a from scale-rotated-ellipse

Percentage Accurate: 2.6% → 53.0%

Time: 2.2min

Alternatives: 14

Speedup: 919.0×

• 30.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 53.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 2.1 \cdot 10^{-74}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(1 + {angle}^{2} \cdot \left(-1.54320987654321 \cdot 10^{-5} \cdot {\pi}^{2} + {angle}^{2} \cdot \left(-4.083499182202519 \cdot 10^{-17} \cdot \left(\left(angle \cdot angle\right) \cdot {\pi}^{6}\right) + 3.969161205100849 \cdot 10^{-11} \cdot {\pi}^{4}\right)\right)\right), b\_m \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 2.1e-74)
   (* b_m y-scale_m)
   (*
    0.25
    (*
     (* x-scale_m (sqrt 8.0))
     (*
      (sqrt 2.0)
      (hypot
       (*
        a
        (+
         1.0
         (*
          (pow angle 2.0)
          (+
           (* -1.54320987654321e-5 (pow PI 2.0))
           (*
            (pow angle 2.0)
            (+
             (* -4.083499182202519e-17 (* (* angle angle) (pow PI 6.0)))
             (* 3.969161205100849e-11 (pow PI 4.0))))))))
       (* b_m (sin (* 0.005555555555555556 (* angle PI))))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 2.1e-74) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * (1.0 + (pow(angle, 2.0) * ((-1.54320987654321e-5 * pow(((double) M_PI), 2.0)) + (pow(angle, 2.0) * ((-4.083499182202519e-17 * ((angle * angle) * pow(((double) M_PI), 6.0))) + (3.969161205100849e-11 * pow(((double) M_PI), 4.0)))))))), (b_m * sin((0.005555555555555556 * (angle * ((double) M_PI))))))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 2.1e-74) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * Math.hypot((a * (1.0 + (Math.pow(angle, 2.0) * ((-1.54320987654321e-5 * Math.pow(Math.PI, 2.0)) + (Math.pow(angle, 2.0) * ((-4.083499182202519e-17 * ((angle * angle) * Math.pow(Math.PI, 6.0))) + (3.969161205100849e-11 * Math.pow(Math.PI, 4.0)))))))), (b_m * Math.sin((0.005555555555555556 * (angle * Math.PI)))))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 2.1e-74:
		tmp = b_m * y_45_scale_m
	else:
		tmp = 0.25 * ((x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * math.hypot((a * (1.0 + (math.pow(angle, 2.0) * ((-1.54320987654321e-5 * math.pow(math.pi, 2.0)) + (math.pow(angle, 2.0) * ((-4.083499182202519e-17 * ((angle * angle) * math.pow(math.pi, 6.0))) + (3.969161205100849e-11 * math.pow(math.pi, 4.0)))))))), (b_m * math.sin((0.005555555555555556 * (angle * math.pi)))))))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 2.1e-74)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * hypot(Float64(a * Float64(1.0 + Float64((angle ^ 2.0) * Float64(Float64(-1.54320987654321e-5 * (pi ^ 2.0)) + Float64((angle ^ 2.0) * Float64(Float64(-4.083499182202519e-17 * Float64(Float64(angle * angle) * (pi ^ 6.0))) + Float64(3.969161205100849e-11 * (pi ^ 4.0)))))))), Float64(b_m * sin(Float64(0.005555555555555556 * Float64(angle * pi))))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 2.1e-74)
		tmp = b_m * y_45_scale_m;
	else
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * (1.0 + ((angle ^ 2.0) * ((-1.54320987654321e-5 * (pi ^ 2.0)) + ((angle ^ 2.0) * ((-4.083499182202519e-17 * ((angle * angle) * (pi ^ 6.0))) + (3.969161205100849e-11 * (pi ^ 4.0)))))))), (b_m * sin((0.005555555555555556 * (angle * pi)))))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 2.1e-74], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[(1.0 + N[(N[Power[angle, 2.0], $MachinePrecision] * N[(N[(-1.54320987654321e-5 * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Power[angle, 2.0], $MachinePrecision] * N[(N[(-4.083499182202519e-17 * N[(N[(angle * angle), $MachinePrecision] * N[Power[Pi, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(3.969161205100849e-11 * N[Power[Pi, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b$95$m * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.1e-74

