
	} else {
		tmp = 0.25 * (b_m * (y_45_scale_m * 4.0));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = Float64(t_1 * t_1)
	t_3 = cos(t_0)
	tmp = 0.0
	if (b_m <= 4.4e+24)
		tmp = Float64(0.25 * fma(b_m, Float64(y_45_scale_m * Float64(t_3 * 4.0)), Float64(Float64(Float64(a * a) * Float64(y_45_scale_m * Float64(t_2 * (8.0 ^ 0.5)))) / Float64(b_m * Float64(t_3 * (2.0 ^ 0.5))))));
	elseif (b_m <= 1.1e+30)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * Float64(y_45_scale_m * (8.0 ^ 0.5)))) * (Float64(Float64(t_2 / Float64(y_45_scale_m * y_45_scale_m)) + Float64(Float64((t_1 ^ 2.0) / (y_45_scale_m ^ 2.0)) + Float64((sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi))) ^ 2.0) / (x_45_scale_m ^ 2.0)))) ^ 0.5)));
	else
		tmp = Float64(0.25 * Float64(b_m * Float64(y_45_scale_m * 4.0)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[b$95$m, 4.4e+24], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * N[(t$95$3 * 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * N[(y$45$scale$95$m * N[(t$95$2 * N[Power[8.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * N[(t$95$3 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.1e+30], N[(0.25 * N[(N[(b$95$m * N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Power[8.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(t$95$2 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.40000000000000003e24

   1. Initial program 1.8%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 20.4%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 20.3%

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

   9. Taylor expanded in a around 0

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

       1. lower-fma.f64 N/A

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

   11. Applied rewrites 21.1%

       \[\leadsto 0.25 \cdot \mathsf{fma}\left(b, \color{blue}{y-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right)}, \frac{\left(a \cdot a\right) \cdot \left(y-scale \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot {8}^{0.5}\right)\right)}{b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right) \]

   if 4.40000000000000003e24 < b < 1.1e30

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 0.0%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{0.5}\right)\right)\right) \cdot {\left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right), y-scale \cdot y-scale, \left(x-scale \cdot x-scale\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}^{0.5}\right)} \]

   6. Taylor expanded in x-scale around inf

      \[\leadsto \frac{1}{4} \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{\frac{1}{2}}\right)\right)\right) \cdot {\left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{\frac{1}{2}} + \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{\frac{1}{2}}\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{\frac{1}{2}}\right)\right)\right) \cdot {\left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{\frac{1}{2}} + \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{\frac{1}{2}}\right) \]

   8. Applied rewrites 0.0%

      \[\leadsto 0.25 \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{0.5}\right)\right)\right) \cdot {\left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{0.5}\right) \]

   9. Taylor expanded in x-scale around inf

      \[\leadsto \frac{1}{4} \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{\frac{1}{2}}\right)\right)\right) \cdot {\left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}} + \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\mathsf{fma}\left(\frac{1}{180}, angle \cdot \pi, \frac{1}{2} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{\frac{1}{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 0.0%

       \[\leadsto 0.25 \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{0.5}\right)\right)\right) \cdot {\left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale} + \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{0.5}\right) \]

   if 1.1e30 < b

   1. Initial program 4.1%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 28.1%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{+24}:\\ \;\;\;\;0.25 \cdot \mathsf{fma}\left(b, y-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right), \frac{\left(a \cdot a\right) \cdot \left(y-scale \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot {8}^{0.5}\right)\right)}{b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+30}:\\ \;\;\;\;0.25 \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{0.5}\right)\right)\right) \cdot {\left(\frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale} + \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}} + \frac{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}}\right)\right)}^{0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 36.0% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;b\_m \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;0.25 \cdot \mathsf{fma}\left(b\_m, y-scale\_m \cdot \left(t\_2 \cdot 4\right), \frac{\left(a \cdot a\right) \cdot \left(y-scale\_m \cdot \left(\left(t\_1 \cdot t\_1\right) \cdot {8}^{0.5}\right)\right)}{b\_m \cdot \left(t\_2 \cdot {2}^{0.5}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(y-scale\_m \cdot 4\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (if (<= b_m 2.7e+23)
     (*
      0.25
      (fma
       b_m
       (* y-scale_m (* t_2 4.0))
       (/
        (* (* a a) (* y-scale_m (* (* t_1 t_1) (pow 8.0 0.5))))
        (* b_m (* t_2 (pow 2.0 0.5))))))
     (* 0.25 (* b_m (* y-scale_m 4.0))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double tmp;
	if (b_m <= 2.7e+23) {
		tmp = 0.25 * fma(b_m, (y_45_scale_m * (t_2 * 4.0)), (((a * a) * (y_45_scale_m * ((t_1 * t_1) * pow(8.0, 0.5)))) / (b_m * (t_2 * pow(2.0, 0.5)))));
	} else {
		tmp = 0.25 * (b_m * (y_45_scale_m * 4.0));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	tmp = 0.0
	if (b_m <= 2.7e+23)
		tmp = Float64(0.25 * fma(b_m, Float64(y_45_scale_m * Float64(t_2 * 4.0)), Float64(Float64(Float64(a * a) * Float64(y_45_scale_m * Float64(Float64(t_1 * t_1) * (8.0 ^ 0.5)))) / Float64(b_m * Float64(t_2 * (2.0 ^ 0.5))))));
	else
		tmp = Float64(0.25 * Float64(b_m * Float64(y_45_scale_m * 4.0)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(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[b$95$m, 2.7e+23], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * N[(t$95$2 * 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * N[(y$45$scale$95$m * N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[Power[8.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * N[(t$95$2 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6999999999999999e23

   1. Initial program 1.8%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 20.5%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 20.4%

