a from scale-rotated-ellipse

Percentage Accurate: 2.8% → 56.7%

Time: 31.0s

Alternatives: 8

Speedup: 919.0×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. a = -3.5901897726954e+245
  2. b = -2.3839556038147577e-138
  3. angle = -2.4310771845451752e+86
  4. x-scale = 3.418502812391112e+53
  5. y-scale = -2.6944507909374682e-194

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: 2.8% 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: 56.7% accurate, 6.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{angle}{\frac{180}{\pi}}\\ \mathbf{if}\;y-scale\_m \leq 2.5 \cdot 10^{-13}:\\ \;\;\;\;0.25 \cdot \left(\left(a\_m \cdot x-scale\_m\right) \cdot \left(\left({\left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right)}^{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot \mathsf{hypot}\left(a\_m \cdot \sin t\_0, b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ angle (/ 180.0 PI))))
   (if (<= y-scale_m 2.5e-13)
     (*
      0.25
      (*
       (* a_m x-scale_m)
       (*
        (* (pow (+ 0.5 (* 0.5 (cos (* 2.0 t_0)))) 0.5) (sqrt 2.0))
        (sqrt 8.0))))
     (* y-scale_m (hypot (* a_m (sin t_0)) b)))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = angle / (180.0 / ((double) M_PI));
	double tmp;
	if (y_45_scale_m <= 2.5e-13) {
		tmp = 0.25 * ((a_m * x_45_scale_m) * ((pow((0.5 + (0.5 * cos((2.0 * t_0)))), 0.5) * sqrt(2.0)) * sqrt(8.0)));
	} else {
		tmp = y_45_scale_m * hypot((a_m * sin(t_0)), b);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = angle / (180.0 / Math.PI);
	double tmp;
	if (y_45_scale_m <= 2.5e-13) {
		tmp = 0.25 * ((a_m * x_45_scale_m) * ((Math.pow((0.5 + (0.5 * Math.cos((2.0 * t_0)))), 0.5) * Math.sqrt(2.0)) * Math.sqrt(8.0)));
	} else {
		tmp = y_45_scale_m * Math.hypot((a_m * Math.sin(t_0)), b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	t_0 = angle / (180.0 / math.pi)
	tmp = 0
	if y_45_scale_m <= 2.5e-13:
		tmp = 0.25 * ((a_m * x_45_scale_m) * ((math.pow((0.5 + (0.5 * math.cos((2.0 * t_0)))), 0.5) * math.sqrt(2.0)) * math.sqrt(8.0)))
	else:
		tmp = y_45_scale_m * math.hypot((a_m * math.sin(t_0)), b)
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(angle / Float64(180.0 / pi))
	tmp = 0.0
	if (y_45_scale_m <= 2.5e-13)
		tmp = Float64(0.25 * Float64(Float64(a_m * x_45_scale_m) * Float64(Float64((Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0)))) ^ 0.5) * sqrt(2.0)) * sqrt(8.0))));
	else
		tmp = Float64(y_45_scale_m * hypot(Float64(a_m * sin(t_0)), b));
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	t_0 = angle / (180.0 / pi);
	tmp = 0.0;
	if (y_45_scale_m <= 2.5e-13)
		tmp = 0.25 * ((a_m * x_45_scale_m) * ((((0.5 + (0.5 * cos((2.0 * t_0)))) ^ 0.5) * sqrt(2.0)) * sqrt(8.0)));
	else
		tmp = y_45_scale_m * hypot((a_m * sin(t_0)), b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 2.5e-13], N[(0.25 * N[(N[(a$95$m * x$45$scale$95$m), $MachinePrecision] * N[(N[(N[Power[N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * N[Sqrt[N[(a$95$m * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.49999999999999995e-13

   1. Initial program 3.1%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 3.5%

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

   6. Taylor expanded in x-scale around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(a \cdot \left(x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(a \cdot x-scale\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot x-scale\right), \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \left(\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \left(\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      13. sqrt-lowering-sqrt.f64 20.2%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   8. Simplified 20.2%

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

      1. unpow1 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{1}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