   1. Initial program 1.4%

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

   3. Simplified 0.9%

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

   5. Taylor expanded in angle around 0 22.0%

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

      1. pow1 22.0%

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

      2. sqrt-unprod 22.2%

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

      3. metadata-eval 22.2%

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

      4. metadata-eval 22.2%

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

   7. Applied egg-rr 22.2%

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

      1. unpow1 22.2%

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

   9. Simplified 22.2%

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

   10. Taylor expanded in b around 0 22.2%

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

   if 2.1e-74 < x-scale

   1. Initial program 4.5%

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

   3. Simplified 4.6%

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

   5. Taylor expanded in y-scale around 0 51.9%

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

      1. pow1/2 51.9%

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

      2. distribute-lft-out 51.9%

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

      3. unpow-prod-down 51.8%

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

      4. pow1/2 51.8%

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

      5. pow-prod-down 51.8%

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

      6. pow-prod-down 54.5%

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

   7. Applied egg-rr 54.5%

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

      1. unpow1/2 54.5%

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

      2. unpow2 54.5%

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

      3. unpow2 54.5%

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

      4. hypot-define 57.3%

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

   9. Simplified 57.3%

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

   10. Taylor expanded in angle around 0 64.1%

       \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \color{blue}{\left(1 + {angle}^{2} \cdot \left(-1.54320987654321 \cdot 10^{-5} \cdot {\pi}^{2} + {angle}^{2} \cdot \left(-4.083499182202519 \cdot 10^{-17} \cdot \left({angle}^{2} \cdot {\pi}^{6}\right) + 3.969161205100849 \cdot 10^{-11} \cdot {\pi}^{4}\right)\right)\right)}, b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. unpow2 64.1%

          \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(1 + {angle}^{2} \cdot \left(-1.54320987654321 \cdot 10^{-5} \cdot {\pi}^{2} + {angle}^{2} \cdot \left(-4.083499182202519 \cdot 10^{-17} \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{6}\right) + 3.969161205100849 \cdot 10^{-11} \cdot {\pi}^{4}\right)\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right) \]

   12. Applied egg-rr 64.1%

       \[\leadsto 0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \left(1 + {angle}^{2} \cdot \left(-1.54320987654321 \cdot 10^{-5} \cdot {\pi}^{2} + {angle}^{2} \cdot \left(-4.083499182202519 \cdot 10^{-17} \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{6}\right) + 3.969161205100849 \cdot 10^{-11} \cdot {\pi}^{4}\right)\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 52.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 1.25 \cdot 10^{+35}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right), b\_m \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 1.25e+35)
   (* b_m y-scale_m)
   (*
    0.25
    (*
     (* x-scale_m (sqrt 8.0))
     (*
      (sqrt 2.0)
      (hypot
       (*
        a
        (cos
         (* 0.005555555555555556 (* angle (* (cbrt PI) (pow (cbrt PI) 2.0))))))
       (* b_m (* 0.005555555555555556 (* angle PI)))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 1.25e+35) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * cos((0.005555555555555556 * (angle * (cbrt(((double) M_PI)) * pow(cbrt(((double) M_PI)), 2.0)))))), (b_m * (0.005555555555555556 * (angle * ((double) M_PI)))))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 1.25e+35) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * Math.hypot((a * Math.cos((0.005555555555555556 * (angle * (Math.cbrt(Math.PI) * Math.pow(Math.cbrt(Math.PI), 2.0)))))), (b_m * (0.005555555555555556 * (angle * Math.PI))))));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 1.25e+35)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * hypot(Float64(a * cos(Float64(0.005555555555555556 * Float64(angle * Float64(cbrt(pi) * (cbrt(pi) ^ 2.0)))))), Float64(b_m * Float64(0.005555555555555556 * Float64(angle * pi)))))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 1.25e+35], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Cos[N[(0.005555555555555556 * N[(angle * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b$95$m * N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.25000000000000005e35

   1. Initial program 1.4%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in angle around 0 21.6%

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

      1. pow1 21.6%

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

      2. sqrt-unprod 21.8%

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

      3. metadata-eval 21.8%

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

      4. metadata-eval 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. unpow1 21.8%