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

   9. Taylor expanded in a around 0

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

       1. lower-fma.f64 N/A

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

   11. Applied rewrites 21.2%

       \[\leadsto 0.25 \cdot \mathsf{fma}\left(b, \color{blue}{y-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right)}, \frac{\left(a \cdot a\right) \cdot \left(y-scale \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot {8}^{0.5}\right)\right)}{b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right) \]

   if 2.6999999999999999e23 < b

   1. Initial program 3.9%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 26.8%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;0.25 \cdot \mathsf{fma}\left(b, y-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right), \frac{\left(a \cdot a\right) \cdot \left(y-scale \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot {8}^{0.5}\right)\right)}{b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 32.3% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := t\_1 \cdot {2}^{0.5}\\ t_3 := {b\_m}^{2} \cdot \left(y-scale\_m \cdot \left({\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2} \cdot \sqrt{8}\right)\right)\\ t_4 := \sin t\_0\\ t_5 := t\_4 \cdot \sqrt{2}\\ t_6 := t\_1 \cdot 4\\ t_7 := \left(a \cdot a\right) \cdot \left(y-scale\_m \cdot \left(\left(t\_4 \cdot t\_4\right) \cdot {8}^{0.5}\right)\right)\\ t_8 := t\_4 \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\\ \mathbf{if}\;angle \leq 1.4 \cdot 10^{+30}:\\ \;\;\;\;0.25 \cdot \mathsf{fma}\left(b\_m, y-scale\_m \cdot t\_6, \frac{t\_7}{b\_m \cdot t\_2}\right)\\ \mathbf{elif}\;angle \leq 8 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \left(\left(-1 \cdot b\_m\right) \cdot \mathsf{fma}\left(y-scale\_m, t\_6, \frac{t\_7}{\left(b\_m \cdot b\_m\right) \cdot t\_2}\right)\right)\\ \mathbf{elif}\;angle \leq 8.5 \cdot 10^{+160}:\\ \;\;\;\;0.25 \cdot \left(\left(-1 \cdot a\right) \cdot \mathsf{fma}\left(y-scale\_m, t\_8, \frac{t\_3}{{a}^{2} \cdot t\_5}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \mathsf{fma}\left(a, y-scale\_m \cdot t\_8, \frac{t\_3}{a \cdot t\_5}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (* t_1 (pow 2.0 0.5)))
        (t_3
         (*
          (pow b_m 2.0)
          (*
           y-scale_m
           (*
            (pow (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))) 2.0)
            (sqrt 8.0)))))
        (t_4 (sin t_0))
        (t_5 (* t_4 (sqrt 2.0)))
        (t_6 (* t_1 4.0))
        (t_7 (* (* a a) (* y-scale_m (* (* t_4 t_4) (pow 8.0 0.5)))))
        (t_8 (* t_4 (* (sqrt 2.0) (sqrt 8.0)))))
   (if (<= angle 1.4e+30)
     (* 0.25 (fma b_m (* y-scale_m t_6) (/ t_7 (* b_m t_2))))
     (if (<= angle 8e+75)
       (*
        0.25
        (* (* -1.0 b_m) (fma y-scale_m t_6 (/ t_7 (* (* b_m b_m) t_2)))))
       (if (<= angle 8.5e+160)
         (*
          0.25
          (* (* -1.0 a) (fma y-scale_m t_8 (/ t_3 (* (pow a 2.0) t_5)))))
         (* 0.25 (fma a (* y-scale_m t_8) (/ t_3 (* a t_5)))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = t_1 * pow(2.0, 0.5);
	double t_3 = pow(b_m, 2.0) * (y_45_scale_m * (pow(sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI)))), 2.0) * sqrt(8.0)));
	double t_4 = sin(t_0);
	double t_5 = t_4 * sqrt(2.0);
	double t_6 = t_1 * 4.0;
	double t_7 = (a * a) * (y_45_scale_m * ((t_4 * t_4) * pow(8.0, 0.5)));
	double t_8 = t_4 * (sqrt(2.0) * sqrt(8.0));
	double tmp;
	if (angle <= 1.4e+30) {
		tmp = 0.25 * fma(b_m, (y_45_scale_m * t_6), (t_7 / (b_m * t_2)));
	} else if (angle <= 8e+75) {
		tmp = 0.25 * ((-1.0 * b_m) * fma(y_45_scale_m, t_6, (t_7 / ((b_m * b_m) * t_2))));
	} else if (angle <= 8.5e+160) {
		tmp = 0.25 * ((-1.0 * a) * fma(y_45_scale_m, t_8, (t_3 / (pow(a, 2.0) * t_5))));
	} else {
		tmp = 0.25 * fma(a, (y_45_scale_m * t_8), (t_3 / (a * t_5)));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = Float64(t_1 * (2.0 ^ 0.5))
	t_3 = Float64((b_m ^ 2.0) * Float64(y_45_scale_m * Float64((sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi))) ^ 2.0) * sqrt(8.0))))
	t_4 = sin(t_0)
	t_5 = Float64(t_4 * sqrt(2.0))
	t_6 = Float64(t_1 * 4.0)
	t_7 = Float64(Float64(a * a) * Float64(y_45_scale_m * Float64(Float64(t_4 * t_4) * (8.0 ^ 0.5))))
	t_8 = Float64(t_4 * Float64(sqrt(2.0) * sqrt(8.0)))
	tmp = 0.0
	if (angle <= 1.4e+30)
		tmp = Float64(0.25 * fma(b_m, Float64(y_45_scale_m * t_6), Float64(t_7 / Float64(b_m * t_2))));
	elseif (angle <= 8e+75)
		tmp = Float64(0.25 * Float64(Float64(-1.0 * b_m) * fma(y_45_scale_m, t_6, Float64(t_7 / Float64(Float64(b_m * b_m) * t_2)))));
	elseif (angle <= 8.5e+160)
		tmp = Float64(0.25 * Float64(Float64(-1.0 * a) * fma(y_45_scale_m, t_8, Float64(t_3 / Float64((a ^ 2.0) * t_5)))));
	else
		tmp = Float64(0.25 * fma(a, Float64(y_45_scale_m * t_8), Float64(t_3 / Float64(a * t_5))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] * N[(y$45$scale$95$m * N[(N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$1 * 4.0), $MachinePrecision]}, Block[{t$95$7 = N[(N[(a * a), $MachinePrecision] * N[(y$45$scale$95$m * N[(N[(t$95$4 * t$95$4), $MachinePrecision] * N[Power[8.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$4 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, 1.4e+30], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * t$95$6), $MachinePrecision] + N[(t$95$7 / N[(b$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 8e+75], N[(0.25 * N[(N[(-1.0 * b$95$m), $MachinePrecision] * N[(y$45$scale$95$m * t$95$6 + N[(t$95$7 / N[(N[(b$95$m * b$95$m), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 8.5e+160], N[(0.25 * N[(N[(-1.0 * a), $MachinePrecision] * N[(y$45$scale$95$m * t$95$8 + N[(t$95$3 / N[(N[Power[a, 2.0], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(a * N[(y$45$scale$95$m * t$95$8), $MachinePrecision] + N[(t$95$3 / N[(a * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if angle < 1.39999999999999992e30

   1. Initial program 3.0%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 22.6%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 22.5%

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

   9. Taylor expanded in a around 0

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

       1. lower-fma.f64 N/A

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

   11. Applied rewrites 24.6%

       \[\leadsto 0.25 \cdot \mathsf{fma}\left(b, \color{blue}{y-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right)}, \frac{\left(a \cdot a\right) \cdot \left(y-scale \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot {8}^{0.5}\right)\right)}{b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right) \]

   if 1.39999999999999992e30 < angle < 7.99999999999999941e75

   1. Initial program 0.0%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 23.2%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 23.2%

