a from scale-rotated-ellipse

Percentage Accurate: 2.5% → 18.3%

Time: 42.1s

Alternatives: 18

Speedup: 6.6×

• 69.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.5% accurate, 1.0× speedup

Math

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

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: 18.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)\\ t_1 := 4 \cdot \left(b \cdot a\right)\\ t_2 := y-scale \cdot \left|x-scale\right|\\ t_3 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ t_4 := t\_3 \cdot 8\\ t_5 := \left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_6 := \cos t\_5\\ t_7 := 0.5 - t\_6 \cdot 0.5\\ t_8 := \left|x-scale\right| \cdot y-scale\\ t_9 := \left|t\_8\right|\\ t_10 := 0.5 - t\_0 \cdot 0.5\\ t_11 := \mathsf{fma}\left(t\_6, 0.5, 0.5\right)\\ t_12 := \frac{\mathsf{fma}\left(t\_11 \cdot a, a, \left(t\_7 \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\\ t_13 := \mathsf{fma}\left(0.5, t\_0, 0.5\right)\\ t_14 := \frac{\frac{\mathsf{fma}\left(t\_10, a \cdot a, \left(t\_13 \cdot b\right) \cdot b\right)}{\left|x-scale\right|}}{\left|x-scale\right|}\\ t_15 := \frac{\frac{\mathsf{fma}\left(t\_13, a \cdot a, \left(t\_10 \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\\ t_16 := \frac{\mathsf{fma}\left(t\_7 \cdot a, a, \left(t\_11 \cdot b\right) \cdot b\right)}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ t_17 := \frac{\sin t\_5 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{t\_8}\\ \mathbf{if}\;\left|x-scale\right| \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_4 \cdot \left(t\_3 \cdot \left(t\_15 + \left(t\_16 + \mathsf{hypot}\left(t\_15 - t\_16, t\_17\right)\right)\right)\right)}}{t\_9}}{t\_1}}{b \cdot a} \cdot t\_2\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_4 \cdot \left(t\_3 \cdot \left(t\_12 + \left(t\_14 + \mathsf{hypot}\left(t\_12 - t\_14, t\_17\right)\right)\right)\right)}}{t\_9}}{t\_1}}{b \cdot a} \cdot t\_2\right) \cdot t\_2\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* (* (+ PI PI) angle) 0.005555555555555556)))
        (t_1 (* 4.0 (* b a)))
        (t_2 (* y-scale (fabs x-scale)))
        (t_3 (* (* (* b a) b) (- a)))
        (t_4 (* t_3 8.0))
        (t_5 (* (* 2.0 PI) (* angle 0.005555555555555556)))
        (t_6 (cos t_5))
        (t_7 (- 0.5 (* t_6 0.5)))
        (t_8 (* (fabs x-scale) y-scale))
        (t_9 (fabs t_8))
        (t_10 (- 0.5 (* t_0 0.5)))
        (t_11 (fma t_6 0.5 0.5))
        (t_12 (/ (fma (* t_11 a) a (* (* t_7 b) b)) (* y-scale y-scale)))
        (t_13 (fma 0.5 t_0 0.5))
        (t_14
         (/
          (/ (fma t_10 (* a a) (* (* t_13 b) b)) (fabs x-scale))
          (fabs x-scale)))
        (t_15 (/ (/ (fma t_13 (* a a) (* (* t_10 b) b)) y-scale) y-scale))
        (t_16
         (/
          (fma (* t_7 a) a (* (* t_11 b) b))
          (* (fabs x-scale) (fabs x-scale))))
        (t_17 (/ (* (sin t_5) (* (- b a) (+ b a))) t_8)))
   (if (<= (fabs x-scale) 2e+150)
     (*
      (*
       (/
        (/
         (/
          (sqrt (* t_4 (* t_3 (+ t_15 (+ t_16 (hypot (- t_15 t_16) t_17))))))
          t_9)
         t_1)
        (* b a))
       t_2)
      t_2)
     (*
      (*
       (/
        (/
         (/
          (sqrt (* t_4 (* t_3 (+ t_12 (+ t_14 (hypot (- t_12 t_14) t_17))))))
          t_9)
         t_1)
        (* b a))
       t_2)
      t_2))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos((((((double) M_PI) + ((double) M_PI)) * angle) * 0.005555555555555556));
	double t_1 = 4.0 * (b * a);
	double t_2 = y_45_scale * fabs(x_45_scale);
	double t_3 = ((b * a) * b) * -a;
	double t_4 = t_3 * 8.0;
	double t_5 = (2.0 * ((double) M_PI)) * (angle * 0.005555555555555556);
	double t_6 = cos(t_5);
	double t_7 = 0.5 - (t_6 * 0.5);
	double t_8 = fabs(x_45_scale) * y_45_scale;
	double t_9 = fabs(t_8);
	double t_10 = 0.5 - (t_0 * 0.5);
	double t_11 = fma(t_6, 0.5, 0.5);
	double t_12 = fma((t_11 * a), a, ((t_7 * b) * b)) / (y_45_scale * y_45_scale);
