a from scale-rotated-ellipse

Percentage Accurate: 3.0% → 25.7%

Time: 38.3s

Alternatives: 7

Speedup: 12.0×

• 30.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 3.0% 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: 25.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \frac{1}{{y-scale\_m}^{2}}\\ t_1 := 0.011111111111111112 \cdot \left(angle \cdot \pi\right)\\ t_2 := \cos t\_1\\ t_3 := 0.5 \cdot t\_2\\ t_4 := \frac{1}{{x-scale\_m}^{2}}\\ t_5 := \left|x-scale\_m \cdot y-scale\_m\right|\\ t_6 := \frac{t\_2}{{x-scale\_m}^{2}}\\ t_7 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_8 := 0.5 \cdot \frac{t\_2}{{y-scale\_m}^{2}}\\ \mathbf{if}\;x-scale\_m \leq 4.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{a\_m \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left({y-scale\_m}^{2} \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_3\right)}^{2}} + t\_3\right)\right)}\right)\right)\right)}{t\_5}\\ \mathbf{elif}\;x-scale\_m \leq 3.9 \cdot 10^{+80}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left({x-scale\_m}^{2} \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_7}^{4}}{{x-scale\_m}^{4}}} + \frac{{t\_7}^{2}}{{x-scale\_m}^{2}}}{{x-scale\_m}^{2}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.25 \cdot \frac{a\_m \cdot \left(x-scale\_m \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin t\_1}^{2}}{{x-scale\_m}^{2} \cdot {y-scale\_m}^{2}} + {\left(0.5 \cdot t\_4 - \mathsf{fma}\left(0.5, t\_0, \mathsf{fma}\left(0.5, t\_6, t\_8\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, t\_4, \mathsf{fma}\left(0.5, t\_0, t\_8\right)\right)\right) - 0.5 \cdot t\_6\right)}\right)\right)}{t\_5}\right) \cdot \left(y-scale\_m \cdot x-scale\_m\right)\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (/ 1.0 (pow y-scale_m 2.0)))
        (t_1 (* 0.011111111111111112 (* angle PI)))
        (t_2 (cos t_1))
        (t_3 (* 0.5 t_2))
        (t_4 (/ 1.0 (pow x-scale_m 2.0)))
        (t_5 (fabs (* x-scale_m y-scale_m)))
        (t_6 (/ t_2 (pow x-scale_m 2.0)))
        (t_7 (cos (* 0.005555555555555556 (* angle PI))))
        (t_8 (* 0.5 (/ t_2 (pow y-scale_m 2.0)))))
   (if (<= x-scale_m 4.6e-115)
     (*
      (/ 0.25 a_m)
      (/
       (*
        a_m
        (*
         b_m
         (*
          x-scale_m
          (*
           (pow y-scale_m 2.0)
           (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_3) 2.0)) t_3))))))))
       t_5))
     (if (<= x-scale_m 3.9e+80)
       (*
        0.25
        (*
         b_m
         (*
          (pow x-scale_m 2.0)
          (*
           y-scale_m
           (sqrt
            (*
             8.0
             (/
              (+
               (sqrt (/ (pow t_7 4.0) (pow x-scale_m 4.0)))
               (/ (pow t_7 2.0) (pow x-scale_m 2.0)))
              (pow x-scale_m 2.0))))))))
       (*
        (*
         0.25
         (/
          (*
           a_m
           (*
            x-scale_m
            (*
             y-scale_m
             (sqrt
              (*
               8.0
               (-
                (+
                 (sqrt
                  (+
                   (/
                    (pow (sin t_1) 2.0)
                    (* (pow x-scale_m 2.0) (pow y-scale_m 2.0)))
                   (pow (- (* 0.5 t_4) (fma 0.5 t_0 (fma 0.5 t_6 t_8))) 2.0)))
                 (fma 0.5 t_4 (fma 0.5 t_0 t_8)))
                (* 0.5 t_6)))))))
          t_5))
        (* y-scale_m x-scale_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 1.0 / pow(y_45_scale_m, 2.0);
	double t_1 = 0.011111111111111112 * (angle * ((double) M_PI));
	double t_2 = cos(t_1);
	double t_3 = 0.5 * t_2;