      3. pow-sqr N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}^{\frac{1}{2}}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({\left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}^{\frac{1}{2}}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   10. Applied egg-rr 20.5%

       \[\leadsto 0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\left(\color{blue}{{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right)\right)}^{0.5}} \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right) \]

   if 2.49999999999999995e-13 < y-scale

   1. Initial program 3.2%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 58.6%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}\right), \color{blue}{\left({\left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right) + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}^{\frac{1}{2}}\right)}\right) \]

   7. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(\sin \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot a, \cos \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot b\right)} \]

   9. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), \cos \left(\frac{angle}{\frac{-180}{\pi}}\right) \cdot b\right) \cdot y-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, b\right)\right), y-scale\right) \]
   11. Step-by-step derivation

   12. Simplified 69.3%

       \[\leadsto \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), \color{blue}{1} \cdot b\right) \cdot y-scale \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.5 \cdot 10^{-13}:\\ \;\;\;\;0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\left({\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \frac{angle}{\frac{180}{\pi}}\right)\right)}^{0.5} \cdot \sqrt{2}\right) \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 56.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 8 \cdot 10^{-14}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(a\_m \cdot x-scale\_m\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot \mathsf{hypot}\left(a\_m \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 8e-14)
   (*
    (* 0.25 (* (* a_m x-scale_m) (* (sqrt 2.0) (sqrt 8.0))))
    (sqrt (+ 0.5 (* 0.5 (cos (* (* angle PI) 0.011111111111111112))))))
   (* y-scale_m (hypot (* a_m (sin (/ angle (/ 180.0 PI)))) b))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 8e-14) {
		tmp = (0.25 * ((a_m * x_45_scale_m) * (sqrt(2.0) * sqrt(8.0)))) * sqrt((0.5 + (0.5 * cos(((angle * ((double) M_PI)) * 0.011111111111111112)))));
	} else {
		tmp = y_45_scale_m * hypot((a_m * sin((angle / (180.0 / ((double) M_PI))))), b);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 8e-14) {
		tmp = (0.25 * ((a_m * x_45_scale_m) * (Math.sqrt(2.0) * Math.sqrt(8.0)))) * Math.sqrt((0.5 + (0.5 * Math.cos(((angle * Math.PI) * 0.011111111111111112)))));
	} else {
		tmp = y_45_scale_m * Math.hypot((a_m * Math.sin((angle / (180.0 / Math.PI)))), b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 8e-14:
		tmp = (0.25 * ((a_m * x_45_scale_m) * (math.sqrt(2.0) * math.sqrt(8.0)))) * math.sqrt((0.5 + (0.5 * math.cos(((angle * math.pi) * 0.011111111111111112)))))
	else:
		tmp = y_45_scale_m * math.hypot((a_m * math.sin((angle / (180.0 / math.pi)))), b)
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 8e-14)
		tmp = Float64(Float64(0.25 * Float64(Float64(a_m * x_45_scale_m) * Float64(sqrt(2.0) * sqrt(8.0)))) * sqrt(Float64(0.5 + Float64(0.5 * cos(Float64(Float64(angle * pi) * 0.011111111111111112))))));
	else
		tmp = Float64(y_45_scale_m * hypot(Float64(a_m * sin(Float64(angle / Float64(180.0 / pi)))), b));
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 8e-14)
		tmp = (0.25 * ((a_m * x_45_scale_m) * (sqrt(2.0) * sqrt(8.0)))) * sqrt((0.5 + (0.5 * cos(((angle * pi) * 0.011111111111111112)))));
	else
		tmp = y_45_scale_m * hypot((a_m * sin((angle / (180.0 / pi)))), b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 8e-14], N[(N[(0.25 * N[(N[(a$95$m * x$45$scale$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(0.5 + N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * N[Sqrt[N[(a$95$m * N[Sin[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7.99999999999999999e-14

   1. Initial program 3.1%

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

   4. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\left(\left(-\left(a \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot \frac{8 \cdot \left(-\left(a \cdot \left(b \cdot b\right)\right) \cdot a\right)}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right) \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) + b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)}{x-scale \cdot x-scale} + \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)}{y-scale \cdot y-scale} + \mathsf{hypot}\left(\frac{\left(a \cdot a\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) + b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(b \cdot b - a \cdot a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}{x-scale \cdot y-scale}\right)\right)\right)}}{\left(-\left(a \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot -4} \cdot \left(x-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot y-scale\right)} \]