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

   9. Simplified 21.8%

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

   10. Taylor expanded in b around 0 21.8%

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

   if 1.25000000000000005e35 < x-scale

   1. Initial program 5.6%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in y-scale around 0 60.1%

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

      1. pow1/2 60.1%

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

      2. distribute-lft-out 60.1%

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

      3. unpow-prod-down 60.0%

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

      4. pow1/2 60.0%

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

      5. pow-prod-down 60.0%

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

      6. pow-prod-down 63.6%

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

   7. Applied egg-rr 63.6%

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

      1. unpow1/2 63.6%

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

      2. unpow2 63.6%

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

      3. unpow2 63.6%

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

      4. hypot-define 69.2%

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

   9. Simplified 69.2%

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

       1. add-cube-cbrt 69.2%

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

       2. pow2 69.2%

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

   11. Applied egg-rr 69.2%

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

   12. Taylor expanded in angle around 0 72.8%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.25 \cdot 10^{+35}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt[3]{\pi} \cdot {\left(\sqrt[3]{\pi}\right)}^{2}\right)\right)\right), b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale\_m \leq 1.5 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(b\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot e^{\log 8 \cdot 0.5}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos t\_0, b\_m \cdot \sin t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale_m 1.5e+78)
     (* 0.25 (* y-scale_m (* b_m 4.0)))
     (*
      0.25
      (*
       (* x-scale_m (exp (* (log 8.0) 0.5)))
       (* (sqrt 2.0) (hypot (* a (cos t_0)) (* b_m (sin t_0)))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale_m <= 1.5e+78) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * exp((log(8.0) * 0.5))) * (sqrt(2.0) * hypot((a * cos(t_0)), (b_m * sin(t_0)))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale_m <= 1.5e+78) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * Math.exp((Math.log(8.0) * 0.5))) * (Math.sqrt(2.0) * Math.hypot((a * Math.cos(t_0)), (b_m * Math.sin(t_0)))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale_m <= 1.5e+78:
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0))
	else:
		tmp = 0.25 * ((x_45_scale_m * math.exp((math.log(8.0) * 0.5))) * (math.sqrt(2.0) * math.hypot((a * math.cos(t_0)), (b_m * math.sin(t_0)))))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale_m <= 1.5e+78)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(b_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * exp(Float64(log(8.0) * 0.5))) * Float64(sqrt(2.0) * hypot(Float64(a * cos(t_0)), Float64(b_m * sin(t_0))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale_m <= 1.5e+78)
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	else
		tmp = 0.25 * ((x_45_scale_m * exp((log(8.0) * 0.5))) * (sqrt(2.0) * hypot((a * cos(t_0)), (b_m * sin(t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 1.5e+78], N[(0.25 * N[(y$45$scale$95$m * N[(b$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Exp[N[(N[Log[8.0], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b$95$m * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.49999999999999991e78

   1. Initial program 1.4%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in angle around 0 22.1%

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

      1. pow1 22.1%

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

      2. sqrt-unprod 22.3%

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

      3. metadata-eval 22.3%

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

      4. metadata-eval 22.3%

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

   7. Applied egg-rr 22.3%

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

      1. unpow1 22.3%

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

   9. Simplified 22.3%

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

   10. Taylor expanded in b around 0 22.3%

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

       1. associate-*r* 22.8%

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

   12. Simplified 22.8%

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

   if 1.49999999999999991e78 < x-scale

   1. Initial program 6.4%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in y-scale around 0 66.5%

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

      1. pow1/2 66.5%

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

      2. distribute-lft-out 66.5%

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

      3. unpow-prod-down 66.4%

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

      4. pow1/2 66.4%

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

      5. pow-prod-down 66.4%

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

      6. pow-prod-down 70.6%

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

   7. Applied egg-rr 70.6%

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

      1. unpow1/2 70.6%

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

      2. unpow2 70.6%

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

      3. unpow2 70.6%

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

      4. hypot-define 76.9%

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

   9. Simplified 76.9%

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

       1. pow1/2 76.9%

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

       2. pow-to-exp 77.1%

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

   11. Applied egg-rr 77.1%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.5 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot e^{\log 8 \cdot 0.5}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale\_m \leq 9 \cdot 10^{+77}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(b\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(a \cdot \cos t\_0, b\_m \cdot \sin t\_0\right) \cdot \left(0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale_m 9e+77)
     (* 0.25 (* y-scale_m (* b_m 4.0)))
     (*
      (hypot (* a (cos t_0)) (* b_m (sin t_0)))
      (* 0.25 (* (* x-scale_m (sqrt 8.0)) (sqrt 2.0)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale_m <= 9e+77) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = hypot((a * cos(t_0)), (b_m * sin(t_0))) * (0.25 * ((x_45_scale_m * sqrt(8.0)) * sqrt(2.0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale_m <= 9e+77) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = Math.hypot((a * Math.cos(t_0)), (b_m * Math.sin(t_0))) * (0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * Math.sqrt(2.0)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale_m <= 9e+77:
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0))
	else:
		tmp = math.hypot((a * math.cos(t_0)), (b_m * math.sin(t_0))) * (0.25 * ((x_45_scale_m * math.sqrt(8.0)) * math.sqrt(2.0)))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale_m <= 9e+77)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(b_m * 4.0)));
	else
		tmp = Float64(hypot(Float64(a * cos(t_0)), Float64(b_m * sin(t_0))) * Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * sqrt(2.0))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale_m <= 9e+77)
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	else
		tmp = hypot((a * cos(t_0)), (b_m * sin(t_0))) * (0.25 * ((x_45_scale_m * sqrt(8.0)) * sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 9e+77], N[(0.25 * N[(y$45$scale$95$m * N[(b$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b$95$m * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 9.00000000000000049e77