	double t_4 = 1.0 / pow(x_45_scale_m, 2.0);
	double t_5 = fabs((x_45_scale_m * y_45_scale_m));
	double t_6 = t_2 / pow(x_45_scale_m, 2.0);
	double t_7 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_8 = 0.5 * (t_2 / pow(y_45_scale_m, 2.0));
	double tmp;
	if (x_45_scale_m <= 4.6e-115) {
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (pow(y_45_scale_m, 2.0) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_3), 2.0)) + t_3)))))))) / t_5);
	} else if (x_45_scale_m <= 3.9e+80) {
		tmp = 0.25 * (b_m * (pow(x_45_scale_m, 2.0) * (y_45_scale_m * sqrt((8.0 * ((sqrt((pow(t_7, 4.0) / pow(x_45_scale_m, 4.0))) + (pow(t_7, 2.0) / pow(x_45_scale_m, 2.0))) / pow(x_45_scale_m, 2.0)))))));
	} else {
		tmp = (0.25 * ((a_m * (x_45_scale_m * (y_45_scale_m * sqrt((8.0 * ((sqrt(((pow(sin(t_1), 2.0) / (pow(x_45_scale_m, 2.0) * pow(y_45_scale_m, 2.0))) + pow(((0.5 * t_4) - fma(0.5, t_0, fma(0.5, t_6, t_8))), 2.0))) + fma(0.5, t_4, fma(0.5, t_0, t_8))) - (0.5 * t_6))))))) / t_5)) * (y_45_scale_m * x_45_scale_m);
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(1.0 / (y_45_scale_m ^ 2.0))
	t_1 = Float64(0.011111111111111112 * Float64(angle * pi))
	t_2 = cos(t_1)
	t_3 = Float64(0.5 * t_2)
	t_4 = Float64(1.0 / (x_45_scale_m ^ 2.0))
	t_5 = abs(Float64(x_45_scale_m * y_45_scale_m))
	t_6 = Float64(t_2 / (x_45_scale_m ^ 2.0))
	t_7 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_8 = Float64(0.5 * Float64(t_2 / (y_45_scale_m ^ 2.0)))
	tmp = 0.0
	if (x_45_scale_m <= 4.6e-115)
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(a_m * Float64(b_m * Float64(x_45_scale_m * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_3) ^ 2.0)) + t_3)))))))) / t_5));
	elseif (x_45_scale_m <= 3.9e+80)
		tmp = Float64(0.25 * Float64(b_m * Float64((x_45_scale_m ^ 2.0) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_7 ^ 4.0) / (x_45_scale_m ^ 4.0))) + Float64((t_7 ^ 2.0) / (x_45_scale_m ^ 2.0))) / (x_45_scale_m ^ 2.0))))))));
	else
		tmp = Float64(Float64(0.25 * Float64(Float64(a_m * Float64(x_45_scale_m * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(Float64((sin(t_1) ^ 2.0) / Float64((x_45_scale_m ^ 2.0) * (y_45_scale_m ^ 2.0))) + (Float64(Float64(0.5 * t_4) - fma(0.5, t_0, fma(0.5, t_6, t_8))) ^ 2.0))) + fma(0.5, t_4, fma(0.5, t_0, t_8))) - Float64(0.5 * t_6))))))) / t_5)) * Float64(y_45_scale_m * x_45_scale_m));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(1.0 / N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Abs[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$2 / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[(0.5 * N[(t$95$2 / N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 4.6e-115], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(a$95$m * N[(b$95$m * N[(x$45$scale$95$m * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$3), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 3.9e+80], N[(0.25 * N[(b$95$m * N[(N[Power[x$45$scale$95$m, 2.0], $MachinePrecision] * N[(y$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$7, 4.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$7, 2.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(N[(a$95$m * N[(x$45$scale$95$m * N[(y$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[(N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[x$45$scale$95$m, 2.0], $MachinePrecision] * N[Power[y$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(0.5 * t$95$4), $MachinePrecision] - N[(0.5 * t$95$0 + N[(0.5 * t$95$6 + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(0.5 * t$95$4 + N[(0.5 * t$95$0 + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 4.59999999999999969e-115