   5. Taylor expanded in x-scale around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right)\right)}, \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right), \mathsf{*.f64}\left(\color{blue}{x-scale}, y-scale\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \sqrt{8}\right), \left(\sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right)\right), \mathsf{*.f64}\left(\color{blue}{x-scale}, y-scale\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\sqrt{8}\right)\right), \left(\sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right)\right), \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(8\right)\right), \left(\sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right)\right), \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(8\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}\right)\right)\right), \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(8\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]

   7. Simplified 5.6%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left(\frac{\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale} + \frac{\left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}\right)} \cdot \left(x-scale \cdot y-scale\right) \]

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \left(\left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) \cdot \color{blue}{\sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right), \color{blue}{\left(\sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right), \left(\sqrt{\color{blue}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right), \left(\sqrt{\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot x-scale\right), \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right), \left(\sqrt{\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right), \left(\sqrt{\frac{1}{2} + \color{blue}{\frac{1}{2}} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\left(\sqrt{2}\right), \left(\sqrt{8}\right)\right)\right)\right), \left(\sqrt{\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{8}\right)\right)\right)\right), \left(\sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right), \left(\sqrt{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right)\right) \]

      17. PI-lowering-PI.f64 20.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right), \frac{1}{90}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 20.5%

       \[\leadsto \color{blue}{\left(0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}} \]

   if 7.99999999999999999e-14 < y-scale

   1. Initial program 3.2%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 58.6%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}\right), \color{blue}{\left({\left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right) + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}^{\frac{1}{2}}\right)}\right) \]

   7. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(\sin \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot a, \cos \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot b\right)} \]

   9. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), \cos \left(\frac{angle}{\frac{-180}{\pi}}\right) \cdot b\right) \cdot y-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, b\right)\right), y-scale\right) \]
   11. Step-by-step derivation

   12. Simplified 69.3%

       \[\leadsto \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), \color{blue}{1} \cdot b\right) \cdot y-scale \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 8 \cdot 10^{-14}:\\ \;\;\;\;\left(0.25 \cdot \left(\left(a \cdot x-scale\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.6% accurate, 12.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;y-scale\_m \leq 2.2 \cdot 10^{-14}:\\ \;\;\;\;\left(0.25 \cdot a\_m\right) \cdot \left(\sqrt{8} \cdot \left(x-scale\_m \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot \mathsf{hypot}\left(a\_m \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), b\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= y-scale_m 2.2e-14)
   (* (* 0.25 a_m) (* (sqrt 8.0) (* x-scale_m (sqrt 2.0))))
   (* y-scale_m (hypot (* a_m (sin (/ angle (/ 180.0 PI)))) b))))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.2e-14) {
		tmp = (0.25 * a_m) * (sqrt(8.0) * (x_45_scale_m * sqrt(2.0)));
	} else {
		tmp = y_45_scale_m * hypot((a_m * sin((angle / (180.0 / ((double) M_PI))))), b);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (y_45_scale_m <= 2.2e-14) {
		tmp = (0.25 * a_m) * (Math.sqrt(8.0) * (x_45_scale_m * Math.sqrt(2.0)));
	} else {
		tmp = y_45_scale_m * Math.hypot((a_m * Math.sin((angle / (180.0 / Math.PI)))), b);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if y_45_scale_m <= 2.2e-14:
		tmp = (0.25 * a_m) * (math.sqrt(8.0) * (x_45_scale_m * math.sqrt(2.0)))
	else:
		tmp = y_45_scale_m * math.hypot((a_m * math.sin((angle / (180.0 / math.pi)))), b)
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (y_45_scale_m <= 2.2e-14)
		tmp = Float64(Float64(0.25 * a_m) * Float64(sqrt(8.0) * Float64(x_45_scale_m * sqrt(2.0))));
	else
		tmp = Float64(y_45_scale_m * hypot(Float64(a_m * sin(Float64(angle / Float64(180.0 / pi)))), b));
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (y_45_scale_m <= 2.2e-14)
		tmp = (0.25 * a_m) * (sqrt(8.0) * (x_45_scale_m * sqrt(2.0)));
	else
		tmp = y_45_scale_m * hypot((a_m * sin((angle / (180.0 / pi)))), b);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[y$45$scale$95$m, 2.2e-14], N[(N[(0.25 * a$95$m), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * N[Sqrt[N[(a$95$m * N[Sin[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + b ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 2.2000000000000001e-14