	double t_13 = fma(0.5, t_0, 0.5);
	double t_14 = (fma(t_10, (a * a), ((t_13 * b) * b)) / fabs(x_45_scale)) / fabs(x_45_scale);
	double t_15 = (fma(t_13, (a * a), ((t_10 * b) * b)) / y_45_scale) / y_45_scale;
	double t_16 = fma((t_7 * a), a, ((t_11 * b) * b)) / (fabs(x_45_scale) * fabs(x_45_scale));
	double t_17 = (sin(t_5) * ((b - a) * (b + a))) / t_8;
	double tmp;
	if (fabs(x_45_scale) <= 2e+150) {
		tmp = ((((sqrt((t_4 * (t_3 * (t_15 + (t_16 + hypot((t_15 - t_16), t_17)))))) / t_9) / t_1) / (b * a)) * t_2) * t_2;
	} else {
		tmp = ((((sqrt((t_4 * (t_3 * (t_12 + (t_14 + hypot((t_12 - t_14), t_17)))))) / t_9) / t_1) / (b * a)) * t_2) * t_2;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(Float64(Float64(pi + pi) * angle) * 0.005555555555555556))
	t_1 = Float64(4.0 * Float64(b * a))
	t_2 = Float64(y_45_scale * abs(x_45_scale))
	t_3 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	t_4 = Float64(t_3 * 8.0)
	t_5 = Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))
	t_6 = cos(t_5)
	t_7 = Float64(0.5 - Float64(t_6 * 0.5))
	t_8 = Float64(abs(x_45_scale) * y_45_scale)
	t_9 = abs(t_8)
	t_10 = Float64(0.5 - Float64(t_0 * 0.5))
	t_11 = fma(t_6, 0.5, 0.5)
	t_12 = Float64(fma(Float64(t_11 * a), a, Float64(Float64(t_7 * b) * b)) / Float64(y_45_scale * y_45_scale))
	t_13 = fma(0.5, t_0, 0.5)
	t_14 = Float64(Float64(fma(t_10, Float64(a * a), Float64(Float64(t_13 * b) * b)) / abs(x_45_scale)) / abs(x_45_scale))
	t_15 = Float64(Float64(fma(t_13, Float64(a * a), Float64(Float64(t_10 * b) * b)) / y_45_scale) / y_45_scale)
	t_16 = Float64(fma(Float64(t_7 * a), a, Float64(Float64(t_11 * b) * b)) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	t_17 = Float64(Float64(sin(t_5) * Float64(Float64(b - a) * Float64(b + a))) / t_8)
	tmp = 0.0
	if (abs(x_45_scale) <= 2e+150)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_4 * Float64(t_3 * Float64(t_15 + Float64(t_16 + hypot(Float64(t_15 - t_16), t_17)))))) / t_9) / t_1) / Float64(b * a)) * t_2) * t_2);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_4 * Float64(t_3 * Float64(t_12 + Float64(t_14 + hypot(Float64(t_12 - t_14), t_17)))))) / t_9) / t_1) / Float64(b * a)) * t_2) * t_2);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(N[(N[(Pi + Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * 8.0), $MachinePrecision]}, Block[{t$95$5 = N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Cos[t$95$5], $MachinePrecision]}, Block[{t$95$7 = N[(0.5 - N[(t$95$6 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[Abs[x$45$scale], $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$9 = N[Abs[t$95$8], $MachinePrecision]}, Block[{t$95$10 = N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[(t$95$6 * 0.5 + 0.5), $MachinePrecision]}, Block[{t$95$12 = N[(N[(N[(t$95$11 * a), $MachinePrecision] * a + N[(N[(t$95$7 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$13 = N[(0.5 * t$95$0 + 0.5), $MachinePrecision]}, Block[{t$95$14 = N[(N[(N[(t$95$10 * N[(a * a), $MachinePrecision] + N[(N[(t$95$13 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$15 = N[(N[(N[(t$95$13 * N[(a * a), $MachinePrecision] + N[(N[(t$95$10 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$16 = N[(N[(N[(t$95$7 * a), $MachinePrecision] * a + N[(N[(t$95$11 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$17 = N[(N[(N[Sin[t$95$5], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$8), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 2e+150], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$4 * N[(t$95$3 * N[(t$95$15 + N[(t$95$16 + N[Sqrt[N[(t$95$15 - t$95$16), $MachinePrecision] ^ 2 + t$95$17 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$9), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$4 * N[(t$95$3 * N[(t$95$12 + N[(t$95$14 + N[Sqrt[N[(t$95$12 - t$95$14), $MachinePrecision] ^ 2 + t$95$17 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$9), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2e150