   1. Initial program 3.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 1.5%

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in x-scale around 0

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

   9. Applied rewrites 10.5%

      \[\leadsto \frac{0.25}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)\right)}{\color{blue}{a \cdot \left|x-scale \cdot y-scale\right|}} \]

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 21.6%

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

   if 4.59999999999999969e-115 < x-scale < 3.89999999999999999e80

   1. Initial program 3.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in y-scale around -inf

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 10.6%

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

   if 3.89999999999999999e80 < x-scale

   1. Initial program 3.0%

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

   3. Applied rewrites 6.5%

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

   4. Taylor expanded in b around 0

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

   6. Applied rewrites 3.0%

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{x-scale \cdot \left(y-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{\frac{{a}^{4} \cdot {\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}} - \frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2} \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{y-scale}^{2}} + \frac{{a}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{{x-scale}^{2}}\right)\right)\right)}\right)}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right) \]

   7. Taylor expanded in a around 0

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

   9. Applied rewrites 14.3%

      \[\leadsto \left(0.25 \cdot \frac{a \cdot \left(x-scale \cdot \left(y-scale \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{x-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)}\right)\right)}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 23.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)\\ t_3 := \frac{0.5 - t\_2 \cdot 0.5}{y-scale\_m \cdot y-scale\_m}\\ t_4 := \frac{\mathsf{fma}\left(t\_2, 0.5, 0.5\right)}{x-scale\_m \cdot x-scale\_m}\\ t_5 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;x-scale\_m \leq 4.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{a\_m \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left({y-scale\_m}^{2} \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_5\right)}^{2}} + t\_5\right)\right)}\right)\right)\right)}{\left|x-scale\_m \cdot y-scale\_m\right|}\\ \mathbf{elif}\;x-scale\_m \leq 5 \cdot 10^{+80}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left({x-scale\_m}^{2} \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_1}^{4}}{{x-scale\_m}^{4}}} + \frac{{t\_1}^{2}}{{x-scale\_m}^{2}}}{{x-scale\_m}^{2}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25 \cdot \left(\left(\left(b\_m \cdot x-scale\_m\right) \cdot x-scale\_m\right) \cdot \frac{\left(\frac{\sqrt{\left(8 \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\left(\sin t\_0 \cdot \cos t\_0\right)}^{2}}{\left(\left(y-scale\_m \cdot y-scale\_m\right) \cdot x-scale\_m\right) \cdot x-scale\_m}, {\left(t\_4 - t\_3\right)}^{2}\right)} + \left(t\_4 + t\_3\right)\right)\right) \cdot {a\_m}^{4}}}{\left|y-scale\_m \cdot x-scale\_m\right|} \cdot y-scale\_m\right) \cdot y-scale\_m}{a\_m}\right)}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (cos (* 0.005555555555555556 (* angle PI))))
        (t_2 (cos (* (* (+ PI PI) angle) 0.005555555555555556)))
        (t_3 (/ (- 0.5 (* t_2 0.5)) (* y-scale_m y-scale_m)))
        (t_4 (/ (fma t_2 0.5 0.5) (* x-scale_m x-scale_m)))
        (t_5 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
   (if (<= x-scale_m 4.6e-115)
     (*
      (/ 0.25 a_m)
      (/
       (*
        a_m
        (*
         b_m
         (*
          x-scale_m
          (*
           (pow y-scale_m 2.0)
           (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_5) 2.0)) t_5))))))))
       (fabs (* x-scale_m y-scale_m))))
     (if (<= x-scale_m 5e+80)
       (*
        0.25
        (*
         b_m
         (*
          (pow x-scale_m 2.0)
          (*
           y-scale_m
           (sqrt
            (*
             8.0
             (/
              (+
               (sqrt (/ (pow t_1 4.0) (pow x-scale_m 4.0)))