   1. Initial program 3.1%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 3.5%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot a\right), \color{blue}{\left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \left(\color{blue}{x-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \mathsf{*.f64}\left(\left(x-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 20.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

   8. Simplified 20.3%

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

   if 2.2000000000000001e-14 < y-scale

   1. Initial program 3.2%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 58.6%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}\right), \color{blue}{\left({\left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right) + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}^{\frac{1}{2}}\right)}\right) \]

   7. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(\sin \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot a, \cos \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot b\right)} \]

   9. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), \cos \left(\frac{angle}{\frac{-180}{\pi}}\right) \cdot b\right) \cdot y-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{hypot.f64}\left(\mathsf{*.f64}\left(a, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(angle, \mathsf{/.f64}\left(180, \mathsf{PI.f64}\left(\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, b\right)\right), y-scale\right) \]
   11. Step-by-step derivation

   12. Simplified 69.3%

       \[\leadsto \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), \color{blue}{1} \cdot b\right) \cdot y-scale \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y-scale \leq 2.2 \cdot 10^{-14}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot \mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-6}:\\ \;\;\;\;\left(0.25 \cdot a\_m\right) \cdot \left(\sqrt{8} \cdot \left(x-scale\_m \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 1e-6)
   (* (* 0.25 a_m) (* (sqrt 8.0) (* x-scale_m (sqrt 2.0))))
   (* y-scale_m b)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1e-6) {
		tmp = (0.25 * a_m) * (sqrt(8.0) * (x_45_scale_m * sqrt(2.0)));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Fortran

a_m = abs(a)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b <= 1d-6) then
        tmp = (0.25d0 * a_m) * (sqrt(8.0d0) * (x_45scale_m * sqrt(2.0d0)))
    else
        tmp = y_45scale_m * b
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1e-6) {
		tmp = (0.25 * a_m) * (Math.sqrt(8.0) * (x_45_scale_m * Math.sqrt(2.0)));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 1e-6:
		tmp = (0.25 * a_m) * (math.sqrt(8.0) * (x_45_scale_m * math.sqrt(2.0)))
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 1e-6)
		tmp = Float64(Float64(0.25 * a_m) * Float64(sqrt(8.0) * Float64(x_45_scale_m * sqrt(2.0))));
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 1e-6)
		tmp = (0.25 * a_m) * (sqrt(8.0) * (x_45_scale_m * sqrt(2.0)));
	else
		tmp = y_45_scale_m * b;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 1e-6], N[(N[(0.25 * a$95$m), $MachinePrecision] * N[(N[Sqrt[8.0], $MachinePrecision] * N[(x$45$scale$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.99999999999999955e-7

   1. Initial program 2.6%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 3.4%

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

   6. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{4} \cdot a\right) \cdot \color{blue}{\left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot a\right), \color{blue}{\left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \left(\color{blue}{x-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \left(\left(x-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \mathsf{*.f64}\left(\left(x-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 20.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

   8. Simplified 20.9%

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

   if 9.99999999999999955e-7 < b

   1. Initial program 4.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot b\right), \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\color{blue}{y-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 24.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

   5. Simplified 24.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right), \color{blue}{b}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right), \frac{1}{4}\right), b\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right), \frac{1}{4}\right), b\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2 \cdot 8}\right), \frac{1}{4}\right), b\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{16}\right), \frac{1}{4}\right), b\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot 4\right), \frac{1}{4}\right), b\right) \]

      9. *-lowering-*.f64 24.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \frac{1}{4}\right), b\right) \]

