a from scale-rotated-ellipse

Percentage Accurate: 2.5% → 57.7%

Time: 53.4s

Alternatives: 5

Speedup: 919.0×

• 29.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.5% 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: 57.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;y-scale\_m \leq 130000000:\\ \;\;\;\;x-scale\_m \cdot \mathsf{hypot}\left(a \cdot t\_2, b \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \sqrt{16 \cdot \left({\left(a \cdot t\_1\right)}^{2} + {\left(t\_2 \cdot b\right)}^{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (if (<= y-scale_m 130000000.0)
     (* x-scale_m (hypot (* a t_2) (* b t_1)))
     (*
      0.25
      (*
       y-scale_m
       (sqrt (* 16.0 (+ (pow (* a t_1) 2.0) (pow (* t_2 b) 2.0)))))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double tmp;
	if (y_45_scale_m <= 130000000.0) {
		tmp = x_45_scale_m * hypot((a * t_2), (b * t_1));
	} else {
		tmp = 0.25 * (y_45_scale_m * sqrt((16.0 * (pow((a * t_1), 2.0) + pow((t_2 * b), 2.0)))));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double tmp;
	if (y_45_scale_m <= 130000000.0) {
		tmp = x_45_scale_m * Math.hypot((a * t_2), (b * t_1));
	} else {
		tmp = 0.25 * (y_45_scale_m * Math.sqrt((16.0 * (Math.pow((a * t_1), 2.0) + Math.pow((t_2 * b), 2.0)))));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	tmp = 0
	if y_45_scale_m <= 130000000.0:
		tmp = x_45_scale_m * math.hypot((a * t_2), (b * t_1))
	else:
		tmp = 0.25 * (y_45_scale_m * math.sqrt((16.0 * (math.pow((a * t_1), 2.0) + math.pow((t_2 * b), 2.0)))))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	tmp = 0.0
	if (y_45_scale_m <= 130000000.0)
		tmp = Float64(x_45_scale_m * hypot(Float64(a * t_2), Float64(b * t_1)));
	else
		tmp = Float64(0.25 * Float64(y_45_scale_m * sqrt(Float64(16.0 * Float64((Float64(a * t_1) ^ 2.0) + (Float64(t_2 * b) ^ 2.0))))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	tmp = 0.0;
	if (y_45_scale_m <= 130000000.0)
		tmp = x_45_scale_m * hypot((a * t_2), (b * t_1));
	else
		tmp = 0.25 * (y_45_scale_m * sqrt((16.0 * (((a * t_1) ^ 2.0) + ((t_2 * b) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 130000000.0], N[(x$45$scale$95$m * N[Sqrt[N[(a * t$95$2), $MachinePrecision] ^ 2 + N[(b * t$95$1), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(y$45$scale$95$m * N[Sqrt[N[(16.0 * N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(t$95$2 * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.3e8

   1. Initial program 3.0%

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

   3. Simplified 3.6%

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

   5. Taylor expanded in y-scale around 0 23.2%

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

      1. associate-*l* 23.2%

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

      2. distribute-lft-out 23.2%

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

      3. fma-define 23.2%

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

      4. *-commutative 23.2%

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

   7. Simplified 23.2%

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

      1. pow1 23.2%

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

   9. Applied egg-rr 25.9%

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

       1. unpow1 25.9%

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

       2. associate-*r* 25.9%

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

       3. metadata-eval 25.9%

          \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{16} \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)} \]

   11. Simplified 25.9%

       \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}} \]