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   7. Applied rewrites 15.1%

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

   9. Applied rewrites 15.2%

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

   if 2e150 < x-scale

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   7. Applied rewrites 13.4%

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

   9. Applied rewrites 13.6%

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

Alternative 2: 17.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\left(\left(\pi + \pi\right) \cdot angle\right) \cdot 0.005555555555555556\right)\\ t_1 := 4 \cdot \left(b \cdot a\right)\\ t_2 := y-scale \cdot \left|x-scale\right|\\ t_3 := \frac{\frac{{b}^{2}}{\left|x-scale\right|}}{\left|x-scale\right|}\\ t_4 := \frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_0, 0.5\right), a \cdot a, \left(\left(0.5 - t\_0 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\\ t_5 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ t_6 := t\_5 \cdot 8\\ t_7 := \left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_8 := \cos t\_7\\ t_9 := \mathsf{fma}\left(t\_8, 0.5, 0.5\right)\\ t_10 := 0.5 - t\_8 \cdot 0.5\\ t_11 := \frac{\mathsf{fma}\left(t\_9 \cdot a, a, \left(t\_10 \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\\ t_12 := \left|x-scale\right| \cdot y-scale\\ t_13 := \left|t\_12\right|\\ t_14 := \frac{\mathsf{fma}\left(t\_10 \cdot a, a, \left(t\_9 \cdot b\right) \cdot b\right)}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ t_15 := \frac{\sin t\_7 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{t\_12}\\ \mathbf{if}\;\left|x-scale\right| \leq 2 \cdot 10^{+150}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_6 \cdot \left(t\_5 \cdot \left(t\_4 + \left(t\_14 + \mathsf{hypot}\left(t\_4 - t\_14, t\_15\right)\right)\right)\right)}}{t\_13}}{t\_1}}{b \cdot a} \cdot t\_2\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_6 \cdot \left(t\_5 \cdot \left(t\_11 + \left(t\_3 + \mathsf{hypot}\left(t\_11 - t\_3, t\_15\right)\right)\right)\right)}}{t\_13}}{t\_1}}{b \cdot a} \cdot t\_2\right) \cdot t\_2\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* (* (+ PI PI) angle) 0.005555555555555556)))
        (t_1 (* 4.0 (* b a)))
        (t_2 (* y-scale (fabs x-scale)))
        (t_3 (/ (/ (pow b 2.0) (fabs x-scale)) (fabs x-scale)))
        (t_4
         (/
          (/
           (fma (fma 0.5 t_0 0.5) (* a a) (* (* (- 0.5 (* t_0 0.5)) b) b))
           y-scale)
          y-scale))
        (t_5 (* (* (* b a) b) (- a)))
        (t_6 (* t_5 8.0))
        (t_7 (* (* 2.0 PI) (* angle 0.005555555555555556)))
        (t_8 (cos t_7))
        (t_9 (fma t_8 0.5 0.5))
        (t_10 (- 0.5 (* t_8 0.5)))
        (t_11 (/ (fma (* t_9 a) a (* (* t_10 b) b)) (* y-scale y-scale)))
        (t_12 (* (fabs x-scale) y-scale))
        (t_13 (fabs t_12))
        (t_14
         (/
          (fma (* t_10 a) a (* (* t_9 b) b))
          (* (fabs x-scale) (fabs x-scale))))
        (t_15 (/ (* (sin t_7) (* (- b a) (+ b a))) t_12)))
   (if (<= (fabs x-scale) 2e+150)
     (*
      (*
       (/
        (/
         (/
          (sqrt (* t_6 (* t_5 (+ t_4 (+ t_14 (hypot (- t_4 t_14) t_15))))))
          t_13)
         t_1)
        (* b a))
       t_2)
      t_2)
     (*
      (*
       (/
        (/
         (/
          (sqrt (* t_6 (* t_5 (+ t_11 (+ t_3 (hypot (- t_11 t_3) t_15))))))
          t_13)
         t_1)
        (* b a))
       t_2)
      t_2))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos((((((double) M_PI) + ((double) M_PI)) * angle) * 0.005555555555555556));
	double t_1 = 4.0 * (b * a);
	double t_2 = y_45_scale * fabs(x_45_scale);
	double t_3 = (pow(b, 2.0) / fabs(x_45_scale)) / fabs(x_45_scale);
	double t_4 = (fma(fma(0.5, t_0, 0.5), (a * a), (((0.5 - (t_0 * 0.5)) * b) * b)) / y_45_scale) / y_45_scale;