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

   9. Taylor expanded in b around -inf

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

   11. Applied rewrites 29.7%

       \[\leadsto 0.25 \cdot \left(-1 \cdot \color{blue}{\left(b \cdot \mathsf{fma}\left(y-scale, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4, \frac{\left(a \cdot a\right) \cdot \left(y-scale \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot {8}^{0.5}\right)\right)}{\left(b \cdot b\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right)\right)}\right) \]

   if 7.99999999999999941e75 < angle < 8.49999999999999982e160

   1. Initial program 0.1%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 20.6%

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

   6. Taylor expanded in a around -inf

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

   8. Applied rewrites 26.7%

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

   if 8.49999999999999982e160 < angle

   1. Initial program 0.0%

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

   3. Taylor expanded in x-scale around 0

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

   5. Applied rewrites 24.8%

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

   6. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

   8. Applied rewrites 20.9%

      \[\leadsto 0.25 \cdot \mathsf{fma}\left(a, \color{blue}{y-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}, \frac{{b}^{2} \cdot \left(y-scale \cdot \left({\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2} \cdot \sqrt{8}\right)\right)}{a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{2}\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 1.4 \cdot 10^{+30}:\\ \;\;\;\;0.25 \cdot \mathsf{fma}\left(b, y-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right), \frac{\left(a \cdot a\right) \cdot \left(y-scale \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 {8}^{0.5}\right)\right)}{b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right)\\ \mathbf{elif}\;angle \leq 8 \cdot 10^{+75}:\\ \;\;\;\;0.25 \cdot \left(\left(-1 \cdot b\right) \cdot \mathsf{fma}\left(y-scale, \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4, \frac{\left(a \cdot a\right) \cdot \left(y-scale \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 {8}^{0.5}\right)\right)}{\left(b \cdot b\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right)\right)\\ \mathbf{elif}\;angle \leq 8.5 \cdot 10^{+160}:\\ \;\;\;\;0.25 \cdot \left(\left(-1 \cdot a\right) \cdot \mathsf{fma}\left(y-scale, \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right), \frac{{b}^{2} \cdot \left(y-scale \cdot \left({\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2} \cdot \sqrt{8}\right)\right)}{{a}^{2} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{2}\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \mathsf{fma}\left(a, y-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right), \frac{{b}^{2} \cdot \left(y-scale \cdot \left({\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{2} \cdot \sqrt{8}\right)\right)}{a \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.0% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ 0.25 \cdot \mathsf{fma}\left(b\_m, y-scale\_m \cdot \left(t\_2 \cdot 4\right), \frac{\left(a \cdot a\right) \cdot \left(y-scale\_m \cdot \left(\left(t\_1 \cdot t\_1\right) \cdot {8}^{0.5}\right)\right)}{b\_m \cdot \left(t\_2 \cdot {2}^{0.5}\right)}\right) \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (*
    0.25
    (fma
     b_m
     (* y-scale_m (* t_2 4.0))
     (/
      (* (* a a) (* y-scale_m (* (* t_1 t_1) (pow 8.0 0.5))))
      (* b_m (* t_2 (pow 2.0 0.5))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	return 0.25 * fma(b_m, (y_45_scale_m * (t_2 * 4.0)), (((a * a) * (y_45_scale_m * ((t_1 * t_1) * pow(8.0, 0.5)))) / (b_m * (t_2 * pow(2.0, 0.5)))));
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	return Float64(0.25 * fma(b_m, Float64(y_45_scale_m * Float64(t_2 * 4.0)), Float64(Float64(Float64(a * a) * Float64(y_45_scale_m * Float64(Float64(t_1 * t_1) * (8.0 ^ 0.5)))) / Float64(b_m * Float64(t_2 * (2.0 ^ 0.5))))))
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * N[(t$95$2 * 4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] * N[(y$45$scale$95$m * N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[Power[8.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b$95$m * N[(t$95$2 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 2.3%

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

3. Taylor expanded in x-scale around 0

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

5. Applied rewrites 22.7%

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

6. Taylor expanded in x-scale around 0

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

8. Applied rewrites 22.6%

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

9. Taylor expanded in a around 0

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

    1. lower-fma.f64 N/A

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

11. Applied rewrites 23.7%

    \[\leadsto 0.25 \cdot \mathsf{fma}\left(b, \color{blue}{y-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right)}, \frac{\left(a \cdot a\right) \cdot \left(y-scale \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot {8}^{0.5}\right)\right)}{b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right) \]

12. Final simplification 23.7%

    \[\leadsto 0.25 \cdot \mathsf{fma}\left(b, y-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right), \frac{\left(a \cdot a\right) \cdot \left(y-scale \cdot \left(\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot {8}^{0.5}\right)\right)}{b \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}\right) \]
13. Add Preprocessing