               (/ (pow t_1 2.0) (pow x-scale_m 2.0)))
              (pow x-scale_m 2.0))))))))
       (/
        (*
         0.25
         (*
          (* (* b_m x-scale_m) x-scale_m)
          (/
           (*
            (*
             (/
              (sqrt
               (*
                (*
                 8.0
                 (+
                  (sqrt
                   (fma
                    4.0
                    (/
                     (pow (* (sin t_0) (cos t_0)) 2.0)
                     (* (* (* y-scale_m y-scale_m) x-scale_m) x-scale_m))
                    (pow (- t_4 t_3) 2.0)))
                  (+ t_4 t_3)))
                (pow a_m 4.0)))
              (fabs (* y-scale_m x-scale_m)))
             y-scale_m)
            y-scale_m)
           a_m)))
        a_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_2 = cos((((((double) M_PI) + ((double) M_PI)) * angle) * 0.005555555555555556));
	double t_3 = (0.5 - (t_2 * 0.5)) / (y_45_scale_m * y_45_scale_m);
	double t_4 = fma(t_2, 0.5, 0.5) / (x_45_scale_m * x_45_scale_m);
	double t_5 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (x_45_scale_m <= 4.6e-115) {
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (pow(y_45_scale_m, 2.0) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_5), 2.0)) + t_5)))))))) / fabs((x_45_scale_m * y_45_scale_m)));
	} else if (x_45_scale_m <= 5e+80) {
		tmp = 0.25 * (b_m * (pow(x_45_scale_m, 2.0) * (y_45_scale_m * sqrt((8.0 * ((sqrt((pow(t_1, 4.0) / pow(x_45_scale_m, 4.0))) + (pow(t_1, 2.0) / pow(x_45_scale_m, 2.0))) / pow(x_45_scale_m, 2.0)))))));
	} else {
		tmp = (0.25 * (((b_m * x_45_scale_m) * x_45_scale_m) * ((((sqrt(((8.0 * (sqrt(fma(4.0, (pow((sin(t_0) * cos(t_0)), 2.0) / (((y_45_scale_m * y_45_scale_m) * x_45_scale_m) * x_45_scale_m)), pow((t_4 - t_3), 2.0))) + (t_4 + t_3))) * pow(a_m, 4.0))) / fabs((y_45_scale_m * x_45_scale_m))) * y_45_scale_m) * y_45_scale_m) / a_m))) / a_m;
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_2 = cos(Float64(Float64(Float64(pi + pi) * angle) * 0.005555555555555556))
	t_3 = Float64(Float64(0.5 - Float64(t_2 * 0.5)) / Float64(y_45_scale_m * y_45_scale_m))
	t_4 = Float64(fma(t_2, 0.5, 0.5) / Float64(x_45_scale_m * x_45_scale_m))
	t_5 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (x_45_scale_m <= 4.6e-115)
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(a_m * Float64(b_m * Float64(x_45_scale_m * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_5) ^ 2.0)) + t_5)))))))) / abs(Float64(x_45_scale_m * y_45_scale_m))));
	elseif (x_45_scale_m <= 5e+80)
		tmp = Float64(0.25 * Float64(b_m * Float64((x_45_scale_m ^ 2.0) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_1 ^ 4.0) / (x_45_scale_m ^ 4.0))) + Float64((t_1 ^ 2.0) / (x_45_scale_m ^ 2.0))) / (x_45_scale_m ^ 2.0))))))));
	else
		tmp = Float64(Float64(0.25 * Float64(Float64(Float64(b_m * x_45_scale_m) * x_45_scale_m) * Float64(Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * Float64(sqrt(fma(4.0, Float64((Float64(sin(t_0) * cos(t_0)) ^ 2.0) / Float64(Float64(Float64(y_45_scale_m * y_45_scale_m) * x_45_scale_m) * x_45_scale_m)), (Float64(t_4 - t_3) ^ 2.0))) + Float64(t_4 + t_3))) * (a_m ^ 4.0))) / abs(Float64(y_45_scale_m * x_45_scale_m))) * y_45_scale_m) * y_45_scale_m) / a_m))) / a_m);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(N[(N[(Pi + Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 * 0.5 + 0.5), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 4.6e-115], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(a$95$m * N[(b$95$m * N[(x$45$scale$95$m * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 5e+80], N[(0.25 * N[(b$95$m * N[(N[Power[x$45$scale$95$m, 2.0], $MachinePrecision] * N[(y$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$1, 4.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 * N[(N[(N[(b$95$m * x$45$scale$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[N[(N[(8.0 * N[(N[Sqrt[N[(4.0 * N[(N[Power[N[(N[Sin[t$95$0], $MachinePrecision] * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[(N[(y$45$scale$95$m * y$45$scale$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$4 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(t$95$4 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[a$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] * y$45$scale$95$m), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 4.59999999999999969e-115