	double t_5 = ((b * a) * b) * -a;
	double t_6 = t_5 * 8.0;
	double t_7 = (2.0 * ((double) M_PI)) * (angle * 0.005555555555555556);
	double t_8 = cos(t_7);
	double t_9 = fma(t_8, 0.5, 0.5);
	double t_10 = 0.5 - (t_8 * 0.5);
	double t_11 = fma((t_9 * a), a, ((t_10 * b) * b)) / (y_45_scale * y_45_scale);
	double t_12 = fabs(x_45_scale) * y_45_scale;
	double t_13 = fabs(t_12);
	double t_14 = fma((t_10 * a), a, ((t_9 * b) * b)) / (fabs(x_45_scale) * fabs(x_45_scale));
	double t_15 = (sin(t_7) * ((b - a) * (b + a))) / t_12;
	double tmp;
	if (fabs(x_45_scale) <= 2e+150) {
		tmp = ((((sqrt((t_6 * (t_5 * (t_4 + (t_14 + hypot((t_4 - t_14), t_15)))))) / t_13) / t_1) / (b * a)) * t_2) * t_2;
	} else {
		tmp = ((((sqrt((t_6 * (t_5 * (t_11 + (t_3 + hypot((t_11 - t_3), t_15)))))) / t_13) / t_1) / (b * a)) * t_2) * t_2;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(Float64(Float64(pi + pi) * angle) * 0.005555555555555556))
	t_1 = Float64(4.0 * Float64(b * a))
	t_2 = Float64(y_45_scale * abs(x_45_scale))
	t_3 = Float64(Float64((b ^ 2.0) / abs(x_45_scale)) / abs(x_45_scale))
	t_4 = Float64(Float64(fma(fma(0.5, t_0, 0.5), Float64(a * a), Float64(Float64(Float64(0.5 - Float64(t_0 * 0.5)) * b) * b)) / y_45_scale) / y_45_scale)
	t_5 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	t_6 = Float64(t_5 * 8.0)
	t_7 = Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))
	t_8 = cos(t_7)
	t_9 = fma(t_8, 0.5, 0.5)
	t_10 = Float64(0.5 - Float64(t_8 * 0.5))
	t_11 = Float64(fma(Float64(t_9 * a), a, Float64(Float64(t_10 * b) * b)) / Float64(y_45_scale * y_45_scale))
	t_12 = Float64(abs(x_45_scale) * y_45_scale)
	t_13 = abs(t_12)
	t_14 = Float64(fma(Float64(t_10 * a), a, Float64(Float64(t_9 * b) * b)) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	t_15 = Float64(Float64(sin(t_7) * Float64(Float64(b - a) * Float64(b + a))) / t_12)
	tmp = 0.0
	if (abs(x_45_scale) <= 2e+150)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_6 * Float64(t_5 * Float64(t_4 + Float64(t_14 + hypot(Float64(t_4 - t_14), t_15)))))) / t_13) / t_1) / Float64(b * a)) * t_2) * t_2);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_6 * Float64(t_5 * Float64(t_11 + Float64(t_3 + hypot(Float64(t_11 - t_3), t_15)))))) / t_13) / t_1) / Float64(b * a)) * t_2) * t_2);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(N[(N[(Pi + Pi), $MachinePrecision] * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Power[b, 2.0], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(0.5 * t$95$0 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * 8.0), $MachinePrecision]}, Block[{t$95$7 = N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Cos[t$95$7], $MachinePrecision]}, Block[{t$95$9 = N[(t$95$8 * 0.5 + 0.5), $MachinePrecision]}, Block[{t$95$10 = N[(0.5 - N[(t$95$8 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[(N[(N[(t$95$9 * a), $MachinePrecision] * a + N[(N[(t$95$10 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$12 = N[(N[Abs[x$45$scale], $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$13 = N[Abs[t$95$12], $MachinePrecision]}, Block[{t$95$14 = N[(N[(N[(t$95$10 * a), $MachinePrecision] * a + N[(N[(t$95$9 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$15 = N[(N[(N[Sin[t$95$7], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$12), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 2e+150], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$6 * N[(t$95$5 * N[(t$95$4 + N[(t$95$14 + N[Sqrt[N[(t$95$4 - t$95$14), $MachinePrecision] ^ 2 + t$95$15 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$13), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$6 * N[(t$95$5 * N[(t$95$11 + N[(t$95$3 + N[Sqrt[N[(t$95$11 - t$95$3), $MachinePrecision] ^ 2 + t$95$15 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$13), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2e150