Alternative 7: 24.9% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_3 := t\_2 \cdot {2}^{0.5}\\ t_4 := {t\_3}^{2}\\ t_5 := t\_1 \cdot t\_1\\ t_6 := t\_2 \cdot t\_2\\ t_7 := \frac{t\_5 \cdot t\_6}{y-scale\_m \cdot y-scale\_m}\\ t_8 := \mathsf{fma}\left(-2, t\_7, 4 \cdot t\_7\right)\\ t_9 := \mathsf{fma}\left(0.5, \frac{t\_8}{t\_6}, \frac{t\_5}{y-scale\_m \cdot y-scale\_m}\right)\\ t_10 := \frac{{t\_1}^{4}}{{y-scale\_m}^{4}} - 0.25 \cdot \frac{t\_8 \cdot t\_8}{{t\_2}^{4}}\\ t_11 := 0.5 \cdot \frac{t\_10}{t\_6} - 0.25 \cdot \frac{t\_9 \cdot t\_9}{t\_4}\\ \mathbf{if}\;x-scale\_m \leq 1.9 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(0.25, b\_m \cdot \left(y-scale\_m \cdot \left(t\_2 \cdot 4\right)\right), \left(x-scale\_m \cdot x-scale\_m\right) \cdot \mathsf{fma}\left(0.125, \frac{b\_m \cdot \left(y-scale\_m \cdot \left({8}^{0.5} \cdot t\_9\right)\right)}{t\_3}, \left(x-scale\_m \cdot x-scale\_m\right) \cdot \mathsf{fma}\left(0.125, \frac{b\_m \cdot \left(y-scale\_m \cdot \left({8}^{0.5} \cdot t\_11\right)\right)}{t\_3}, 0.125 \cdot \frac{b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \left({8}^{0.5} \cdot \left(-0.25 \cdot \frac{t\_8 \cdot t\_10}{{t\_2}^{6}} - 0.5 \cdot \frac{t\_9 \cdot t\_11}{t\_4}\right)\right)\right)\right)}{t\_3}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left(t\_0 \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_3 (* t_2 (pow 2.0 0.5)))
        (t_4 (pow t_3 2.0))
        (t_5 (* t_1 t_1))
        (t_6 (* t_2 t_2))
        (t_7 (/ (* t_5 t_6) (* y-scale_m y-scale_m)))
        (t_8 (fma -2.0 t_7 (* 4.0 t_7)))
        (t_9 (fma 0.5 (/ t_8 t_6) (/ t_5 (* y-scale_m y-scale_m))))
        (t_10
         (-
          (/ (pow t_1 4.0) (pow y-scale_m 4.0))
          (* 0.25 (/ (* t_8 t_8) (pow t_2 4.0)))))
        (t_11 (- (* 0.5 (/ t_10 t_6)) (* 0.25 (/ (* t_9 t_9) t_4)))))
   (if (<= x-scale_m 1.9e+142)
     (fma
      0.25
      (* b_m (* y-scale_m (* t_2 4.0)))
      (*
       (* x-scale_m x-scale_m)
       (fma
        0.125
        (/ (* b_m (* y-scale_m (* (pow 8.0 0.5) t_9))) t_3)
        (*
         (* x-scale_m x-scale_m)
         (fma
          0.125
          (/ (* b_m (* y-scale_m (* (pow 8.0 0.5) t_11))) t_3)
          (*
           0.125
           (/
            (*
             b_m
             (*
              (* x-scale_m x-scale_m)
              (*
               y-scale_m
               (*
                (pow 8.0 0.5)
                (-
                 (* -0.25 (/ (* t_8 t_10) (pow t_2 6.0)))
                 (* 0.5 (/ (* t_9 t_11) t_4)))))))
            t_3)))))))
     (* 0.25 (* b_m (* x-scale_m (* t_0 (* (sqrt 2.0) (sqrt 8.0)))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_3 = t_2 * pow(2.0, 0.5);
	double t_4 = pow(t_3, 2.0);
	double t_5 = t_1 * t_1;
	double t_6 = t_2 * t_2;
	double t_7 = (t_5 * t_6) / (y_45_scale_m * y_45_scale_m);
	double t_8 = fma(-2.0, t_7, (4.0 * t_7));
	double t_9 = fma(0.5, (t_8 / t_6), (t_5 / (y_45_scale_m * y_45_scale_m)));
	double t_10 = (pow(t_1, 4.0) / pow(y_45_scale_m, 4.0)) - (0.25 * ((t_8 * t_8) / pow(t_2, 4.0)));
	double t_11 = (0.5 * (t_10 / t_6)) - (0.25 * ((t_9 * t_9) / t_4));
	double tmp;
	if (x_45_scale_m <= 1.9e+142) {
		tmp = fma(0.25, (b_m * (y_45_scale_m * (t_2 * 4.0))), ((x_45_scale_m * x_45_scale_m) * fma(0.125, ((b_m * (y_45_scale_m * (pow(8.0, 0.5) * t_9))) / t_3), ((x_45_scale_m * x_45_scale_m) * fma(0.125, ((b_m * (y_45_scale_m * (pow(8.0, 0.5) * t_11))) / t_3), (0.125 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * (pow(8.0, 0.5) * ((-0.25 * ((t_8 * t_10) / pow(t_2, 6.0))) - (0.5 * ((t_9 * t_11) / t_4))))))) / t_3)))))));
	} else {
		tmp = 0.25 * (b_m * (x_45_scale_m * (t_0 * (sqrt(2.0) * sqrt(8.0)))));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_3 = Float64(t_2 * (2.0 ^ 0.5))
	t_4 = t_3 ^ 2.0
	t_5 = Float64(t_1 * t_1)
	t_6 = Float64(t_2 * t_2)
	t_7 = Float64(Float64(t_5 * t_6) / Float64(y_45_scale_m * y_45_scale_m))
	t_8 = fma(-2.0, t_7, Float64(4.0 * t_7))
	t_9 = fma(0.5, Float64(t_8 / t_6), Float64(t_5 / Float64(y_45_scale_m * y_45_scale_m)))
	t_10 = Float64(Float64((t_1 ^ 4.0) / (y_45_scale_m ^ 4.0)) - Float64(0.25 * Float64(Float64(t_8 * t_8) / (t_2 ^ 4.0))))
	t_11 = Float64(Float64(0.5 * Float64(t_10 / t_6)) - Float64(0.25 * Float64(Float64(t_9 * t_9) / t_4)))
	tmp = 0.0
	if (x_45_scale_m <= 1.9e+142)
		tmp = fma(0.25, Float64(b_m * Float64(y_45_scale_m * Float64(t_2 * 4.0))), Float64(Float64(x_45_scale_m * x_45_scale_m) * fma(0.125, Float64(Float64(b_m * Float64(y_45_scale_m * Float64((8.0 ^ 0.5) * t_9))) / t_3), Float64(Float64(x_45_scale_m * x_45_scale_m) * fma(0.125, Float64(Float64(b_m * Float64(y_45_scale_m * Float64((8.0 ^ 0.5) * t_11))) / t_3), Float64(0.125 * Float64(Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * Float64((8.0 ^ 0.5) * Float64(Float64(-0.25 * Float64(Float64(t_8 * t_10) / (t_2 ^ 6.0))) - Float64(0.5 * Float64(Float64(t_9 * t_11) / t_4))))))) / t_3)))))));
	else
		tmp = Float64(0.25 * Float64(b_m * Float64(x_45_scale_m * Float64(t_0 * Float64(sqrt(2.0) * sqrt(8.0))))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$5 * t$95$6), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(-2.0 * t$95$7 + N[(4.0 * t$95$7), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(0.5 * N[(t$95$8 / t$95$6), $MachinePrecision] + N[(t$95$5 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[(N[Power[t$95$1, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[(N[(t$95$8 * t$95$8), $MachinePrecision] / N[Power[t$95$2, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[(N[(0.5 * N[(t$95$10 / t$95$6), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[(N[(t$95$9 * t$95$9), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.9e+142], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * N[(t$95$2 * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(0.125 * N[(N[(b$95$m * N[(y$45$scale$95$m * N[(N[Power[8.0, 0.5], $MachinePrecision] * t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(0.125 * N[(N[(b$95$m * N[(y$45$scale$95$m * N[(N[Power[8.0, 0.5], $MachinePrecision] * t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(0.125 * N[(N[(b$95$m * N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale$95$m * N[(N[Power[8.0, 0.5], $MachinePrecision] * N[(N[(-0.25 * N[(N[(t$95$8 * t$95$10), $MachinePrecision] / N[Power[t$95$2, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[(t$95$9 * t$95$11), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(b$95$m * N[(x$45$scale$95$m * N[(t$95$0 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.89999999999999995e142