   12. Taylor expanded in x-scale around 0 23.2%

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

       1. unpow2 23.2%

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

       2. unpow2 23.2%

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

       3. swap-sqr 23.2%

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

       4. unpow2 23.2%

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

       5. unpow2 23.2%

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

       6. swap-sqr 25.9%

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

       7. hypot-undefine 28.6%

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

   14. Simplified 28.6%

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

   if 1.3e8 < y-scale

   1. Initial program 3.7%

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

   3. Simplified 2.1%

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

   5. Taylor expanded in x-scale around 0 49.9%

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

      1. associate-*l* 49.9%

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

      2. distribute-lft-out 49.9%

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

      3. fma-define 49.9%

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

      4. *-commutative 49.9%

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

   7. Simplified 49.9%

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

      1. pow1 49.9%

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

   9. Applied egg-rr 55.8%

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

       1. unpow1 55.8%

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

       2. associate-*l* 55.8%

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

       3. associate-*r* 55.8%

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

       4. metadata-eval 55.8%

          \[\leadsto 0.25 \cdot \left(y-scale \cdot \sqrt{\color{blue}{16} \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}\right)}\right) \]

   11. Simplified 55.8%

       \[\leadsto \color{blue}{0.25 \cdot \left(y-scale \cdot \sqrt{16 \cdot \left({\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2}\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 43.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;x-scale\_m \leq 88000000000:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale\_m \cdot 0.25\right) \cdot \left(\mathsf{hypot}\left(b \cdot \sin t\_0, a \cdot \cos t\_0\right) \cdot 4\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* angle (* 0.005555555555555556 PI))))
   (if (<= x-scale_m 88000000000.0)
     (* y-scale_m b)
     (* (* x-scale_m 0.25) (* (hypot (* b (sin t_0)) (* a (cos t_0))) 4.0)))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = angle * (0.005555555555555556 * ((double) M_PI));
	double tmp;
	if (x_45_scale_m <= 88000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (x_45_scale_m * 0.25) * (hypot((b * sin(t_0)), (a * cos(t_0))) * 4.0);
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = angle * (0.005555555555555556 * Math.PI);
	double tmp;
	if (x_45_scale_m <= 88000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = (x_45_scale_m * 0.25) * (Math.hypot((b * Math.sin(t_0)), (a * Math.cos(t_0))) * 4.0);
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = angle * (0.005555555555555556 * math.pi)
	tmp = 0
	if x_45_scale_m <= 88000000000.0:
		tmp = y_45_scale_m * b
	else:
		tmp = (x_45_scale_m * 0.25) * (math.hypot((b * math.sin(t_0)), (a * math.cos(t_0))) * 4.0)
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(angle * Float64(0.005555555555555556 * pi))
	tmp = 0.0
	if (x_45_scale_m <= 88000000000.0)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(Float64(x_45_scale_m * 0.25) * Float64(hypot(Float64(b * sin(t_0)), Float64(a * cos(t_0))) * 4.0));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = angle * (0.005555555555555556 * pi);
	tmp = 0.0;
	if (x_45_scale_m <= 88000000000.0)
		tmp = y_45_scale_m * b;
	else
		tmp = (x_45_scale_m * 0.25) * (hypot((b * sin(t_0)), (a * cos(t_0))) * 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 88000000000.0], N[(y$45$scale$95$m * b), $MachinePrecision], N[(N[(x$45$scale$95$m * 0.25), $MachinePrecision] * N[(N[Sqrt[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.8e10

   1. Initial program 2.8%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in angle around 0 18.1%

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

      1. *-commutative 18.1%

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

   7. Simplified 18.1%

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

      1. pow1 18.1%

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

      2. associate-*r* 18.1%

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

      3. sqrt-unprod 18.3%

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

      4. metadata-eval 18.3%

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

      5. metadata-eval 18.3%

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

   9. Applied egg-rr 18.3%

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

       1. unpow1 18.3%

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

       2. associate-*l* 18.3%

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

   11. Simplified 18.3%

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

   12. Taylor expanded in b around 0 18.3%

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

   if 8.8e10 < x-scale

   1. Initial program 4.4%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in y-scale around 0 59.9%

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

      1. associate-*l* 59.9%

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

      2. distribute-lft-out 59.9%

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

      3. fma-define 59.9%

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

      4. *-commutative 59.9%

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

   7. Simplified 59.9%

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

      1. pow1 59.9%

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

   9. Applied egg-rr 67.4%

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

       1. unpow1 67.4%

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

       2. associate-*r* 67.4%

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

       3. metadata-eval 67.4%

          \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{16} \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)} \]