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   7. Applied rewrites 15.1%

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

   9. Applied rewrites 15.2%

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

   if 2e150 < x-scale

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   7. Applied rewrites 13.4%

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

   9. Applied rewrites 13.6%

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

   10. Taylor expanded in angle around 0

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

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 15.4%

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

   12. Applied rewrites 15.4%

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

   13. Taylor expanded in angle around 0

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

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 15.3%

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

   15. Applied rewrites 15.3%

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

Alternative 3: 17.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ t_2 := t\_1 \cdot 8\\ t_3 := x-scale \cdot \left|y-scale\right|\\ t_4 := 4 \cdot \left(b \cdot a\right)\\ t_5 := \cos t\_0\\ t_6 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_5, 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - t\_5 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{\left|y-scale\right| \cdot \left|y-scale\right|}\\ t_7 := \left|t\_3\right|\\ t_8 := \frac{\frac{{b}^{2}}{x-scale}}{x-scale}\\ t_9 := \left|y-scale\right| \cdot x-scale\\ t_10 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_11 := \mathsf{fma}\left({a}^{2}, 0.5 - t\_10, {b}^{2} \cdot \left(0.5 + t\_10\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 1.1 \cdot 10^{+157}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_2 \cdot \left(t\_1 \cdot \left(t\_6 + \left(t\_8 + \mathsf{hypot}\left(t\_6 - t\_8, \frac{\sin t\_0 \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{t\_3}\right)\right)\right)\right)}}{t\_7}}{t\_4}}{b \cdot a} \cdot t\_9\right) \cdot t\_9\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_2 \cdot \left(t\_1 \cdot \frac{\sqrt{{t\_11}^{2}} + t\_11}{{x-scale}^{2}}\right)}}{t\_7}}{t\_4}}{b \cdot a} \cdot t\_9\right) \cdot t\_9\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* 2.0 PI) (* angle 0.005555555555555556)))
        (t_1 (* (* (* b a) b) (- a)))
        (t_2 (* t_1 8.0))
        (t_3 (* x-scale (fabs y-scale)))
        (t_4 (* 4.0 (* b a)))
        (t_5 (cos t_0))
        (t_6
         (/
          (fma (* (fma t_5 0.5 0.5) a) a (* (* (- 0.5 (* t_5 0.5)) b) b))
          (* (fabs y-scale) (fabs y-scale))))
        (t_7 (fabs t_3))
        (t_8 (/ (/ (pow b 2.0) x-scale) x-scale))
        (t_9 (* (fabs y-scale) x-scale))
        (t_10 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
        (t_11 (fma (pow a 2.0) (- 0.5 t_10) (* (pow b 2.0) (+ 0.5 t_10)))))
   (if (<= (fabs y-scale) 1.1e+157)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            t_2
            (*
             t_1
             (+
              t_6
              (+
               t_8
               (hypot
                (- t_6 t_8)
                (/ (* (sin t_0) (* (- b a) (+ b a))) t_3)))))))
          t_7)
         t_4)
        (* b a))
       t_9)
      t_9)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            t_2
            (* t_1 (/ (+ (sqrt (pow t_11 2.0)) t_11) (pow x-scale 2.0)))))
          t_7)
         t_4)
        (* b a))
       t_9)
      t_9))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (2.0 * ((double) M_PI)) * (angle * 0.005555555555555556);
	double t_1 = ((b * a) * b) * -a;
	double t_2 = t_1 * 8.0;
	double t_3 = x_45_scale * fabs(y_45_scale);
	double t_4 = 4.0 * (b * a);
	double t_5 = cos(t_0);
	double t_6 = fma((fma(t_5, 0.5, 0.5) * a), a, (((0.5 - (t_5 * 0.5)) * b) * b)) / (fabs(y_45_scale) * fabs(y_45_scale));
	double t_7 = fabs(t_3);
	double t_8 = (pow(b, 2.0) / x_45_scale) / x_45_scale;
	double t_9 = fabs(y_45_scale) * x_45_scale;
	double t_10 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_11 = fma(pow(a, 2.0), (0.5 - t_10), (pow(b, 2.0) * (0.5 + t_10)));
	double tmp;
	if (fabs(y_45_scale) <= 1.1e+157) {
		tmp = ((((sqrt((t_2 * (t_1 * (t_6 + (t_8 + hypot((t_6 - t_8), ((sin(t_0) * ((b - a) * (b + a))) / t_3))))))) / t_7) / t_4) / (b * a)) * t_9) * t_9;
	} else {
		tmp = ((((sqrt((t_2 * (t_1 * ((sqrt(pow(t_11, 2.0)) + t_11) / pow(x_45_scale, 2.0))))) / t_7) / t_4) / (b * a)) * t_9) * t_9;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))
	t_1 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	t_2 = Float64(t_1 * 8.0)
	t_3 = Float64(x_45_scale * abs(y_45_scale))
	t_4 = Float64(4.0 * Float64(b * a))
	t_5 = cos(t_0)
	t_6 = Float64(fma(Float64(fma(t_5, 0.5, 0.5) * a), a, Float64(Float64(Float64(0.5 - Float64(t_5 * 0.5)) * b) * b)) / Float64(abs(y_45_scale) * abs(y_45_scale)))
	t_7 = abs(t_3)
	t_8 = Float64(Float64((b ^ 2.0) / x_45_scale) / x_45_scale)
	t_9 = Float64(abs(y_45_scale) * x_45_scale)
	t_10 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_11 = fma((a ^ 2.0), Float64(0.5 - t_10), Float64((b ^ 2.0) * Float64(0.5 + t_10)))
	tmp = 0.0
	if (abs(y_45_scale) <= 1.1e+157)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_2 * Float64(t_1 * Float64(t_6 + Float64(t_8 + hypot(Float64(t_6 - t_8), Float64(Float64(sin(t_0) * Float64(Float64(b - a) * Float64(b + a))) / t_3))))))) / t_7) / t_4) / Float64(b * a)) * t_9) * t_9);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_2 * Float64(t_1 * Float64(Float64(sqrt((t_11 ^ 2.0)) + t_11) / (x_45_scale ^ 2.0))))) / t_7) / t_4) / Float64(b * a)) * t_9) * t_9);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 8.0), $MachinePrecision]}, Block[{t$95$3 = N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(t$95$5 * 0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(0.5 - N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Abs[t$95$3], $MachinePrecision]}, Block[{t$95$8 = N[(N[(N[Power[b, 2.0], $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, Block[{t$95$9 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$10 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[(N[Power[a, 2.0], $MachinePrecision] * N[(0.5 - t$95$10), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * N[(0.5 + t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.1e+157], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$2 * N[(t$95$1 * N[(t$95$6 + N[(t$95$8 + N[Sqrt[N[(t$95$6 - t$95$8), $MachinePrecision] ^ 2 + N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$7), $MachinePrecision] / t$95$4), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$9), $MachinePrecision] * t$95$9), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$2 * N[(t$95$1 * N[(N[(N[Sqrt[N[Power[t$95$11, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$11), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$7), $MachinePrecision] / t$95$4), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$9), $MachinePrecision] * t$95$9), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.1000000000000001e157

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   7. Applied rewrites 13.4%