   1. Initial program 2.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 5.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{0.5}\right)\right)\right) \cdot {\left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right), y-scale \cdot y-scale, \left(x-scale \cdot x-scale\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}^{0.5}\right)} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 9.0%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       6. lift-PI.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       8. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       9. lift-*.f64 10.2

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

   11. Applied rewrites 10.2%

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

   12. Taylor expanded in x-scale around 0

       \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) + \color{blue}{{x-scale}^{2} \cdot \left(\frac{1}{8} \cdot \frac{b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}} + {x-scale}^{2} \cdot \left(\frac{1}{8} \cdot \frac{b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\frac{1}{2} \cdot \frac{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} - \frac{1}{4} \cdot \frac{{\left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}} + \frac{1}{8} \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\frac{-1}{4} \cdot \frac{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right) \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}\right)}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{6}} - \frac{1}{2} \cdot \frac{\left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right) \cdot \left(\frac{1}{2} \cdot \frac{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} - \frac{1}{4} \cdot \frac{{\left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}\right)}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}}\right)\right)} \]

   14. Applied rewrites 16.3%

       \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{b \cdot \left(y-scale \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot 4\right)\right)}, \left(x-scale \cdot x-scale\right) \cdot \mathsf{fma}\left(0.125, \frac{b \cdot \left(y-scale \cdot \left({8}^{0.5} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}}, \left(x-scale \cdot x-scale\right) \cdot \mathsf{fma}\left(0.125, \frac{b \cdot \left(y-scale \cdot \left({8}^{0.5} \cdot \left(0.5 \cdot \frac{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} - 0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)} - 0.25 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}}, 0.125 \cdot \frac{b \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \left({8}^{0.5} \cdot \left(-0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} - 0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{6}} - 0.5 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(0.5 \cdot \frac{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} - 0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)} - 0.25 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}^{2}}\right)}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}^{2}}\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}}\right)\right)\right) \]

   if 1.89999999999999995e142 < x-scale

   1. Initial program 3.6%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 5.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{0.5}\right)\right)\right) \cdot {\left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right), y-scale \cdot y-scale, \left(x-scale \cdot x-scale\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}^{0.5}\right)} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 20.3%

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

   9. Taylor expanded in angle around 0

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

       1. lift-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left(x-scale \cdot \left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left(x-scale \cdot \left(\left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) \]