   7. Applied egg-rr 24.6%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot \left(4 \cdot \frac{1}{4}\right)\right), b\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot 1\right), b\right) \]

      3. *-rgt-identity 24.6%

         \[\leadsto \mathsf{*.f64}\left(y-scale, b\right) \]

   9. Applied egg-rr 24.6%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-6}:\\ \;\;\;\;\left(0.25 \cdot a\right) \cdot \left(\sqrt{8} \cdot \left(x-scale \cdot \sqrt{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 41.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;0.25 \cdot \left(\left(a\_m \cdot x-scale\_m\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 1.75e-6)
   (* 0.25 (* (* a_m x-scale_m) (* (sqrt 2.0) (sqrt 8.0))))
   (* y-scale_m b)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1.75e-6) {
		tmp = 0.25 * ((a_m * x_45_scale_m) * (sqrt(2.0) * sqrt(8.0)));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Fortran

a_m = abs(a)
x-scale_m = abs(x_45scale)
y-scale_m = abs(y_45scale)
real(8) function code(a_m, b, angle, x_45scale_m, y_45scale_m)
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    real(8) :: tmp
    if (b <= 1.75d-6) then
        tmp = 0.25d0 * ((a_m * x_45scale_m) * (sqrt(2.0d0) * sqrt(8.0d0)))
    else
        tmp = y_45scale_m * b
    end if
    code = tmp
end function

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1.75e-6) {
		tmp = 0.25 * ((a_m * x_45_scale_m) * (Math.sqrt(2.0) * Math.sqrt(8.0)));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 1.75e-6:
		tmp = 0.25 * ((a_m * x_45_scale_m) * (math.sqrt(2.0) * math.sqrt(8.0)))
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 1.75e-6)
		tmp = Float64(0.25 * Float64(Float64(a_m * x_45_scale_m) * Float64(sqrt(2.0) * sqrt(8.0))));
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 1.75e-6)
		tmp = 0.25 * ((a_m * x_45_scale_m) * (sqrt(2.0) * sqrt(8.0)));
	else
		tmp = y_45_scale_m * b;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 1.75e-6], N[(0.25 * N[(N[(a$95$m * x$45$scale$95$m), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.74999999999999997e-6

   1. Initial program 2.6%

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

   4. Applied egg-rr 1.8%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\left(\left(-\left(a \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot \frac{8 \cdot \left(-\left(a \cdot \left(b \cdot b\right)\right) \cdot a\right)}{y-scale \cdot \left(x-scale \cdot \left(x-scale \cdot y-scale\right)\right)}\right) \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) + b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)}{x-scale \cdot x-scale} + \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)}{y-scale \cdot y-scale} + \mathsf{hypot}\left(\frac{\left(a \cdot a\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) + b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right) + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(b \cdot b - a \cdot a\right) \cdot \sin \left(2 \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}{x-scale \cdot y-scale}\right)\right)\right)}}{\left(-\left(a \cdot \left(b \cdot b\right)\right) \cdot a\right) \cdot -4} \cdot \left(x-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot y-scale\right)} \]

   5. Taylor expanded in x-scale around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot \left(\sqrt{8} \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right)\right)}, \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right), \mathsf{*.f64}\left(\color{blue}{x-scale}, y-scale\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \sqrt{8}\right), \left(\sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right)\right), \mathsf{*.f64}\left(\color{blue}{x-scale}, y-scale\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\sqrt{8}\right)\right), \left(\sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right)\right), \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]

      4. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(8\right)\right), \left(\sqrt{2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}}\right)\right), \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(8\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + 2 \cdot \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}\right)\right)\right), \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]

      6. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(8\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\frac{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}} + \frac{{b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}{{y-scale}^{2}}\right)\right)\right)\right), \mathsf{*.f64}\left(x-scale, y-scale\right)\right) \]

   7. Simplified 6.3%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \sqrt{8}\right) \cdot \sqrt{2 \cdot \left(\frac{\left(b \cdot b\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale} + \frac{\left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{y-scale \cdot y-scale}\right)}\right)} \cdot \left(x-scale \cdot y-scale\right) \]