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

   9. Applied rewrites 13.6%

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

   10. Taylor expanded in angle around 0

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

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 15.4%

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

   12. Applied rewrites 15.4%

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

   13. Taylor expanded in angle around 0

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

       1. lower-/.f64 N/A

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

       2. lower-pow.f64 15.3%

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

   15. Applied rewrites 15.3%

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

   if 1.1000000000000001e157 < y-scale

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 10.3%

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

Alternative 4: 16.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := 4 \cdot \left(b \cdot a\right)\\ t_1 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ t_2 := \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_3 := \left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\\ t_4 := 0.5 \cdot t\_2\\ t_5 := y-scale \cdot \left|x-scale\right|\\ t_6 := \mathsf{fma}\left({a}^{2}, 0.5 - t\_4, {b}^{2} \cdot \left(0.5 + t\_4\right)\right)\\ t_7 := \left|\left|x-scale\right| \cdot y-scale\right|\\ t_8 := \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\left|x-scale\right| \leq 7.7 \cdot 10^{+121}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_1 \cdot 8\right) \cdot \left(t\_1 \cdot \frac{\sqrt{{t\_6}^{2}} + t\_6}{{\left(\left|x-scale\right|\right)}^{2}}\right)}}{t\_7}}{t\_0}}{b \cdot a} \cdot t\_5\right) \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_3 \cdot \left(8 \cdot \left(\left(\frac{\left|\mathsf{fma}\left(0.5 - t\_2 \cdot 0.5, b \cdot b, \mathsf{fma}\left(t\_2, 0.5, 0.5\right) \cdot \left(a \cdot a\right)\right)\right|}{y-scale \cdot y-scale} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_8, 0.5\right), a \cdot a, \left(\left(0.5 - t\_8 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\right) \cdot t\_3\right)\right)}}{t\_7}}{t\_0}}{b \cdot a} \cdot t\_5\right) \cdot t\_5\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b a)))
        (t_1 (* (* (* b a) b) (- a)))
        (t_2 (cos (* 0.011111111111111112 (* angle PI))))
        (t_3 (* (* (* (- a) b) a) b))
        (t_4 (* 0.5 t_2))
        (t_5 (* y-scale (fabs x-scale)))
        (t_6 (fma (pow a 2.0) (- 0.5 t_4) (* (pow b 2.0) (+ 0.5 t_4))))
        (t_7 (fabs (* (fabs x-scale) y-scale)))
        (t_8 (cos (* (* angle (+ PI PI)) 0.005555555555555556))))
   (if (<= (fabs x-scale) 7.7e+121)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            (* t_1 8.0)
            (* t_1 (/ (+ (sqrt (pow t_6 2.0)) t_6) (pow (fabs x-scale) 2.0)))))
          t_7)
         t_0)
        (* b a))
       t_5)
      t_5)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            t_3
            (*
             8.0
             (*
              (+
               (/
                (fabs
                 (fma
                  (- 0.5 (* t_2 0.5))
                  (* b b)
                  (* (fma t_2 0.5 0.5) (* a a))))
                (* y-scale y-scale))
               (/
                (fma (fma 0.5 t_8 0.5) (* a a) (* (* (- 0.5 (* t_8 0.5)) b) b))
                (* y-scale y-scale)))
              t_3))))
          t_7)
         t_0)
        (* b a))
       t_5)
      t_5))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 4.0 * (b * a);
	double t_1 = ((b * a) * b) * -a;
	double t_2 = cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_3 = ((-a * b) * a) * b;
	double t_4 = 0.5 * t_2;
	double t_5 = y_45_scale * fabs(x_45_scale);
	double t_6 = fma(pow(a, 2.0), (0.5 - t_4), (pow(b, 2.0) * (0.5 + t_4)));
	double t_7 = fabs((fabs(x_45_scale) * y_45_scale));
	double t_8 = cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556));
	double tmp;
	if (fabs(x_45_scale) <= 7.7e+121) {
		tmp = ((((sqrt(((t_1 * 8.0) * (t_1 * ((sqrt(pow(t_6, 2.0)) + t_6) / pow(fabs(x_45_scale), 2.0))))) / t_7) / t_0) / (b * a)) * t_5) * t_5;
	} else {
		tmp = ((((sqrt((t_3 * (8.0 * (((fabs(fma((0.5 - (t_2 * 0.5)), (b * b), (fma(t_2, 0.5, 0.5) * (a * a)))) / (y_45_scale * y_45_scale)) + (fma(fma(0.5, t_8, 0.5), (a * a), (((0.5 - (t_8 * 0.5)) * b) * b)) / (y_45_scale * y_45_scale))) * t_3)))) / t_7) / t_0) / (b * a)) * t_5) * t_5;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(4.0 * Float64(b * a))
	t_1 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	t_2 = cos(Float64(0.011111111111111112 * Float64(angle * pi)))
	t_3 = Float64(Float64(Float64(Float64(-a) * b) * a) * b)
	t_4 = Float64(0.5 * t_2)
	t_5 = Float64(y_45_scale * abs(x_45_scale))
	t_6 = fma((a ^ 2.0), Float64(0.5 - t_4), Float64((b ^ 2.0) * Float64(0.5 + t_4)))
	t_7 = abs(Float64(abs(x_45_scale) * y_45_scale))
	t_8 = cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556))
	tmp = 0.0
	if (abs(x_45_scale) <= 7.7e+121)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_1 * 8.0) * Float64(t_1 * Float64(Float64(sqrt((t_6 ^ 2.0)) + t_6) / (abs(x_45_scale) ^ 2.0))))) / t_7) / t_0) / Float64(b * a)) * t_5) * t_5);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_3 * Float64(8.0 * Float64(Float64(Float64(abs(fma(Float64(0.5 - Float64(t_2 * 0.5)), Float64(b * b), Float64(fma(t_2, 0.5, 0.5) * Float64(a * a)))) / Float64(y_45_scale * y_45_scale)) + Float64(fma(fma(0.5, t_8, 0.5), Float64(a * a), Float64(Float64(Float64(0.5 - Float64(t_8 * 0.5)) * b) * b)) / Float64(y_45_scale * y_45_scale))) * t_3)))) / t_7) / t_0) / Float64(b * a)) * t_5) * t_5);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[((-a) * b), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[a, 2.0], $MachinePrecision] * N[(0.5 - t$95$4), $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] * N[(0.5 + t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * y$45$scale), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 7.7e+121], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$1 * 8.0), $MachinePrecision] * N[(t$95$1 * N[(N[(N[Sqrt[N[Power[t$95$6, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$6), $MachinePrecision] / N[Power[N[Abs[x$45$scale], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$7), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * t$95$5), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$3 * N[(8.0 * N[(N[(N[(N[Abs[N[(N[(0.5 - N[(t$95$2 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(t$95$2 * 0.5 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 * t$95$8 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(0.5 - N[(t$95$8 * 0.5), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$7), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * t$95$5), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 7.7000000000000003e121