   1. Initial program 1.4%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in angle around 0 22.1%

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

      1. pow1 22.1%

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

      2. sqrt-unprod 22.3%

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

      3. metadata-eval 22.3%

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

      4. metadata-eval 22.3%

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

   7. Applied egg-rr 22.3%

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

      1. unpow1 22.3%

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

   9. Simplified 22.3%

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

   10. Taylor expanded in b around 0 22.3%

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

       1. associate-*r* 22.8%

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

   12. Simplified 22.8%

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

   if 9.00000000000000049e77 < x-scale

   1. Initial program 6.4%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in y-scale around 0 66.5%

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

      1. pow1/2 66.5%

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

      2. distribute-lft-out 66.5%

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

      3. unpow-prod-down 66.4%

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

      4. pow1/2 66.4%

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

      5. pow-prod-down 66.4%

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

      6. pow-prod-down 70.6%

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

   7. Applied egg-rr 70.6%

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

      1. unpow1/2 70.6%

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

      2. unpow2 70.6%

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

      3. unpow2 70.6%

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

      4. hypot-define 76.9%

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

   9. Simplified 76.9%

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

       1. add-cube-cbrt 76.8%

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

       2. pow2 76.8%

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

   11. Applied egg-rr 76.8%

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

       1. pow1 76.8%

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

   13. Applied egg-rr 76.9%

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

       1. unpow1 76.9%

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

       2. associate-*r* 76.9%

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

       3. *-commutative 76.9%

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

   15. Simplified 76.9%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 9 \cdot 10^{+77}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \sqrt{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.7% accurate, 5.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;x-scale\_m \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(b\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos t\_0, b\_m \cdot \sin t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= x-scale_m 2.6e+77)
     (* 0.25 (* y-scale_m (* b_m 4.0)))
     (*
      0.25
      (*
       (* x-scale_m (sqrt 8.0))
       (* (sqrt 2.0) (hypot (* a (cos t_0)) (* b_m (sin t_0)))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (x_45_scale_m <= 2.6e+77) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * cos(t_0)), (b_m * sin(t_0)))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (x_45_scale_m <= 2.6e+77) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * Math.hypot((a * Math.cos(t_0)), (b_m * Math.sin(t_0)))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if x_45_scale_m <= 2.6e+77:
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0))
	else:
		tmp = 0.25 * ((x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * math.hypot((a * math.cos(t_0)), (b_m * math.sin(t_0)))))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (x_45_scale_m <= 2.6e+77)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(b_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * hypot(Float64(a * cos(t_0)), Float64(b_m * sin(t_0))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (x_45_scale_m <= 2.6e+77)
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	else
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * hypot((a * cos(t_0)), (b_m * sin(t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 2.6e+77], N[(0.25 * N[(y$45$scale$95$m * N[(b$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(b$95$m * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.6000000000000002e77

   1. Initial program 1.4%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in angle around 0 22.1%

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

      1. pow1 22.1%

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

      2. sqrt-unprod 22.3%

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

      3. metadata-eval 22.3%

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

      4. metadata-eval 22.3%

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

   7. Applied egg-rr 22.3%

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

      1. unpow1 22.3%

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

   9. Simplified 22.3%

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

   10. Taylor expanded in b around 0 22.3%

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

       1. associate-*r* 22.8%

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

   12. Simplified 22.8%

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

   if 2.6000000000000002e77 < x-scale

   1. Initial program 6.4%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in y-scale around 0 66.5%

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

      1. pow1/2 66.5%

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

      2. distribute-lft-out 66.5%

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

      3. unpow-prod-down 66.4%

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

      4. pow1/2 66.4%

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

      5. pow-prod-down 66.4%

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

      6. pow-prod-down 70.6%

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

   7. Applied egg-rr 70.6%

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

      1. unpow1/2 70.6%

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

      2. unpow2 70.6%

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

      3. unpow2 70.6%

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

      4. hypot-define 76.9%

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

   9. Simplified 76.9%

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

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot \mathsf{hypot}\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 4.8 \cdot 10^{+77}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(b\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale\_m \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 4.8e+77)
   (* 0.25 (* y-scale_m (* b_m 4.0)))
   (* 0.25 (* (* x-scale_m (sqrt 8.0)) (* (sqrt 2.0) a)))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 4.8e+77) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	}
	return tmp;
}