       3. lift-*.f64 25.4

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

   11. Applied rewrites 25.4%

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

Alternative 8: 24.8% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_2 := t\_1 \cdot {2}^{0.5}\\ t_3 := {t\_2}^{2}\\ t_4 := t\_0 \cdot t\_0\\ t_5 := t\_1 \cdot t\_1\\ t_6 := \frac{t\_4 \cdot t\_5}{y-scale\_m \cdot y-scale\_m}\\ t_7 := \mathsf{fma}\left(-2, t\_6, 4 \cdot t\_6\right)\\ t_8 := \mathsf{fma}\left(0.5, \frac{t\_7}{t\_5}, \frac{t\_4}{y-scale\_m \cdot y-scale\_m}\right)\\ t_9 := \frac{{t\_0}^{4}}{{y-scale\_m}^{4}} - 0.25 \cdot \frac{t\_7 \cdot t\_7}{{t\_1}^{4}}\\ t_10 := 0.5 \cdot \frac{t\_9}{t\_5} - 0.25 \cdot \frac{t\_8 \cdot t\_8}{t\_3}\\ \mathbf{if}\;x-scale\_m \leq 1.75 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(0.25, b\_m \cdot \left(y-scale\_m \cdot \left(t\_1 \cdot 4\right)\right), \left(x-scale\_m \cdot x-scale\_m\right) \cdot \mathsf{fma}\left(0.125, \frac{b\_m \cdot \left(y-scale\_m \cdot \left({8}^{0.5} \cdot t\_8\right)\right)}{t\_2}, \left(x-scale\_m \cdot x-scale\_m\right) \cdot \mathsf{fma}\left(0.125, \frac{b\_m \cdot \left(y-scale\_m \cdot \left({8}^{0.5} \cdot t\_10\right)\right)}{t\_2}, 0.125 \cdot \frac{b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \left({8}^{0.5} \cdot \left(-0.25 \cdot \frac{t\_7 \cdot t\_9}{{t\_1}^{6}} - 0.5 \cdot \frac{t\_8 \cdot t\_10}{t\_3}\right)\right)\right)\right)}{t\_2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI))))
        (t_1 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_2 (* t_1 (pow 2.0 0.5)))
        (t_3 (pow t_2 2.0))
        (t_4 (* t_0 t_0))
        (t_5 (* t_1 t_1))
        (t_6 (/ (* t_4 t_5) (* y-scale_m y-scale_m)))
        (t_7 (fma -2.0 t_6 (* 4.0 t_6)))
        (t_8 (fma 0.5 (/ t_7 t_5) (/ t_4 (* y-scale_m y-scale_m))))
        (t_9
         (-
          (/ (pow t_0 4.0) (pow y-scale_m 4.0))
          (* 0.25 (/ (* t_7 t_7) (pow t_1 4.0)))))
        (t_10 (- (* 0.5 (/ t_9 t_5)) (* 0.25 (/ (* t_8 t_8) t_3)))))
   (if (<= x-scale_m 1.75e+142)
     (fma
      0.25
      (* b_m (* y-scale_m (* t_1 4.0)))
      (*
       (* x-scale_m x-scale_m)
       (fma
        0.125
        (/ (* b_m (* y-scale_m (* (pow 8.0 0.5) t_8))) t_2)
        (*
         (* x-scale_m x-scale_m)
         (fma
          0.125
          (/ (* b_m (* y-scale_m (* (pow 8.0 0.5) t_10))) t_2)
          (*
           0.125
           (/
            (*
             b_m
             (*
              (* x-scale_m x-scale_m)
              (*
               y-scale_m
               (*
                (pow 8.0 0.5)
                (-
                 (* -0.25 (/ (* t_7 t_9) (pow t_1 6.0)))
                 (* 0.5 (/ (* t_8 t_10) t_3)))))))
            t_2)))))))
     (*
      0.25
      (*
       0.005555555555555556
       (* angle (* b_m (* x-scale_m (* PI (* (sqrt 2.0) (sqrt 8.0)))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_2 = t_1 * pow(2.0, 0.5);
	double t_3 = pow(t_2, 2.0);
	double t_4 = t_0 * t_0;
	double t_5 = t_1 * t_1;
	double t_6 = (t_4 * t_5) / (y_45_scale_m * y_45_scale_m);
	double t_7 = fma(-2.0, t_6, (4.0 * t_6));
	double t_8 = fma(0.5, (t_7 / t_5), (t_4 / (y_45_scale_m * y_45_scale_m)));
	double t_9 = (pow(t_0, 4.0) / pow(y_45_scale_m, 4.0)) - (0.25 * ((t_7 * t_7) / pow(t_1, 4.0)));
	double t_10 = (0.5 * (t_9 / t_5)) - (0.25 * ((t_8 * t_8) / t_3));
	double tmp;
	if (x_45_scale_m <= 1.75e+142) {
		tmp = fma(0.25, (b_m * (y_45_scale_m * (t_1 * 4.0))), ((x_45_scale_m * x_45_scale_m) * fma(0.125, ((b_m * (y_45_scale_m * (pow(8.0, 0.5) * t_8))) / t_2), ((x_45_scale_m * x_45_scale_m) * fma(0.125, ((b_m * (y_45_scale_m * (pow(8.0, 0.5) * t_10))) / t_2), (0.125 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * (pow(8.0, 0.5) * ((-0.25 * ((t_7 * t_9) / pow(t_1, 6.0))) - (0.5 * ((t_8 * t_10) / t_3))))))) / t_2)))))));
	} else {
		tmp = 0.25 * (0.005555555555555556 * (angle * (b_m * (x_45_scale_m * (((double) M_PI) * (sqrt(2.0) * sqrt(8.0)))))));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_2 = Float64(t_1 * (2.0 ^ 0.5))
	t_3 = t_2 ^ 2.0
	t_4 = Float64(t_0 * t_0)
	t_5 = Float64(t_1 * t_1)
	t_6 = Float64(Float64(t_4 * t_5) / Float64(y_45_scale_m * y_45_scale_m))
	t_7 = fma(-2.0, t_6, Float64(4.0 * t_6))
	t_8 = fma(0.5, Float64(t_7 / t_5), Float64(t_4 / Float64(y_45_scale_m * y_45_scale_m)))
	t_9 = Float64(Float64((t_0 ^ 4.0) / (y_45_scale_m ^ 4.0)) - Float64(0.25 * Float64(Float64(t_7 * t_7) / (t_1 ^ 4.0))))
	t_10 = Float64(Float64(0.5 * Float64(t_9 / t_5)) - Float64(0.25 * Float64(Float64(t_8 * t_8) / t_3)))
	tmp = 0.0
	if (x_45_scale_m <= 1.75e+142)
		tmp = fma(0.25, Float64(b_m * Float64(y_45_scale_m * Float64(t_1 * 4.0))), Float64(Float64(x_45_scale_m * x_45_scale_m) * fma(0.125, Float64(Float64(b_m * Float64(y_45_scale_m * Float64((8.0 ^ 0.5) * t_8))) / t_2), Float64(Float64(x_45_scale_m * x_45_scale_m) * fma(0.125, Float64(Float64(b_m * Float64(y_45_scale_m * Float64((8.0 ^ 0.5) * t_10))) / t_2), Float64(0.125 * Float64(Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * Float64((8.0 ^ 0.5) * Float64(Float64(-0.25 * Float64(Float64(t_7 * t_9) / (t_1 ^ 6.0))) - Float64(0.5 * Float64(Float64(t_8 * t_10) / t_3))))))) / t_2)))))));
	else
		tmp = Float64(0.25 * Float64(0.005555555555555556 * Float64(angle * Float64(b_m * Float64(x_45_scale_m * Float64(pi * Float64(sqrt(2.0) * sqrt(8.0))))))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$4 * t$95$5), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(-2.0 * t$95$6 + N[(4.0 * t$95$6), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(0.5 * N[(t$95$7 / t$95$5), $MachinePrecision] + N[(t$95$4 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[Power[t$95$0, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[(N[(t$95$7 * t$95$7), $MachinePrecision] / N[Power[t$95$1, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[(0.5 * N[(t$95$9 / t$95$5), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[(N[(t$95$8 * t$95$8), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.75e+142], N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * N[(t$95$1 * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(0.125 * N[(N[(b$95$m * N[(y$45$scale$95$m * N[(N[Power[8.0, 0.5], $MachinePrecision] * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(0.125 * N[(N[(b$95$m * N[(y$45$scale$95$m * N[(N[Power[8.0, 0.5], $MachinePrecision] * t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(0.125 * N[(N[(b$95$m * N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale$95$m * N[(N[Power[8.0, 0.5], $MachinePrecision] * N[(N[(-0.25 * N[(N[(t$95$7 * t$95$9), $MachinePrecision] / N[Power[t$95$1, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[(t$95$8 * t$95$10), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(0.005555555555555556 * N[(angle * N[(b$95$m * N[(x$45$scale$95$m * N[(Pi * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.74999999999999999e142

   1. Initial program 2.1%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 5.3%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{0.5}\right)\right)\right) \cdot {\left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right), y-scale \cdot y-scale, \left(x-scale \cdot x-scale\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}^{0.5}\right)} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 9.0%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       6. lift-PI.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       8. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       9. lift-*.f64 10.2

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

   11. Applied rewrites 10.2%

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

   12. Taylor expanded in x-scale around 0

       \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) + \color{blue}{{x-scale}^{2} \cdot \left(\frac{1}{8} \cdot \frac{b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}} + {x-scale}^{2} \cdot \left(\frac{1}{8} \cdot \frac{b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\frac{1}{2} \cdot \frac{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} - \frac{1}{4} \cdot \frac{{\left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}} + \frac{1}{8} \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\frac{-1}{4} \cdot \frac{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right) \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}\right)}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{6}} - \frac{1}{2} \cdot \frac{\left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right) \cdot \left(\frac{1}{2} \cdot \frac{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} - \frac{1}{4} \cdot \frac{{\left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}\right)}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}}\right)\right)} \]