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in x-scale around 0

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

   8. Applied rewrites 10.3%

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

   if 7.7000000000000003e121 < x-scale

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in y-scale around 0

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.2%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left({a}^{2}, 0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Applied rewrites 10.6%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\color{blue}{\left(\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\right) \cdot \left(8 \cdot \left(\left(\frac{\left|\mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, b \cdot b, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\right)\right|}{y-scale \cdot y-scale} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right), 0.5\right), a \cdot a, \left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\right)\right)\right)}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 14.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{{b}^{2}}{{x-scale}^{2}}\\ t_1 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ t_2 := \left|x-scale \cdot \left|y-scale\right|\right|\\ t_3 := \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right)\\ t_4 := \left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\\ t_5 := \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_6 := \left|y-scale\right| \cdot \left|y-scale\right|\\ t_7 := 4 \cdot \left(b \cdot a\right)\\ t_8 := \left|y-scale\right| \cdot x-scale\\ t_9 := \frac{{a}^{2}}{{\left(\left|y-scale\right|\right)}^{2}}\\ \mathbf{if}\;\left|y-scale\right| \leq 3 \cdot 10^{+89}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_4 \cdot \left(8 \cdot \left(\left(\frac{\left|\mathsf{fma}\left(0.5 - t\_5 \cdot 0.5, b \cdot b, \mathsf{fma}\left(t\_5, 0.5, 0.5\right) \cdot \left(a \cdot a\right)\right)\right|}{t\_6} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_3, 0.5\right), a \cdot a, \left(\left(0.5 - t\_3 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{t\_6}\right) \cdot t\_4\right)\right)}}{t\_2}}{t\_7}}{b \cdot a} \cdot t\_8\right) \cdot t\_8\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_1 \cdot 8\right) \cdot \left(t\_1 \cdot \left(\sqrt{{\left(t\_9 - t\_0\right)}^{2}} + \left(t\_9 + t\_0\right)\right)\right)}}{t\_2}}{t\_7}}{b \cdot a} \cdot t\_8\right) \cdot t\_8\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (pow b 2.0) (pow x-scale 2.0)))
        (t_1 (* (* (* b a) b) (- a)))
        (t_2 (fabs (* x-scale (fabs y-scale))))
        (t_3 (cos (* (* angle (+ PI PI)) 0.005555555555555556)))
        (t_4 (* (* (* (- a) b) a) b))
        (t_5 (cos (* 0.011111111111111112 (* angle PI))))
        (t_6 (* (fabs y-scale) (fabs y-scale)))
        (t_7 (* 4.0 (* b a)))
        (t_8 (* (fabs y-scale) x-scale))
        (t_9 (/ (pow a 2.0) (pow (fabs y-scale) 2.0))))
   (if (<= (fabs y-scale) 3e+89)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            t_4
            (*
             8.0
             (*
              (+
               (/
                (fabs
                 (fma
                  (- 0.5 (* t_5 0.5))
                  (* b b)
                  (* (fma t_5 0.5 0.5) (* a a))))
                t_6)
               (/
                (fma (fma 0.5 t_3 0.5) (* a a) (* (* (- 0.5 (* t_3 0.5)) b) b))
                t_6))
              t_4))))
          t_2)
         t_7)
        (* b a))
       t_8)
      t_8)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            (* t_1 8.0)
            (* t_1 (+ (sqrt (pow (- t_9 t_0) 2.0)) (+ t_9 t_0)))))
          t_2)
         t_7)
        (* b a))
       t_8)
      t_8))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(b, 2.0) / pow(x_45_scale, 2.0);
	double t_1 = ((b * a) * b) * -a;
	double t_2 = fabs((x_45_scale * fabs(y_45_scale)));
	double t_3 = cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556));
	double t_4 = ((-a * b) * a) * b;
	double t_5 = cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_6 = fabs(y_45_scale) * fabs(y_45_scale);
	double t_7 = 4.0 * (b * a);
	double t_8 = fabs(y_45_scale) * x_45_scale;
	double t_9 = pow(a, 2.0) / pow(fabs(y_45_scale), 2.0);
	double tmp;
	if (fabs(y_45_scale) <= 3e+89) {
		tmp = ((((sqrt((t_4 * (8.0 * (((fabs(fma((0.5 - (t_5 * 0.5)), (b * b), (fma(t_5, 0.5, 0.5) * (a * a)))) / t_6) + (fma(fma(0.5, t_3, 0.5), (a * a), (((0.5 - (t_3 * 0.5)) * b) * b)) / t_6)) * t_4)))) / t_2) / t_7) / (b * a)) * t_8) * t_8;