   1. Initial program 3.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 1.5%

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in x-scale around 0

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

   9. Applied rewrites 10.5%

      \[\leadsto \frac{0.25}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)\right)}{\color{blue}{a \cdot \left|x-scale \cdot y-scale\right|}} \]

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 21.6%

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

   if 4.59999999999999969e-115 < x-scale < 4.99999999999999961e80

   1. Initial program 3.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in y-scale around -inf

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 10.6%

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

   if 4.99999999999999961e80 < x-scale

   1. Initial program 3.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 1.5%

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

   6. Applied rewrites 10.2%

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

   8. Applied rewrites 14.2%

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

Alternative 3: 23.7% accurate, 3.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left|x-scale\_m \cdot y-scale\_m\right|\\ t_1 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;x-scale\_m \leq 4.6 \cdot 10^{-115}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{a\_m \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left({y-scale\_m}^{2} \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_2\right)}^{2}} + t\_2\right)\right)}\right)\right)\right)}{t\_0}\\ \mathbf{elif}\;x-scale\_m \leq 6.1 \cdot 10^{+80}:\\ \;\;\;\;0.25 \cdot \left(b\_m \cdot \left({x-scale\_m}^{2} \cdot \left(y-scale\_m \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{{t\_1}^{4}}{{x-scale\_m}^{4}}} + \frac{{t\_1}^{2}}{{x-scale\_m}^{2}}}{{x-scale\_m}^{2}}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\frac{b\_m \cdot \left(x-scale\_m \cdot \left({y-scale\_m}^{2} \cdot \sqrt{16 \cdot {a\_m}^{4}}\right)\right)}{t\_0}}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (fabs (* x-scale_m y-scale_m)))
        (t_1 (cos (* 0.005555555555555556 (* angle PI))))
        (t_2 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
   (if (<= x-scale_m 4.6e-115)
     (*
      (/ 0.25 a_m)
      (/
       (*
        a_m
        (*
         b_m
         (*
          x-scale_m
          (*
           (pow y-scale_m 2.0)
           (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_2) 2.0)) t_2))))))))
       t_0))
     (if (<= x-scale_m 6.1e+80)
       (*
        0.25
        (*
         b_m
         (*
          (pow x-scale_m 2.0)
          (*
           y-scale_m
           (sqrt
            (*
             8.0
             (/
              (+
               (sqrt (/ (pow t_1 4.0) (pow x-scale_m 4.0)))
               (/ (pow t_1 2.0) (pow x-scale_m 2.0)))
              (pow x-scale_m 2.0))))))))
       (*
        (/ 0.25 a_m)
        (/
         (/
          (*
           b_m
           (* x-scale_m (* (pow y-scale_m 2.0) (sqrt (* 16.0 (pow a_m 4.0))))))
          t_0)
         a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = fabs((x_45_scale_m * y_45_scale_m));
	double t_1 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_2 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (x_45_scale_m <= 4.6e-115) {
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (pow(y_45_scale_m, 2.0) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_2), 2.0)) + t_2)))))))) / t_0);
	} else if (x_45_scale_m <= 6.1e+80) {
		tmp = 0.25 * (b_m * (pow(x_45_scale_m, 2.0) * (y_45_scale_m * sqrt((8.0 * ((sqrt((pow(t_1, 4.0) / pow(x_45_scale_m, 4.0))) + (pow(t_1, 2.0) / pow(x_45_scale_m, 2.0))) / pow(x_45_scale_m, 2.0)))))));
	} else {
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * (pow(y_45_scale_m, 2.0) * sqrt((16.0 * pow(a_m, 4.0)))))) / t_0) / a_m);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.abs((x_45_scale_m * y_45_scale_m));
	double t_1 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double t_2 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double tmp;
	if (x_45_scale_m <= 4.6e-115) {
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (Math.pow(y_45_scale_m, 2.0) * Math.sqrt((8.0 * (0.5 + (Math.sqrt(Math.pow((0.5 + t_2), 2.0)) + t_2)))))))) / t_0);
	} else if (x_45_scale_m <= 6.1e+80) {
		tmp = 0.25 * (b_m * (Math.pow(x_45_scale_m, 2.0) * (y_45_scale_m * Math.sqrt((8.0 * ((Math.sqrt((Math.pow(t_1, 4.0) / Math.pow(x_45_scale_m, 4.0))) + (Math.pow(t_1, 2.0) / Math.pow(x_45_scale_m, 2.0))) / Math.pow(x_45_scale_m, 2.0)))))));
	} else {
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * (Math.pow(y_45_scale_m, 2.0) * Math.sqrt((16.0 * Math.pow(a_m, 4.0)))))) / t_0) / a_m);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.fabs((x_45_scale_m * y_45_scale_m))
	t_1 = math.cos((0.005555555555555556 * (angle * math.pi)))
	t_2 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	tmp = 0
	if x_45_scale_m <= 4.6e-115:
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (math.pow(y_45_scale_m, 2.0) * math.sqrt((8.0 * (0.5 + (math.sqrt(math.pow((0.5 + t_2), 2.0)) + t_2)))))))) / t_0)
	elif x_45_scale_m <= 6.1e+80:
		tmp = 0.25 * (b_m * (math.pow(x_45_scale_m, 2.0) * (y_45_scale_m * math.sqrt((8.0 * ((math.sqrt((math.pow(t_1, 4.0) / math.pow(x_45_scale_m, 4.0))) + (math.pow(t_1, 2.0) / math.pow(x_45_scale_m, 2.0))) / math.pow(x_45_scale_m, 2.0)))))))
	else:
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * (math.pow(y_45_scale_m, 2.0) * math.sqrt((16.0 * math.pow(a_m, 4.0)))))) / t_0) / a_m)
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = abs(Float64(x_45_scale_m * y_45_scale_m))
	t_1 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_2 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (x_45_scale_m <= 4.6e-115)
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(a_m * Float64(b_m * Float64(x_45_scale_m * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_2) ^ 2.0)) + t_2)))))))) / t_0));
	elseif (x_45_scale_m <= 6.1e+80)
		tmp = Float64(0.25 * Float64(b_m * Float64((x_45_scale_m ^ 2.0) * Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64((t_1 ^ 4.0) / (x_45_scale_m ^ 4.0))) + Float64((t_1 ^ 2.0) / (x_45_scale_m ^ 2.0))) / (x_45_scale_m ^ 2.0))))))));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(b_m * Float64(x_45_scale_m * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(16.0 * (a_m ^ 4.0)))))) / t_0) / a_m));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = abs((x_45_scale_m * y_45_scale_m));
	t_1 = cos((0.005555555555555556 * (angle * pi)));
	t_2 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if (x_45_scale_m <= 4.6e-115)
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * ((y_45_scale_m ^ 2.0) * sqrt((8.0 * (0.5 + (sqrt(((0.5 + t_2) ^ 2.0)) + t_2)))))))) / t_0);
	elseif (x_45_scale_m <= 6.1e+80)
		tmp = 0.25 * (b_m * ((x_45_scale_m ^ 2.0) * (y_45_scale_m * sqrt((8.0 * ((sqrt(((t_1 ^ 4.0) / (x_45_scale_m ^ 4.0))) + ((t_1 ^ 2.0) / (x_45_scale_m ^ 2.0))) / (x_45_scale_m ^ 2.0)))))));
	else
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * ((y_45_scale_m ^ 2.0) * sqrt((16.0 * (a_m ^ 4.0)))))) / t_0) / a_m);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Abs[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 4.6e-115], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(a$95$m * N[(b$95$m * N[(x$45$scale$95$m * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale$95$m, 6.1e+80], N[(0.25 * N[(b$95$m * N[(N[Power[x$45$scale$95$m, 2.0], $MachinePrecision] * N[(y$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[Power[t$95$1, 4.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(b$95$m * N[(x$45$scale$95$m * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(16.0 * N[Power[a$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 4.59999999999999969e-115