   8. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(a \cdot \left(x-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(a \cdot x-scale\right) \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot x-scale\right), \color{blue}{\left(\sqrt{2} \cdot \sqrt{8}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \left(\color{blue}{\sqrt{2}} \cdot \sqrt{8}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\left(\sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 20.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   10. Simplified 20.8%

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

   if 1.74999999999999997e-6 < b

   1. Initial program 4.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot b\right), \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\color{blue}{y-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 24.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

   5. Simplified 24.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right), \color{blue}{b}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right), \frac{1}{4}\right), b\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right), \frac{1}{4}\right), b\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2 \cdot 8}\right), \frac{1}{4}\right), b\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{16}\right), \frac{1}{4}\right), b\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot 4\right), \frac{1}{4}\right), b\right) \]

      9. *-lowering-*.f64 24.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \frac{1}{4}\right), b\right) \]

   7. Applied egg-rr 24.6%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot \left(4 \cdot \frac{1}{4}\right)\right), b\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot 1\right), b\right) \]

      3. *-rgt-identity 24.6%

         \[\leadsto \mathsf{*.f64}\left(y-scale, b\right) \]

   9. Applied egg-rr 24.6%

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

Alternative 6: 39.8% accurate, 23.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;0.25 \cdot \left(x-scale\_m \cdot \left(a\_m \cdot \left(\cos \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 1.2e-6)
   (* 0.25 (* x-scale_m (* a_m (* (cos (/ angle (/ 180.0 PI))) 4.0))))
   (* y-scale_m b)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1.2e-6) {
		tmp = 0.25 * (x_45_scale_m * (a_m * (cos((angle / (180.0 / ((double) M_PI)))) * 4.0)));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1.2e-6) {
		tmp = 0.25 * (x_45_scale_m * (a_m * (Math.cos((angle / (180.0 / Math.PI))) * 4.0)));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 1.2e-6:
		tmp = 0.25 * (x_45_scale_m * (a_m * (math.cos((angle / (180.0 / math.pi))) * 4.0)))
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 1.2e-6)
		tmp = Float64(0.25 * Float64(x_45_scale_m * Float64(a_m * Float64(cos(Float64(angle / Float64(180.0 / pi))) * 4.0))));
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 1.2e-6)
		tmp = 0.25 * (x_45_scale_m * (a_m * (cos((angle / (180.0 / pi))) * 4.0)));
	else
		tmp = y_45_scale_m * b;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 1.2e-6], N[(0.25 * N[(x$45$scale$95$m * N[(a$95$m * N[(N[Cos[N[(angle / N[(180.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.1999999999999999e-6

   1. Initial program 2.6%

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

   3. Taylor expanded in b around 0

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

   5. Simplified 3.4%

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

   6. Taylor expanded in x-scale around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(a \cdot \left(x-scale \cdot \left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(a \cdot x-scale\right) \cdot \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(a \cdot x-scale\right), \color{blue}{\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \left(\color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \left(\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right)\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI}\left(\right)\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \left(\sqrt{2}\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right)\right) \]

      13. sqrt-lowering-sqrt.f64 21.1%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(a, x-scale\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{180}, \mathsf{*.f64}\left(angle, \mathsf{PI.f64}\left(\right)\right)\right)\right), \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right)\right) \]

   8. Simplified 21.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \color{blue}{\left(a \cdot x-scale\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \left(\left(\left(\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right) \cdot \color{blue}{x-scale}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\left(\left(\left(\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot a\right), \color{blue}{x-scale}\right)\right) \]

   10. Applied egg-rr 21.4%

       \[\leadsto 0.25 \cdot \color{blue}{\left(\left(\left(\cos \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot 4\right) \cdot a\right) \cdot x-scale\right)} \]

   if 1.1999999999999999e-6 < b

   1. Initial program 4.6%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot b\right), \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\color{blue}{y-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 24.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

   5. Simplified 24.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right), \color{blue}{b}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right), \frac{1}{4}\right), b\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right), \frac{1}{4}\right), b\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2 \cdot 8}\right), \frac{1}{4}\right), b\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{16}\right), \frac{1}{4}\right), b\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot 4\right), \frac{1}{4}\right), b\right) \]