   11. Simplified 67.4%

       \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}} \]
   12. Step-by-step derivation

       1. add-cbrt-cube 67.4%

          \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)} \]

       2. pow3 67.4%

          \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)} \]

   13. Applied egg-rr 67.4%

       \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\sqrt[3]{{\pi}^{3}}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)} \]

   15. Applied egg-rr 72.6%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 88000000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(x-scale \cdot 0.25\right) \cdot \left(\mathsf{hypot}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), a \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot 4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.6% accurate, 8.6× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale\_m \leq 5000000000:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot \mathsf{hypot}\left(a \cdot \cos t\_0, b \cdot \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale_m 5000000000.0)
     (* y-scale_m b)
     (* x-scale_m (hypot (* a (cos t_0)) (* b (sin t_0)))))))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale_m <= 5000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = x_45_scale_m * hypot((a * cos(t_0)), (b * sin(t_0)));
	}
	return tmp;
}

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale_m <= 5000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = x_45_scale_m * Math.hypot((a * Math.cos(t_0)), (b * Math.sin(t_0)));
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale_m <= 5000000000.0:
		tmp = y_45_scale_m * b
	else:
		tmp = x_45_scale_m * math.hypot((a * math.cos(t_0)), (b * math.sin(t_0)))
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale_m <= 5000000000.0)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(x_45_scale_m * hypot(Float64(a * cos(t_0)), Float64(b * sin(t_0))));
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale_m <= 5000000000.0)
		tmp = y_45_scale_m * b;
	else
		tmp = x_45_scale_m * hypot((a * cos(t_0)), (b * sin(t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 5000000000.0], N[(y$45$scale$95$m * b), $MachinePrecision], N[(x$45$scale$95$m * N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 5e9

   1. Initial program 2.8%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in angle around 0 18.1%

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

      1. *-commutative 18.1%

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

   7. Simplified 18.1%

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

      1. pow1 18.1%

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

      2. associate-*r* 18.1%

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

      3. sqrt-unprod 18.3%

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

      4. metadata-eval 18.3%

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

      5. metadata-eval 18.3%

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

   9. Applied egg-rr 18.3%

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

       1. unpow1 18.3%

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

       2. associate-*l* 18.3%

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

   11. Simplified 18.3%

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

   12. Taylor expanded in b around 0 18.3%

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

   if 5e9 < x-scale

   1. Initial program 4.4%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in y-scale around 0 59.9%

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

      1. associate-*l* 59.9%

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

      2. distribute-lft-out 59.9%

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

      3. fma-define 59.9%

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

      4. *-commutative 59.9%

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

   7. Simplified 59.9%

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

      1. pow1 59.9%

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

   9. Applied egg-rr 67.4%

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

       1. unpow1 67.4%

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

       2. associate-*r* 67.4%

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

       3. metadata-eval 67.4%

          \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{16} \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)} \]

   11. Simplified 67.4%

       \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}} \]

   12. Taylor expanded in x-scale around 0 60.0%

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

       1. unpow2 60.0%

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

       2. unpow2 60.0%

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

       3. swap-sqr 60.0%

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

       4. unpow2 60.0%

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

       5. unpow2 60.0%

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

       6. swap-sqr 67.4%

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

       7. hypot-undefine 72.6%

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

   14. Simplified 72.6%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 5000000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 23.7% accurate, 344.0× speedup

Math

\[\begin{array}{l} x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 49000000000:\\ \;\;\;\;y-scale\_m \cdot b\\ \mathbf{else}:\\ \;\;\;\;x-scale\_m \cdot a\\ \end{array} \end{array} \]