   1. Initial program 3.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 1.5%

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in x-scale around 0

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

   9. Applied rewrites 10.5%

      \[\leadsto \frac{0.25}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)\right)}{\color{blue}{a \cdot \left|x-scale \cdot y-scale\right|}} \]

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 21.6%

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

   if 4.59999999999999969e-115 < x-scale < 6.09999999999999975e80

   1. Initial program 3.0%

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

   2. Taylor expanded in a around 0

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in y-scale around -inf

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

   7. Applied rewrites 0.3%

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

   8. Taylor expanded in b around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

   10. Applied rewrites 10.6%

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

   if 6.09999999999999975e80 < x-scale

   1. Initial program 3.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 1.5%

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in x-scale around 0

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

   9. Applied rewrites 17.7%

      \[\leadsto \frac{0.25}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4}}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4}}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]

       2. lower-pow.f64 17.7

          \[\leadsto \frac{0.25}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]

   12. Applied rewrites 17.7%

       \[\leadsto \frac{0.25}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 23.5% accurate, 4.6× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \left|x-scale\_m \cdot y-scale\_m\right|\\ \mathbf{if}\;y-scale\_m \leq 1.24 \cdot 10^{-120}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\left(b\_m \cdot \left(x-scale\_m \cdot x-scale\_m\right)\right) \cdot \frac{y-scale\_m \cdot \sqrt{8 \cdot \left({a\_m}^{4} \cdot \left(\left(0.5 + \sqrt{{\left(t\_0 - 0.5\right)}^{2}}\right) - t\_0\right)\right)}}{t\_1}}{a\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{a\_m \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left({y-scale\_m}^{2} \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_0\right)}^{2}} + t\_0\right)\right)}\right)\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
        (t_1 (fabs (* x-scale_m y-scale_m))))
   (if (<= y-scale_m 1.24e-120)
     (*
      (/ 0.25 a_m)
      (/
       (*
        (* b_m (* x-scale_m x-scale_m))
        (/
         (*
          y-scale_m
          (sqrt
           (*
            8.0
            (* (pow a_m 4.0) (- (+ 0.5 (sqrt (pow (- t_0 0.5) 2.0))) t_0)))))
         t_1))
       a_m))
     (*
      (/ 0.25 a_m)
      (/
       (*
        a_m
        (*
         b_m
         (*
          x-scale_m
          (*
           (pow y-scale_m 2.0)
           (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_0) 2.0)) t_0))))))))
       t_1)))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_1 = fabs((x_45_scale_m * y_45_scale_m));
	double tmp;
	if (y_45_scale_m <= 1.24e-120) {
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * x_45_scale_m)) * ((y_45_scale_m * sqrt((8.0 * (pow(a_m, 4.0) * ((0.5 + sqrt(pow((t_0 - 0.5), 2.0))) - t_0))))) / t_1)) / a_m);
	} else {
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (pow(y_45_scale_m, 2.0) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_0), 2.0)) + t_0)))))))) / t_1);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double t_1 = Math.abs((x_45_scale_m * y_45_scale_m));
	double tmp;
	if (y_45_scale_m <= 1.24e-120) {
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * x_45_scale_m)) * ((y_45_scale_m * Math.sqrt((8.0 * (Math.pow(a_m, 4.0) * ((0.5 + Math.sqrt(Math.pow((t_0 - 0.5), 2.0))) - t_0))))) / t_1)) / a_m);
	} else {
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (Math.pow(y_45_scale_m, 2.0) * Math.sqrt((8.0 * (0.5 + (Math.sqrt(Math.pow((0.5 + t_0), 2.0)) + t_0)))))))) / t_1);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	t_1 = math.fabs((x_45_scale_m * y_45_scale_m))
	tmp = 0
	if y_45_scale_m <= 1.24e-120:
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * x_45_scale_m)) * ((y_45_scale_m * math.sqrt((8.0 * (math.pow(a_m, 4.0) * ((0.5 + math.sqrt(math.pow((t_0 - 0.5), 2.0))) - t_0))))) / t_1)) / a_m)
	else:
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (math.pow(y_45_scale_m, 2.0) * math.sqrt((8.0 * (0.5 + (math.sqrt(math.pow((0.5 + t_0), 2.0)) + t_0)))))))) / t_1)
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_1 = abs(Float64(x_45_scale_m * y_45_scale_m))
	tmp = 0.0
	if (y_45_scale_m <= 1.24e-120)
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(b_m * Float64(x_45_scale_m * x_45_scale_m)) * Float64(Float64(y_45_scale_m * sqrt(Float64(8.0 * Float64((a_m ^ 4.0) * Float64(Float64(0.5 + sqrt((Float64(t_0 - 0.5) ^ 2.0))) - t_0))))) / t_1)) / a_m));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(a_m * Float64(b_m * Float64(x_45_scale_m * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_0) ^ 2.0)) + t_0)))))))) / t_1));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	t_1 = abs((x_45_scale_m * y_45_scale_m));
	tmp = 0.0;
	if (y_45_scale_m <= 1.24e-120)
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * x_45_scale_m)) * ((y_45_scale_m * sqrt((8.0 * ((a_m ^ 4.0) * ((0.5 + sqrt(((t_0 - 0.5) ^ 2.0))) - t_0))))) / t_1)) / a_m);
	else
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * ((y_45_scale_m ^ 2.0) * sqrt((8.0 * (0.5 + (sqrt(((0.5 + t_0) ^ 2.0)) + t_0)))))))) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$45$scale$95$m, 1.24e-120], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(b$95$m * N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[(y$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(0.5 + N[Sqrt[N[Power[N[(t$95$0 - 0.5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(a$95$m * N[(b$95$m * N[(x$45$scale$95$m * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.2399999999999999e-120

   1. Initial program 3.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 1.5%

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 9.2%

      \[\leadsto \frac{0.25}{a} \cdot \frac{\left(b \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \frac{y-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\left(0.5 + \sqrt{{\left(0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) - 0.5\right)}^{2}}\right) - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{a} \]

   if 1.2399999999999999e-120 < y-scale

   1. Initial program 3.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 1.5%

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in x-scale around 0

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

   9. Applied rewrites 10.5%

      \[\leadsto \frac{0.25}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)\right)}{\color{blue}{a \cdot \left|x-scale \cdot y-scale\right|}} \]