Fortran

b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    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 <= 4.8d+77) then
        tmp = 0.25d0 * (y_45scale_m * (b_m * 4.0d0))
    else
        tmp = 0.25d0 * ((x_45scale_m * sqrt(8.0d0)) * (sqrt(2.0d0) * a))
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 4.8e+77) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * ((x_45_scale_m * Math.sqrt(8.0)) * (Math.sqrt(2.0) * a));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 4.8e+77:
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0))
	else:
		tmp = 0.25 * ((x_45_scale_m * math.sqrt(8.0)) * (math.sqrt(2.0) * a))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 4.8e+77)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(b_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(x_45_scale_m * sqrt(8.0)) * Float64(sqrt(2.0) * a)));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 4.8e+77)
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	else
		tmp = 0.25 * ((x_45_scale_m * sqrt(8.0)) * (sqrt(2.0) * a));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 4.8e+77], N[(0.25 * N[(y$45$scale$95$m * N[(b$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(x$45$scale$95$m * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 4.7999999999999997e77

   1. Initial program 1.4%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in angle around 0 22.1%

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

      1. pow1 22.1%

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

      2. sqrt-unprod 22.3%

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

      3. metadata-eval 22.3%

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

      4. metadata-eval 22.3%

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

   7. Applied egg-rr 22.3%

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

      1. unpow1 22.3%

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

   9. Simplified 22.3%

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

   10. Taylor expanded in b around 0 22.3%

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

       1. associate-*r* 22.8%

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

   12. Simplified 22.8%

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

   if 4.7999999999999997e77 < x-scale

   1. Initial program 6.4%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in y-scale around 0 66.5%

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

   6. Taylor expanded in angle around 0 33.9%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 4.8 \cdot 10^{+77}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(x-scale \cdot \sqrt{8}\right) \cdot \left(\sqrt{2} \cdot a\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.8% accurate, 12.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 3.2 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(b\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale\_m \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 3.2e+78)
   (* 0.25 (* y-scale_m (* b_m 4.0)))
   (* 0.25 (* a (* x-scale_m (* (sqrt 8.0) (sqrt 2.0)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 3.2e+78) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	}
	return tmp;
}

Fortran

b_m = abs(b)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a, b_m, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    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 <= 3.2d+78) then
        tmp = 0.25d0 * (y_45scale_m * (b_m * 4.0d0))
    else
        tmp = 0.25d0 * (a * (x_45scale_m * (sqrt(8.0d0) * sqrt(2.0d0))))
    end if
    code = tmp
end function

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 3.2e+78) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * (a * (x_45_scale_m * (Math.sqrt(8.0) * Math.sqrt(2.0))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 3.2e+78:
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0))
	else:
		tmp = 0.25 * (a * (x_45_scale_m * (math.sqrt(8.0) * math.sqrt(2.0))))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 3.2e+78)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(b_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(a * Float64(x_45_scale_m * Float64(sqrt(8.0) * sqrt(2.0)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 3.2e+78)
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	else
		tmp = 0.25 * (a * (x_45_scale_m * (sqrt(8.0) * sqrt(2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 3.2e+78], N[(0.25 * N[(y$45$scale$95$m * N[(b$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(a * N[(x$45$scale$95$m * N[(N[Sqrt[8.0], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 3.19999999999999994e78

   1. Initial program 1.4%

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

   3. Simplified 1.0%

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

   5. Taylor expanded in angle around 0 22.1%

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

      1. pow1 22.1%

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

      2. sqrt-unprod 22.3%

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

      3. metadata-eval 22.3%

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

      4. metadata-eval 22.3%

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

   7. Applied egg-rr 22.3%

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

      1. unpow1 22.3%

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

   9. Simplified 22.3%

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

   10. Taylor expanded in b around 0 22.3%

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

       1. associate-*r* 22.8%

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

   12. Simplified 22.8%

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

   if 3.19999999999999994e78 < x-scale

   1. Initial program 6.4%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in b around 0 2.5%

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

   6. Taylor expanded in angle around 0 33.8%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 3.2 \cdot 10^{+78}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{8} \cdot \sqrt{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.6% accurate, 13.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 1.25 \cdot 10^{+132}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(b\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(b\_m \cdot y-scale\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 1.25e+132)
   (* 0.25 (* y-scale_m (* b_m 4.0)))
   (log1p (expm1 (* b_m y-scale_m)))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 1.25e+132) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = log1p(expm1((b_m * y_45_scale_m)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 1.25e+132) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = Math.log1p(Math.expm1((b_m * y_45_scale_m)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 1.25e+132:
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0))
	else:
		tmp = math.log1p(math.expm1((b_m * y_45_scale_m)))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 1.25e+132)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(b_m * 4.0)));
	else
		tmp = log1p(expm1(Float64(b_m * y_45_scale_m)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 1.25e+132], N[(0.25 * N[(y$45$scale$95$m * N[(b$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(b$95$m * y$45$scale$95$m), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.25e132