FPCore

x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 49000000000.0) (* y-scale_m b) (* x-scale_m a)))

C

x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 49000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Fortran

x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (x_45scale_m <= 49000000000.0d0) then
        tmp = y_45scale_m * b
    else
        tmp = x_45scale_m * a
    end if
    code = tmp
end function

Java

x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 49000000000.0) {
		tmp = y_45_scale_m * b;
	} else {
		tmp = x_45_scale_m * a;
	}
	return tmp;
}

Python

x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 49000000000.0:
		tmp = y_45_scale_m * b
	else:
		tmp = x_45_scale_m * a
	return tmp

Julia

x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 49000000000.0)
		tmp = Float64(y_45_scale_m * b);
	else
		tmp = Float64(x_45_scale_m * a);
	end
	return tmp
end

MATLAB

x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 49000000000.0)
		tmp = y_45_scale_m * b;
	else
		tmp = x_45_scale_m * a;
	end
	tmp_2 = tmp;
end

Wolfram

x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 49000000000.0], N[(y$45$scale$95$m * b), $MachinePrecision], N[(x$45$scale$95$m * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.9e10

   1. Initial program 2.8%

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

   3. Simplified 2.2%

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

   5. Taylor expanded in angle around 0 18.1%

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

      1. *-commutative 18.1%

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

   7. Simplified 18.1%

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

      1. pow1 18.1%

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

      2. associate-*r* 18.1%

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

      3. sqrt-unprod 18.3%

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

      4. metadata-eval 18.3%

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

      5. metadata-eval 18.3%

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

   9. Applied egg-rr 18.3%

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

       1. unpow1 18.3%

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

       2. associate-*l* 18.3%

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

   11. Simplified 18.3%

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

   12. Taylor expanded in b around 0 18.3%

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

   if 4.9e10 < x-scale

   1. Initial program 4.4%

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

   3. Simplified 5.9%

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

   5. Taylor expanded in y-scale around 0 59.9%

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

      1. associate-*l* 59.9%

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

      2. distribute-lft-out 59.9%

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

      3. fma-define 59.9%

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

      4. *-commutative 59.9%

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

   7. Simplified 59.9%

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

      1. pow1 59.9%

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

   9. Applied egg-rr 67.4%

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

       1. unpow1 67.4%

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

       2. associate-*r* 67.4%

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

       3. metadata-eval 67.4%

          \[\leadsto \left(0.25 \cdot x-scale\right) \cdot \sqrt{\color{blue}{16} \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)} \]

   11. Simplified 67.4%

       \[\leadsto \color{blue}{\left(0.25 \cdot x-scale\right) \cdot \sqrt{16 \cdot \left({\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\right)}} \]

   12. Taylor expanded in angle around 0 26.6%

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

       1. *-commutative 26.6%

          \[\leadsto \color{blue}{x-scale \cdot a} \]

   14. Simplified 26.6%

       \[\leadsto \color{blue}{x-scale \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 49000000000:\\ \;\;\;\;y-scale \cdot b\\ \mathbf{else}:\\ \;\;\;\;x-scale \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 17.5% accurate, 919.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.2%

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

3. Simplified 3.2%

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

5. Taylor expanded in angle around 0 15.1%

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

   1. *-commutative 15.1%

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

7. Simplified 15.1%

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

   1. pow1 15.1%

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

   2. associate-*r* 15.1%

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

   3. sqrt-unprod 15.3%

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

   4. metadata-eval 15.3%

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

   5. metadata-eval 15.3%

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

9. Applied egg-rr 15.3%

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

    1. unpow1 15.3%

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

    2. associate-*l* 15.3%

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

11. Simplified 15.3%

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

12. Taylor expanded in b around 0 15.3%

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

13. Final simplification 15.3%

    \[\leadsto y-scale \cdot b \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(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))))