   14. Applied rewrites 16.3%

       \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{b \cdot \left(y-scale \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot 4\right)\right)}, \left(x-scale \cdot x-scale\right) \cdot \mathsf{fma}\left(0.125, \frac{b \cdot \left(y-scale \cdot \left({8}^{0.5} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}}, \left(x-scale \cdot x-scale\right) \cdot \mathsf{fma}\left(0.125, \frac{b \cdot \left(y-scale \cdot \left({8}^{0.5} \cdot \left(0.5 \cdot \frac{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} - 0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)} - 0.25 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}}, 0.125 \cdot \frac{b \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \left({8}^{0.5} \cdot \left(-0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} - 0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{6}} - 0.5 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(0.5 \cdot \frac{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} - 0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)} - 0.25 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}^{2}}\right)}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}^{2}}\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}}\right)\right)\right) \]

   if 1.74999999999999999e142 < x-scale

   1. Initial program 3.6%

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

   3. Taylor expanded in b around inf

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

   5. Applied rewrites 5.9%

      \[\leadsto \color{blue}{0.25 \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{0.5}\right)\right)\right) \cdot {\left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right), y-scale \cdot y-scale, \left(x-scale \cdot x-scale\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}^{0.5}\right)} \]

   6. Taylor expanded in x-scale around inf

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

   8. Applied rewrites 20.3%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       6. lift-PI.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       7. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       8. lift-sqrt.f64 N/A

          \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

       9. lift-*.f64 33.8

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

   11. Applied rewrites 33.8%

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

Alternative 9: 20.1% accurate, N/A× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\\ t_2 := t\_1 \cdot {2}^{0.5}\\ t_3 := {t\_2}^{2}\\ t_4 := t\_0 \cdot t\_0\\ t_5 := t\_1 \cdot t\_1\\ t_6 := \frac{t\_4 \cdot t\_5}{y-scale\_m \cdot y-scale\_m}\\ t_7 := \mathsf{fma}\left(-2, t\_6, 4 \cdot t\_6\right)\\ t_8 := \mathsf{fma}\left(0.5, \frac{t\_7}{t\_5}, \frac{t\_4}{y-scale\_m \cdot y-scale\_m}\right)\\ t_9 := \frac{{t\_0}^{4}}{{y-scale\_m}^{4}} - 0.25 \cdot \frac{t\_7 \cdot t\_7}{{t\_1}^{4}}\\ t_10 := 0.5 \cdot \frac{t\_9}{t\_5} - 0.25 \cdot \frac{t\_8 \cdot t\_8}{t\_3}\\ \mathsf{fma}\left(0.25, b\_m \cdot \left(y-scale\_m \cdot \left(t\_1 \cdot 4\right)\right), \left(x-scale\_m \cdot x-scale\_m\right) \cdot \mathsf{fma}\left(0.125, \frac{b\_m \cdot \left(y-scale\_m \cdot \left({8}^{0.5} \cdot t\_8\right)\right)}{t\_2}, \left(x-scale\_m \cdot x-scale\_m\right) \cdot \mathsf{fma}\left(0.125, \frac{b\_m \cdot \left(y-scale\_m \cdot \left({8}^{0.5} \cdot t\_10\right)\right)}{t\_2}, 0.125 \cdot \frac{b\_m \cdot \left(\left(x-scale\_m \cdot x-scale\_m\right) \cdot \left(y-scale\_m \cdot \left({8}^{0.5} \cdot \left(-0.25 \cdot \frac{t\_7 \cdot t\_9}{{t\_1}^{6}} - 0.5 \cdot \frac{t\_8 \cdot t\_10}{t\_3}\right)\right)\right)\right)}{t\_2}\right)\right)\right) \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI))))
        (t_1 (sin (fma 0.005555555555555556 (* angle PI) (* 0.5 PI))))
        (t_2 (* t_1 (pow 2.0 0.5)))
        (t_3 (pow t_2 2.0))
        (t_4 (* t_0 t_0))
        (t_5 (* t_1 t_1))
        (t_6 (/ (* t_4 t_5) (* y-scale_m y-scale_m)))
        (t_7 (fma -2.0 t_6 (* 4.0 t_6)))
        (t_8 (fma 0.5 (/ t_7 t_5) (/ t_4 (* y-scale_m y-scale_m))))
        (t_9
         (-
          (/ (pow t_0 4.0) (pow y-scale_m 4.0))
          (* 0.25 (/ (* t_7 t_7) (pow t_1 4.0)))))
        (t_10 (- (* 0.5 (/ t_9 t_5)) (* 0.25 (/ (* t_8 t_8) t_3)))))
   (fma
    0.25
    (* b_m (* y-scale_m (* t_1 4.0)))
    (*
     (* x-scale_m x-scale_m)
     (fma
      0.125
      (/ (* b_m (* y-scale_m (* (pow 8.0 0.5) t_8))) t_2)
      (*
       (* x-scale_m x-scale_m)
       (fma
        0.125
        (/ (* b_m (* y-scale_m (* (pow 8.0 0.5) t_10))) t_2)
        (*
         0.125
         (/
          (*
           b_m
           (*
            (* x-scale_m x-scale_m)
            (*
             y-scale_m
             (*
              (pow 8.0 0.5)
              (-
               (* -0.25 (/ (* t_7 t_9) (pow t_1 6.0)))
               (* 0.5 (/ (* t_8 t_10) t_3)))))))
          t_2)))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = sin(fma(0.005555555555555556, (angle * ((double) M_PI)), (0.5 * ((double) M_PI))));
	double t_2 = t_1 * pow(2.0, 0.5);
	double t_3 = pow(t_2, 2.0);
	double t_4 = t_0 * t_0;
	double t_5 = t_1 * t_1;
	double t_6 = (t_4 * t_5) / (y_45_scale_m * y_45_scale_m);
	double t_7 = fma(-2.0, t_6, (4.0 * t_6));
	double t_8 = fma(0.5, (t_7 / t_5), (t_4 / (y_45_scale_m * y_45_scale_m)));
	double t_9 = (pow(t_0, 4.0) / pow(y_45_scale_m, 4.0)) - (0.25 * ((t_7 * t_7) / pow(t_1, 4.0)));
	double t_10 = (0.5 * (t_9 / t_5)) - (0.25 * ((t_8 * t_8) / t_3));
	return fma(0.25, (b_m * (y_45_scale_m * (t_1 * 4.0))), ((x_45_scale_m * x_45_scale_m) * fma(0.125, ((b_m * (y_45_scale_m * (pow(8.0, 0.5) * t_8))) / t_2), ((x_45_scale_m * x_45_scale_m) * fma(0.125, ((b_m * (y_45_scale_m * (pow(8.0, 0.5) * t_10))) / t_2), (0.125 * ((b_m * ((x_45_scale_m * x_45_scale_m) * (y_45_scale_m * (pow(8.0, 0.5) * ((-0.25 * ((t_7 * t_9) / pow(t_1, 6.0))) - (0.5 * ((t_8 * t_10) / t_3))))))) / t_2)))))));
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = sin(fma(0.005555555555555556, Float64(angle * pi), Float64(0.5 * pi)))
	t_2 = Float64(t_1 * (2.0 ^ 0.5))
	t_3 = t_2 ^ 2.0
	t_4 = Float64(t_0 * t_0)
	t_5 = Float64(t_1 * t_1)
	t_6 = Float64(Float64(t_4 * t_5) / Float64(y_45_scale_m * y_45_scale_m))
	t_7 = fma(-2.0, t_6, Float64(4.0 * t_6))
	t_8 = fma(0.5, Float64(t_7 / t_5), Float64(t_4 / Float64(y_45_scale_m * y_45_scale_m)))
	t_9 = Float64(Float64((t_0 ^ 4.0) / (y_45_scale_m ^ 4.0)) - Float64(0.25 * Float64(Float64(t_7 * t_7) / (t_1 ^ 4.0))))
	t_10 = Float64(Float64(0.5 * Float64(t_9 / t_5)) - Float64(0.25 * Float64(Float64(t_8 * t_8) / t_3)))
	return fma(0.25, Float64(b_m * Float64(y_45_scale_m * Float64(t_1 * 4.0))), Float64(Float64(x_45_scale_m * x_45_scale_m) * fma(0.125, Float64(Float64(b_m * Float64(y_45_scale_m * Float64((8.0 ^ 0.5) * t_8))) / t_2), Float64(Float64(x_45_scale_m * x_45_scale_m) * fma(0.125, Float64(Float64(b_m * Float64(y_45_scale_m * Float64((8.0 ^ 0.5) * t_10))) / t_2), Float64(0.125 * Float64(Float64(b_m * Float64(Float64(x_45_scale_m * x_45_scale_m) * Float64(y_45_scale_m * Float64((8.0 ^ 0.5) * Float64(Float64(-0.25 * Float64(Float64(t_7 * t_9) / (t_1 ^ 6.0))) - Float64(0.5 * Float64(Float64(t_8 * t_10) / t_3))))))) / t_2)))))))
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Power[2.0, 0.5], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$1 * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$4 * t$95$5), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(-2.0 * t$95$6 + N[(4.0 * t$95$6), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(0.5 * N[(t$95$7 / t$95$5), $MachinePrecision] + N[(t$95$4 / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[Power[t$95$0, 4.0], $MachinePrecision] / N[Power[y$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[(N[(t$95$7 * t$95$7), $MachinePrecision] / N[Power[t$95$1, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[(0.5 * N[(t$95$9 / t$95$5), $MachinePrecision]), $MachinePrecision] - N[(0.25 * N[(N[(t$95$8 * t$95$8), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(0.25 * N[(b$95$m * N[(y$45$scale$95$m * N[(t$95$1 * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(0.125 * N[(N[(b$95$m * N[(y$45$scale$95$m * N[(N[Power[8.0, 0.5], $MachinePrecision] * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(0.125 * N[(N[(b$95$m * N[(y$45$scale$95$m * N[(N[Power[8.0, 0.5], $MachinePrecision] * t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(0.125 * N[(N[(b$95$m * N[(N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision] * N[(y$45$scale$95$m * N[(N[Power[8.0, 0.5], $MachinePrecision] * N[(N[(-0.25 * N[(N[(t$95$7 * t$95$9), $MachinePrecision] / N[Power[t$95$1, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(N[(t$95$8 * t$95$10), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Initial program 2.3%