   1. Initial program 1.9%

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

   3. Simplified 1.4%

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

   5. Taylor expanded in angle around 0 21.6%

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

      1. pow1 21.6%

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

      2. sqrt-unprod 21.8%

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

      3. metadata-eval 21.8%

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

      4. metadata-eval 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. unpow1 21.8%

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

   9. Simplified 21.8%

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

   10. Taylor expanded in b around 0 21.8%

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

       1. associate-*r* 22.3%

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

   12. Simplified 22.3%

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

   if 1.25e132 < x-scale

   1. Initial program 4.9%

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

   3. Simplified 5.0%

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

   5. Taylor expanded in angle around 0 11.1%

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

      1. pow1 11.1%

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

      2. sqrt-unprod 11.2%

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

      3. metadata-eval 11.2%

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

      4. metadata-eval 11.2%

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

   7. Applied egg-rr 11.2%

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

      1. unpow1 11.2%

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

   9. Simplified 11.2%

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

   10. Taylor expanded in b around 0 9.3%

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

       1. log1p-expm1-u 15.8%

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

   12. Applied egg-rr 15.8%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 1.25 \cdot 10^{+132}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(b \cdot y-scale\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.4% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := b\_m \cdot \left(y-scale\_m \cdot 4\right)\\ \mathbf{if}\;x-scale\_m \leq 2.7 \cdot 10^{+172}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(b\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt[3]{t\_0 \cdot \left(t\_0 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* b_m (* y-scale_m 4.0))))
   (if (<= x-scale_m 2.7e+172)
     (* 0.25 (* y-scale_m (* b_m 4.0)))
     (* 0.25 (cbrt (* t_0 (* t_0 t_0)))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if (x_45_scale_m <= 2.7e+172) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * cbrt((t_0 * (t_0 * t_0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = b_m * (y_45_scale_m * 4.0);
	double tmp;
	if (x_45_scale_m <= 2.7e+172) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * Math.cbrt((t_0 * (t_0 * t_0)));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(b_m * Float64(y_45_scale_m * 4.0))
	tmp = 0.0
	if (x_45_scale_m <= 2.7e+172)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(b_m * 4.0)));
	else
		tmp = Float64(0.25 * cbrt(Float64(t_0 * Float64(t_0 * t_0))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(b$95$m * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 2.7e+172], N[(0.25 * N[(y$45$scale$95$m * N[(b$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[Power[N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.7e172

   1. Initial program 2.2%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in angle around 0 21.2%

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

      1. pow1 21.2%

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

      2. sqrt-unprod 21.4%

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

      3. metadata-eval 21.4%

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

      4. metadata-eval 21.4%

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

   7. Applied egg-rr 21.4%

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

      1. unpow1 21.4%

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

   9. Simplified 21.4%

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

   10. Taylor expanded in b around 0 21.4%

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

       1. associate-*r* 21.8%

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

   12. Simplified 21.8%

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

   if 2.7e172 < x-scale

   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 3.3%

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

   5. Taylor expanded in angle around 0 10.9%

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

      1. add-cbrt-cube 13.9%

         \[\leadsto 0.25 \cdot \color{blue}{\sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)}} \]

      2. sqrt-unprod 13.9%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]

      3. metadata-eval 13.9%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]

      4. metadata-eval 13.9%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]

      5. sqrt-unprod 13.9%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]

      6. metadata-eval 13.9%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]

      7. metadata-eval 13.9%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)} \]

      8. sqrt-unprod 13.9%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right)\right)} \]

      9. metadata-eval 13.9%

         \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right)} \]

      10. metadata-eval 13.9%

          \[\leadsto 0.25 \cdot \sqrt[3]{\left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right) \cdot \left(b \cdot \left(y-scale \cdot \color{blue}{4}\right)\right)} \]