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 21.6%

       \[\leadsto \frac{0.25}{a} \cdot \frac{a \cdot \left(b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right)\right)\right)}{\left|x-scale \cdot y-scale\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 23.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \begin{array}{l} t_0 := \left|x-scale\_m \cdot y-scale\_m\right|\\ t_1 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 4 \cdot 10^{-71}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{a\_m \cdot \left(b\_m \cdot \left(x-scale\_m \cdot \left({y-scale\_m}^{2} \cdot \sqrt{8 \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_1\right)}^{2}} + t\_1\right)\right)}\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\frac{b\_m \cdot \left(x-scale\_m \cdot \left({y-scale\_m}^{2} \cdot \sqrt{16 \cdot {a\_m}^{4}}\right)\right)}{t\_0}}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (let* ((t_0 (fabs (* x-scale_m y-scale_m)))
        (t_1 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
   (if (<= a_m 4e-71)
     (*
      (/ 0.25 a_m)
      (/
       (*
        a_m
        (*
         b_m
         (*
          x-scale_m
          (*
           (pow y-scale_m 2.0)
           (sqrt (* 8.0 (+ 0.5 (+ (sqrt (pow (+ 0.5 t_1) 2.0)) t_1))))))))
       t_0))
     (*
      (/ 0.25 a_m)
      (/
       (/
        (*
         b_m
         (* x-scale_m (* (pow y-scale_m 2.0) (sqrt (* 16.0 (pow a_m 4.0))))))
        t_0)
       a_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = fabs((x_45_scale_m * y_45_scale_m));
	double t_1 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 4e-71) {
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (pow(y_45_scale_m, 2.0) * sqrt((8.0 * (0.5 + (sqrt(pow((0.5 + t_1), 2.0)) + t_1)))))))) / t_0);
	} else {
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * (pow(y_45_scale_m, 2.0) * sqrt((16.0 * pow(a_m, 4.0)))))) / t_0) / a_m);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	double t_0 = Math.abs((x_45_scale_m * y_45_scale_m));
	double t_1 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double tmp;
	if (a_m <= 4e-71) {
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (Math.pow(y_45_scale_m, 2.0) * Math.sqrt((8.0 * (0.5 + (Math.sqrt(Math.pow((0.5 + t_1), 2.0)) + t_1)))))))) / t_0);
	} else {
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * (Math.pow(y_45_scale_m, 2.0) * Math.sqrt((16.0 * Math.pow(a_m, 4.0)))))) / t_0) / a_m);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	t_0 = math.fabs((x_45_scale_m * y_45_scale_m))
	t_1 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	tmp = 0
	if a_m <= 4e-71:
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * (math.pow(y_45_scale_m, 2.0) * math.sqrt((8.0 * (0.5 + (math.sqrt(math.pow((0.5 + t_1), 2.0)) + t_1)))))))) / t_0)
	else:
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * (math.pow(y_45_scale_m, 2.0) * math.sqrt((16.0 * math.pow(a_m, 4.0)))))) / t_0) / a_m)
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = abs(Float64(x_45_scale_m * y_45_scale_m))
	t_1 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (a_m <= 4e-71)
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(a_m * Float64(b_m * Float64(x_45_scale_m * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(8.0 * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_1) ^ 2.0)) + t_1)))))))) / t_0));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(b_m * Float64(x_45_scale_m * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(16.0 * (a_m ^ 4.0)))))) / t_0) / a_m));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	t_0 = abs((x_45_scale_m * y_45_scale_m));
	t_1 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if (a_m <= 4e-71)
		tmp = (0.25 / a_m) * ((a_m * (b_m * (x_45_scale_m * ((y_45_scale_m ^ 2.0) * sqrt((8.0 * (0.5 + (sqrt(((0.5 + t_1) ^ 2.0)) + t_1)))))))) / t_0);
	else
		tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * ((y_45_scale_m ^ 2.0) * sqrt((16.0 * (a_m ^ 4.0)))))) / t_0) / a_m);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := Block[{t$95$0 = N[Abs[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 4e-71], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(a$95$m * N[(b$95$m * N[(x$45$scale$95$m * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$1), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(b$95$m * N[(x$45$scale$95$m * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(16.0 * N[Power[a$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.9999999999999997e-71

   1. Initial program 3.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 1.5%

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in x-scale around 0

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

   9. Applied rewrites 10.5%

      \[\leadsto \frac{0.25}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)\right)}{\color{blue}{a \cdot \left|x-scale \cdot y-scale\right|}} \]

   10. Taylor expanded in a around 0

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

       1. lower-/.f64 N/A

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

   12. Applied rewrites 21.6%

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

   if 3.9999999999999997e-71 < a

   1. Initial program 3.0%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 1.5%

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

   6. Applied rewrites 10.2%

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

   7. Taylor expanded in x-scale around 0

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

   9. Applied rewrites 17.7%

      \[\leadsto \frac{0.25}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]

   10. Taylor expanded in angle around 0

       \[\leadsto \frac{\frac{1}{4}}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{4}}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]

       2. lower-pow.f64 17.7

          \[\leadsto \frac{0.25}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]

   12. Applied rewrites 17.7%

       \[\leadsto \frac{0.25}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 17.7% accurate, 12.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \frac{0.25}{a\_m} \cdot \frac{\frac{b\_m \cdot \left(x-scale\_m \cdot \left({y-scale\_m}^{2} \cdot \sqrt{16 \cdot {a\_m}^{4}}\right)\right)}{\left|x-scale\_m \cdot y-scale\_m\right|}}{a\_m} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  (/ 0.25 a_m)
  (/
   (/
    (* b_m (* x-scale_m (* (pow y-scale_m 2.0) (sqrt (* 16.0 (pow a_m 4.0))))))
    (fabs (* x-scale_m y-scale_m)))
   a_m)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (0.25 / a_m) * (((b_m * (x_45_scale_m * (pow(y_45_scale_m, 2.0) * sqrt((16.0 * pow(a_m, 4.0)))))) / fabs((x_45_scale_m * y_45_scale_m))) / a_m);
}

Fortran

a_m =     private
b_m =     private
x-scale_m =     private
y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = (0.25d0 / a_m) * (((b_m * (x_45scale_m * ((y_45scale_m ** 2.0d0) * sqrt((16.0d0 * (a_m ** 4.0d0)))))) / abs((x_45scale_m * y_45scale_m))) / a_m)
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (0.25 / a_m) * (((b_m * (x_45_scale_m * (Math.pow(y_45_scale_m, 2.0) * Math.sqrt((16.0 * Math.pow(a_m, 4.0)))))) / Math.abs((x_45_scale_m * y_45_scale_m))) / a_m);
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return (0.25 / a_m) * (((b_m * (x_45_scale_m * (math.pow(y_45_scale_m, 2.0) * math.sqrt((16.0 * math.pow(a_m, 4.0)))))) / math.fabs((x_45_scale_m * y_45_scale_m))) / a_m)