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

3. Taylor expanded in b around inf

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

5. Applied rewrites 5.4%

   \[\leadsto \color{blue}{0.25 \cdot \left(\left(b \cdot \left(x-scale \cdot \left(y-scale \cdot {8}^{0.5}\right)\right)\right) \cdot {\left({\left(\mathsf{fma}\left(4, \frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} \cdot \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}, \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right)}{x-scale \cdot x-scale} - \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}^{0.5} + \frac{\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, \frac{\pi}{2}\right)\right), y-scale \cdot y-scale, \left(x-scale \cdot x-scale\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right)}^{0.5}\right)} \]

6. Taylor expanded in x-scale around inf

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

8. Applied rewrites 10.5%

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

9. Taylor expanded in angle around 0

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

    1. lower-*.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

    6. lift-PI.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

    7. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

    8. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{4} \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \left(b \cdot \left(x-scale \cdot \left(\pi \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)\right)\right) \]

    9. lift-*.f64 13.3

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

11. Applied rewrites 13.3%

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

12. Taylor expanded in x-scale around 0

    \[\leadsto \frac{1}{4} \cdot \left(b \cdot \left(y-scale \cdot \left(\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) + \color{blue}{{x-scale}^{2} \cdot \left(\frac{1}{8} \cdot \frac{b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}} + {x-scale}^{2} \cdot \left(\frac{1}{8} \cdot \frac{b \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\frac{1}{2} \cdot \frac{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} - \frac{1}{4} \cdot \frac{{\left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}} + \frac{1}{8} \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left(y-scale \cdot \left(\sqrt{8} \cdot \left(\frac{-1}{4} \cdot \frac{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right) \cdot \left(\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}\right)}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{6}} - \frac{1}{2} \cdot \frac{\left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right) \cdot \left(\frac{1}{2} \cdot \frac{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}}{{y-scale}^{4}} - \frac{1}{4} \cdot \frac{{\left(-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{4}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} - \frac{1}{4} \cdot \frac{{\left(\frac{1}{2} \cdot \frac{-2 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}} + 4 \cdot \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{y-scale}^{2}}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}\right)}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}\right)\right)\right)\right)}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2}}\right)\right)} \]

14. Applied rewrites 14.5%

    \[\leadsto \mathsf{fma}\left(0.25, \color{blue}{b \cdot \left(y-scale \cdot \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot 4\right)\right)}, \left(x-scale \cdot x-scale\right) \cdot \mathsf{fma}\left(0.125, \frac{b \cdot \left(y-scale \cdot \left({8}^{0.5} \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}}, \left(x-scale \cdot x-scale\right) \cdot \mathsf{fma}\left(0.125, \frac{b \cdot \left(y-scale \cdot \left({8}^{0.5} \cdot \left(0.5 \cdot \frac{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} - 0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)} - 0.25 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}^{2}}\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}}, 0.125 \cdot \frac{b \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(y-scale \cdot \left({8}^{0.5} \cdot \left(-0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} - 0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{6}} - 0.5 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \left(0.5 \cdot \frac{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}} - 0.25 \cdot \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}^{4}}}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)} - 0.25 \cdot \frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right) \cdot \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(-2, \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}, 4 \cdot \frac{\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 \left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right)}, \frac{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{y-scale \cdot y-scale}\right)}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}^{2}}\right)}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}\right)}^{2}}\right)\right)\right)\right)}{\sin \left(\mathsf{fma}\left(0.005555555555555556, angle \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot {2}^{0.5}}\right)\right)\right) \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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))))