      9. *-lowering-*.f64 24.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \frac{1}{4}\right), b\right) \]

   7. Applied egg-rr 24.6%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot \left(4 \cdot \frac{1}{4}\right)\right), b\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot 1\right), b\right) \]

      3. *-rgt-identity 24.6%

         \[\leadsto \mathsf{*.f64}\left(y-scale, b\right) \]

   9. Applied egg-rr 24.6%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;0.25 \cdot \left(x-scale \cdot \left(a \cdot \left(\cos \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot 4\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 24.8% accurate, 86.1× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+150}:\\ \;\;\;\;y-scale\_m \cdot \left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(-3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left(a\_m \cdot a\_m\right)\right)\right)\right)}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale\_m \cdot b\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b angle x-scale_m y-scale_m)
 :precision binary64
 (if (<= b 1.6e+150)
   (*
    y-scale_m
    (+
     b
     (/
      (*
       0.5
       (*
        (* angle angle)
        (*
         (* PI PI)
         (+
          (* -3.08641975308642e-5 (* b b))
          (* 3.08641975308642e-5 (* a_m a_m))))))
      b)))
   (* y-scale_m b)))

C

a_m = fabs(a);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1.6e+150) {
		tmp = y_45_scale_m * (b + ((0.5 * ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * ((-3.08641975308642e-5 * (b * b)) + (3.08641975308642e-5 * (a_m * a_m)))))) / b));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Java

a_m = Math.abs(a);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b, double angle, double x_45_scale_m, double y_45_scale_m) {
	double tmp;
	if (b <= 1.6e+150) {
		tmp = y_45_scale_m * (b + ((0.5 * ((angle * angle) * ((Math.PI * Math.PI) * ((-3.08641975308642e-5 * (b * b)) + (3.08641975308642e-5 * (a_m * a_m)))))) / b));
	} else {
		tmp = y_45_scale_m * b;
	}
	return tmp;
}

Python

a_m = math.fabs(a)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b, angle, x_45_scale_m, y_45_scale_m):
	tmp = 0
	if b <= 1.6e+150:
		tmp = y_45_scale_m * (b + ((0.5 * ((angle * angle) * ((math.pi * math.pi) * ((-3.08641975308642e-5 * (b * b)) + (3.08641975308642e-5 * (a_m * a_m)))))) / b))
	else:
		tmp = y_45_scale_m * b
	return tmp

Julia

a_m = abs(a)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0
	if (b <= 1.6e+150)
		tmp = Float64(y_45_scale_m * Float64(b + Float64(Float64(0.5 * Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * Float64(Float64(-3.08641975308642e-5 * Float64(b * b)) + Float64(3.08641975308642e-5 * Float64(a_m * a_m)))))) / b)));
	else
		tmp = Float64(y_45_scale_m * b);
	end
	return tmp
end

MATLAB

a_m = abs(a);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b, angle, x_45_scale_m, y_45_scale_m)
	tmp = 0.0;
	if (b <= 1.6e+150)
		tmp = y_45_scale_m * (b + ((0.5 * ((angle * angle) * ((pi * pi) * ((-3.08641975308642e-5 * (b * b)) + (3.08641975308642e-5 * (a_m * a_m)))))) / b));
	else
		tmp = y_45_scale_m * b;
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := If[LessEqual[b, 1.6e+150], N[(y$45$scale$95$m * N[(b + N[(N[(0.5 * N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(-3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(3.08641975308642e-5 * N[(a$95$m * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$45$scale$95$m * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.60000000000000008e150

   1. Initial program 2.3%

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

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right), \color{blue}{\left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(y-scale \cdot \sqrt{8}\right)\right), \left(\sqrt{\color{blue}{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \left(\sqrt{8}\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + \color{blue}{2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \left(\sqrt{2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \color{blue}{\left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)}}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right) + 2 \cdot \left({b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\left(2 \cdot \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(8\right)\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({a}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {b}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)\right)\right) \]

   5. Simplified 24.5%

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

      1. sqrt-prod N/A

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

      2. pow1/2 N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\frac{1}{4} \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}\right), \color{blue}{\left({\left({\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(b \cdot b\right) + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)}^{\frac{1}{2}}\right)}\right) \]