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(b_m * Float64(x_45_scale_m * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(16.0 * (a_m ^ 4.0)))))) / abs(Float64(x_45_scale_m * y_45_scale_m))) / a_m))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = (0.25 / a_m) * (((b_m * (x_45_scale_m * ((y_45_scale_m ^ 2.0) * sqrt((16.0 * (a_m ^ 4.0)))))) / abs((x_45_scale_m * y_45_scale_m))) / a_m);
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(b$95$m * N[(x$45$scale$95$m * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(16.0 * N[Power[a$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 1.5%

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

6. Applied rewrites 10.2%

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

7. Taylor expanded in x-scale around 0

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

9. Applied rewrites 17.7%

   \[\leadsto \frac{0.25}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]

10. Taylor expanded in angle around 0

    \[\leadsto \frac{\frac{1}{4}}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{4}}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]

    2. lower-pow.f64 17.7

       \[\leadsto \frac{0.25}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]

12. Applied rewrites 17.7%

    \[\leadsto \frac{0.25}{a} \cdot \frac{\frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{\left|x-scale \cdot y-scale\right|}}{a} \]
13. Add Preprocessing

Alternative 7: 10.5% accurate, 12.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ y-scale_m = \left|y-scale\right| \\ \frac{0.25}{a\_m} \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \left({y-scale\_m}^{2} \cdot \sqrt{16 \cdot {a\_m}^{4}}\right)\right)}{a\_m \cdot \left|x-scale\_m \cdot y-scale\_m\right|} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
y-scale_m = (fabs.f64 y-scale)
(FPCore (a_m b_m angle x-scale_m y-scale_m)
 :precision binary64
 (*
  (/ 0.25 a_m)
  (/
   (* b_m (* x-scale_m (* (pow y-scale_m 2.0) (sqrt (* 16.0 (pow a_m 4.0))))))
   (* a_m (fabs (* x-scale_m y-scale_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
y-scale_m = fabs(y_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (0.25 / a_m) * ((b_m * (x_45_scale_m * (pow(y_45_scale_m, 2.0) * sqrt((16.0 * pow(a_m, 4.0)))))) / (a_m * fabs((x_45_scale_m * y_45_scale_m))));
}

Fortran

a_m =     private
b_m =     private
x-scale_m =     private
y-scale_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a_m, b_m, angle, x_45scale_m, y_45scale_m)
use fmin_fmax_functions
    real(8), intent (in) :: a_m
    real(8), intent (in) :: b_m
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale_m
    real(8), intent (in) :: y_45scale_m
    code = (0.25d0 / a_m) * ((b_m * (x_45scale_m * ((y_45scale_m ** 2.0d0) * sqrt((16.0d0 * (a_m ** 4.0d0)))))) / (a_m * abs((x_45scale_m * y_45scale_m))))
end function

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
y-scale_m = Math.abs(y_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale_m) {
	return (0.25 / a_m) * ((b_m * (x_45_scale_m * (Math.pow(y_45_scale_m, 2.0) * Math.sqrt((16.0 * Math.pow(a_m, 4.0)))))) / (a_m * Math.abs((x_45_scale_m * y_45_scale_m))));
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
y-scale_m = math.fabs(y_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m):
	return (0.25 / a_m) * ((b_m * (x_45_scale_m * (math.pow(y_45_scale_m, 2.0) * math.sqrt((16.0 * math.pow(a_m, 4.0)))))) / (a_m * math.fabs((x_45_scale_m * y_45_scale_m))))

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
y-scale_m = abs(y_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	return Float64(Float64(0.25 / a_m) * Float64(Float64(b_m * Float64(x_45_scale_m * Float64((y_45_scale_m ^ 2.0) * sqrt(Float64(16.0 * (a_m ^ 4.0)))))) / Float64(a_m * abs(Float64(x_45_scale_m * y_45_scale_m)))))
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
y-scale_m = abs(y_45_scale);
function tmp = code(a_m, b_m, angle, x_45_scale_m, y_45_scale_m)
	tmp = (0.25 / a_m) * ((b_m * (x_45_scale_m * ((y_45_scale_m ^ 2.0) * sqrt((16.0 * (a_m ^ 4.0)))))) / (a_m * abs((x_45_scale_m * y_45_scale_m))));
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
y-scale_m = N[Abs[y$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale$95$m_] := N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(b$95$m * N[(x$45$scale$95$m * N[(N[Power[y$45$scale$95$m, 2.0], $MachinePrecision] * N[Sqrt[N[(16.0 * N[Power[a$95$m, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a$95$m * N[Abs[N[(x$45$scale$95$m * y$45$scale$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.0%

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

2. Taylor expanded in b around inf

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

4. Applied rewrites 1.5%

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

6. Applied rewrites 10.2%

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

7. Taylor expanded in x-scale around 0

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

9. Applied rewrites 10.5%

   \[\leadsto \frac{0.25}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}\right)\right)}{\color{blue}{a \cdot \left|x-scale \cdot y-scale\right|}} \]

10. Taylor expanded in angle around 0

    \[\leadsto \frac{\frac{1}{4}}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{a \cdot \left|x-scale \cdot y-scale\right|} \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{4}}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{a \cdot \left|x-scale \cdot y-scale\right|} \]

    2. lower-pow.f64 10.5

       \[\leadsto \frac{0.25}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{a \cdot \left|x-scale \cdot y-scale\right|} \]

12. Applied rewrites 10.5%

    \[\leadsto \frac{0.25}{a} \cdot \frac{b \cdot \left(x-scale \cdot \left({y-scale}^{2} \cdot \sqrt{16 \cdot {a}^{4}}\right)\right)}{a \cdot \left|x-scale \cdot y-scale\right|} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(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))))