   7. Applied egg-rr 13.9%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 2.7 \cdot 10^{+172}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt[3]{\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(\left(b \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(b \cdot \left(y-scale \cdot 4\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.7% accurate, 23.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+20}:\\ \;\;\;\;b\_m \cdot y-scale\_m\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \sqrt[3]{\left(y-scale\_m \cdot 4\right) \cdot \left(\left(y-scale\_m \cdot 4\right) \cdot \left(y-scale\_m \cdot 4\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= a 1.08e+20)
   (* b_m y-scale_m)
   (*
    0.25
    (*
     b_m
     (cbrt (* (* y-scale_m 4.0) (* (* y-scale_m 4.0) (* y-scale_m 4.0))))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.08e+20) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * (b_m * cbrt(((y_45_scale_m * 4.0) * ((y_45_scale_m * 4.0) * (y_45_scale_m * 4.0)))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (a <= 1.08e+20) {
		tmp = b_m * y_45_scale_m;
	} else {
		tmp = 0.25 * (b_m * Math.cbrt(((y_45_scale_m * 4.0) * ((y_45_scale_m * 4.0) * (y_45_scale_m * 4.0)))));
	}
	return tmp;
}

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (a <= 1.08e+20)
		tmp = Float64(b_m * y_45_scale_m);
	else
		tmp = Float64(0.25 * Float64(b_m * cbrt(Float64(Float64(y_45_scale_m * 4.0) * Float64(Float64(y_45_scale_m * 4.0) * Float64(y_45_scale_m * 4.0))))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[a, 1.08e+20], N[(b$95$m * y$45$scale$95$m), $MachinePrecision], N[(0.25 * N[(b$95$m * N[Power[N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(N[(y$45$scale$95$m * 4.0), $MachinePrecision] * N[(y$45$scale$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.08e20

   1. Initial program 2.3%

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

   3. Simplified 1.9%

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

   5. Taylor expanded in angle around 0 22.0%

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

      1. pow1 22.0%

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

      2. sqrt-unprod 22.2%

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

      3. metadata-eval 22.2%

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

      4. metadata-eval 22.2%

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

   7. Applied egg-rr 22.2%

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

      1. unpow1 22.2%

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

   9. Simplified 22.2%

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

   10. Taylor expanded in b around 0 22.2%

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

   if 1.08e20 < a

   1. Initial program 2.4%

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

   3. Simplified 2.4%

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

   5. Taylor expanded in angle around 0 13.2%

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

      1. add-cbrt-cube 17.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \color{blue}{\sqrt[3]{\left(\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}}\right) \]

      2. sqrt-unprod 17.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      3. metadata-eval 17.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot \sqrt{\color{blue}{16}}\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      4. metadata-eval 17.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot \color{blue}{4}\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      5. sqrt-unprod 17.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      6. metadata-eval 17.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      7. metadata-eval 17.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)\right) \cdot \left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      8. sqrt-unprod 17.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(y-scale \cdot \color{blue}{\sqrt{2 \cdot 8}}\right)}\right) \]

      9. metadata-eval 17.7%

         \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(y-scale \cdot \sqrt{\color{blue}{16}}\right)}\right) \]

      10. metadata-eval 17.7%

          \[\leadsto 0.25 \cdot \left(b \cdot \sqrt[3]{\left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot 4\right)\right) \cdot \left(y-scale \cdot \color{blue}{4}\right)}\right) \]

   7. Applied egg-rr 17.7%

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

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.08 \cdot 10^{+20}:\\ \;\;\;\;b \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \sqrt[3]{\left(y-scale \cdot 4\right) \cdot \left(\left(y-scale \cdot 4\right) \cdot \left(y-scale \cdot 4\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.9% accurate, 23.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;x-scale\_m \leq 8.5 \cdot 10^{+172}:\\ \;\;\;\;0.25 \cdot \left(y-scale\_m \cdot \left(b\_m \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot -4\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a b_m angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= x-scale_m 8.5e+172)
   (* 0.25 (* y-scale_m (* b_m 4.0)))
   (*
    0.25
    (*
     b_m
     (* x-scale_m (* (sin (* 0.005555555555555556 (* angle PI))) -4.0))))))

C

b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 8.5e+172) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * (b_m * (x_45_scale_m * (sin((0.005555555555555556 * (angle * ((double) M_PI)))) * -4.0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (x_45_scale_m <= 8.5e+172) {
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	} else {
		tmp = 0.25 * (b_m * (x_45_scale_m * (Math.sin((0.005555555555555556 * (angle * Math.PI))) * -4.0)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a, b_m, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if x_45_scale_m <= 8.5e+172:
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0))
	else:
		tmp = 0.25 * (b_m * (x_45_scale_m * (math.sin((0.005555555555555556 * (angle * math.pi))) * -4.0)))
	return tmp