   7. Applied egg-rr 26.3%

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \left(y-scale \cdot \sqrt{8}\right)\right) \cdot \sqrt{2}\right) \cdot \mathsf{hypot}\left(\sin \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot a, \cos \left(\frac{angle}{\frac{180}{\pi}}\right) \cdot b\right)} \]

   9. Applied egg-rr 26.4%

      \[\leadsto \color{blue}{\mathsf{hypot}\left(a \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right), \cos \left(\frac{angle}{\frac{-180}{\pi}}\right) \cdot b\right) \cdot y-scale} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(b + \frac{1}{2} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{b}\right)}, y-scale\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{1}{2} \cdot \frac{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}{b}\right)\right), y-scale\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \left(\frac{\frac{1}{2} \cdot \left({angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}{b}\right)\right), y-scale\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(b, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left({angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right), b\right)\right), y-scale\right) \]

   12. Simplified 19.9%

       \[\leadsto \color{blue}{\left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(-3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\right)}{b}\right)} \cdot y-scale \]

   if 1.60000000000000008e150 < b

   1. Initial program 7.7%

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

   3. Taylor expanded in angle around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot b\right), \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\color{blue}{y-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

      7. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 34.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

   5. Simplified 34.1%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right), \color{blue}{b}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right), \frac{1}{4}\right), b\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right), \frac{1}{4}\right), b\right) \]

      6. sqrt-unprod N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2 \cdot 8}\right), \frac{1}{4}\right), b\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{16}\right), \frac{1}{4}\right), b\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot 4\right), \frac{1}{4}\right), b\right) \]

      9. *-lowering-*.f64 34.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \frac{1}{4}\right), b\right) \]

   7. Applied egg-rr 34.3%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot \left(4 \cdot \frac{1}{4}\right)\right), b\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot 1\right), b\right) \]

      3. *-rgt-identity 34.3%

         \[\leadsto \mathsf{*.f64}\left(y-scale, b\right) \]

   9. Applied egg-rr 34.3%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.6 \cdot 10^{+150}:\\ \;\;\;\;y-scale \cdot \left(b + \frac{0.5 \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \left(-3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right) + 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right)\right)}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;y-scale \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 19.4% accurate, 919.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 3.1%

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

3. Taylor expanded in angle around 0

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

   1. associate-*r* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{4} \cdot b\right), \color{blue}{\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\color{blue}{y-scale} \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \left(\left(y-scale \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{8}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2}\right), \color{blue}{\left(\sqrt{8}\right)}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \left(\sqrt{2}\right)\right), \left(\sqrt{\color{blue}{8}}\right)\right)\right) \]

   7. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \left(\sqrt{8}\right)\right)\right) \]

   8. sqrt-lowering-sqrt.f64 19.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, \mathsf{sqrt.f64}\left(2\right)\right), \mathsf{sqrt.f64}\left(8\right)\right)\right) \]

5. Simplified 19.4%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right) \cdot \frac{1}{4}\right), \color{blue}{b}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(y-scale \cdot \sqrt{2}\right) \cdot \sqrt{8}\right), \frac{1}{4}\right), b\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \left(\sqrt{2} \cdot \sqrt{8}\right)\right), \frac{1}{4}\right), b\right) \]

   6. sqrt-unprod N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{2 \cdot 8}\right), \frac{1}{4}\right), b\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot \sqrt{16}\right), \frac{1}{4}\right), b\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y-scale \cdot 4\right), \frac{1}{4}\right), b\right) \]

   9. *-lowering-*.f64 19.6%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y-scale, 4\right), \frac{1}{4}\right), b\right) \]

7. Applied egg-rr 19.6%

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

   1. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot \left(4 \cdot \frac{1}{4}\right)\right), b\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left(y-scale \cdot 1\right), b\right) \]

   3. *-rgt-identity 19.6%

      \[\leadsto \mathsf{*.f64}\left(y-scale, b\right) \]

9. Applied egg-rr 19.6%

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

Reproduce

herbie shell --seed 2024192 
(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))))