Julia

b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (x_45_scale_m <= 8.5e+172)
		tmp = Float64(0.25 * Float64(y_45_scale_m * Float64(b_m * 4.0)));
	else
		tmp = Float64(0.25 * Float64(b_m * Float64(x_45_scale_m * Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * -4.0))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (x_45_scale_m <= 8.5e+172)
		tmp = 0.25 * (y_45_scale_m * (b_m * 4.0));
	else
		tmp = 0.25 * (b_m * (x_45_scale_m * (sin((0.005555555555555556 * (angle * pi))) * -4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[x$45$scale$95$m, 8.5e+172], N[(0.25 * N[(y$45$scale$95$m * N[(b$95$m * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(b$95$m * N[(x$45$scale$95$m * N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.50000000000000053e172

   1. Initial program 2.2%

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

   3. Simplified 1.8%

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

   5. Taylor expanded in angle around 0 21.2%

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

      1. pow1 21.2%

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

      2. sqrt-unprod 21.4%

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

      3. metadata-eval 21.4%

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

      4. metadata-eval 21.4%

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

   7. Applied egg-rr 21.4%

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

      1. unpow1 21.4%

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

   9. Simplified 21.4%

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

   10. Taylor expanded in b around 0 21.4%

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

       1. associate-*r* 21.8%

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

   12. Simplified 21.8%

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

   if 8.50000000000000053e172 < x-scale

   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 3.3%

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

   5. Taylor expanded in x-scale around inf 9.0%

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

      1. distribute-lft-out 9.0%

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

      2. +-commutative 9.0%

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

      3. associate-/l* 9.0%

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

      4. associate-/l* 9.0%

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

   7. Simplified 9.0%

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

   8. Taylor expanded in b around -inf 16.6%

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

      1. mul-1-neg 16.6%

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

   10. Simplified 16.6%

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

       1. pow1 16.6%

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

       2. distribute-rgt-neg-out 16.6%

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

       3. associate-*r* 16.5%

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

       4. sqrt-unprod 16.5%

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

       5. metadata-eval 16.5%

          \[\leadsto {\left(-0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{16}}\right)\right)\right)}^{1} \]

       6. metadata-eval 16.5%

          \[\leadsto {\left(-0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{4}\right)\right)\right)}^{1} \]

   12. Applied egg-rr 16.5%

       \[\leadsto \color{blue}{{\left(-0.25 \cdot \left(\left(b \cdot x-scale\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot 4\right)\right)\right)}^{1}} \]
   13. Step-by-step derivation

       1. unpow1 16.5%

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

       2. distribute-rgt-neg-in 16.5%

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

       3. associate-*l* 16.6%

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

       4. distribute-rgt-neg-out 16.6%

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

       5. distribute-rgt-neg-in 16.6%

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

       6. distribute-rgt-neg-in 16.6%

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

       7. metadata-eval 16.6%

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

   14. Simplified 16.6%

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

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x-scale \leq 8.5 \cdot 10^{+172}:\\ \;\;\;\;0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(b \cdot \left(x-scale \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot -4\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.0% accurate, 393.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 2.4%

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

3. Simplified 2.0%

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

5. Taylor expanded in angle around 0 19.8%

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

   1. pow1 19.8%

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

   2. sqrt-unprod 20.0%

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

   3. metadata-eval 20.0%

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

   4. metadata-eval 20.0%

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

7. Applied egg-rr 20.0%

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

   1. unpow1 20.0%

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

9. Simplified 20.0%

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

10. Taylor expanded in b around 0 20.0%

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

    1. associate-*r* 20.4%

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

12. Simplified 20.4%

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

13. Final simplification 20.4%

    \[\leadsto 0.25 \cdot \left(y-scale \cdot \left(b \cdot 4\right)\right) \]
14. Add Preprocessing

Alternative 13: 33.0% accurate, 393.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 2.4%

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

3. Simplified 2.0%

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

5. Taylor expanded in angle around 0 19.8%

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

   1. pow1 19.8%

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

   2. sqrt-unprod 20.0%

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

   3. metadata-eval 20.0%

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

   4. metadata-eval 20.0%

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

7. Applied egg-rr 20.0%

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

   1. unpow1 20.0%

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

9. Simplified 20.0%

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

Alternative 14: 33.0% accurate, 919.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 2.4%

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

3. Simplified 2.0%

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

5. Taylor expanded in angle around 0 19.8%

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

   1. pow1 19.8%

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

   2. sqrt-unprod 20.0%

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

   3. metadata-eval 20.0%

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

   4. metadata-eval 20.0%

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

7. Applied egg-rr 20.0%

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

   1. unpow1 20.0%

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

9. Simplified 20.0%

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

10. Taylor expanded in b around 0 19.7%

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

Reproduce

herbie shell --seed 2024143 
(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))))