	} else {
		tmp = ((((sqrt(((t_1 * 8.0) * (t_1 * (sqrt(pow((t_9 - t_0), 2.0)) + (t_9 + t_0))))) / t_2) / t_7) / (b * a)) * t_8) * t_8;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64((b ^ 2.0) / (x_45_scale ^ 2.0))
	t_1 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	t_2 = abs(Float64(x_45_scale * abs(y_45_scale)))
	t_3 = cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556))
	t_4 = Float64(Float64(Float64(Float64(-a) * b) * a) * b)
	t_5 = cos(Float64(0.011111111111111112 * Float64(angle * pi)))
	t_6 = Float64(abs(y_45_scale) * abs(y_45_scale))
	t_7 = Float64(4.0 * Float64(b * a))
	t_8 = Float64(abs(y_45_scale) * x_45_scale)
	t_9 = Float64((a ^ 2.0) / (abs(y_45_scale) ^ 2.0))
	tmp = 0.0
	if (abs(y_45_scale) <= 3e+89)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_4 * Float64(8.0 * Float64(Float64(Float64(abs(fma(Float64(0.5 - Float64(t_5 * 0.5)), Float64(b * b), Float64(fma(t_5, 0.5, 0.5) * Float64(a * a)))) / t_6) + Float64(fma(fma(0.5, t_3, 0.5), Float64(a * a), Float64(Float64(Float64(0.5 - Float64(t_3 * 0.5)) * b) * b)) / t_6)) * t_4)))) / t_2) / t_7) / Float64(b * a)) * t_8) * t_8);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_1 * 8.0) * Float64(t_1 * Float64(sqrt((Float64(t_9 - t_0) ^ 2.0)) + Float64(t_9 + t_0))))) / t_2) / t_7) / Float64(b * a)) * t_8) * t_8);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Power[b, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[((-a) * b), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$9 = N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 3e+89], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$4 * N[(8.0 * N[(N[(N[(N[Abs[N[(N[(0.5 - N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(t$95$5 * 0.5 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision] + N[(N[(N[(0.5 * t$95$3 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$7), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$8), $MachinePrecision] * t$95$8), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$1 * 8.0), $MachinePrecision] * N[(t$95$1 * N[(N[Sqrt[N[Power[N[(t$95$9 - t$95$0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(t$95$9 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$7), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$8), $MachinePrecision] * t$95$8), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 3.0000000000000001e89

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in y-scale around 0

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.2%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left({a}^{2}, 0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   10. Applied rewrites 10.6%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\color{blue}{\left(\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\right) \cdot \left(8 \cdot \left(\left(\frac{\left|\mathsf{fma}\left(0.5 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot 0.5, b \cdot b, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5, 0.5\right) \cdot \left(a \cdot a\right)\right)\right|}{y-scale \cdot y-scale} + \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right), 0.5\right), a \cdot a, \left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale}\right) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\right)\right)\right)}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 3.0000000000000001e89 < y-scale

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 13.1%

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

Alternative 6: 14.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ t_1 := \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_2 := {\left(\left|y-scale\right|\right)}^{2}\\ t_3 := \frac{{a}^{2}}{t\_2}\\ t_4 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ t_5 := t\_4 \cdot 8\\ t_6 := \left|x-scale \cdot \left|y-scale\right|\right|\\ t_7 := 4 \cdot \left(b \cdot a\right)\\ t_8 := \frac{{b}^{2}}{{x-scale}^{2}}\\ \mathbf{if}\;\left|y-scale\right| \leq 4.2 \cdot 10^{+103}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_5 \cdot \left(t\_4 \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_1, 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - t\_1 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{\left|y-scale\right| \cdot \left|y-scale\right|} + \frac{\sqrt{{a}^{4}}}{t\_2}\right)\right)}}{t\_6}}{t\_7}}{b \cdot a} \cdot t\_0\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{t\_5 \cdot \left(t\_4 \cdot \left(\sqrt{{\left(t\_3 - t\_8\right)}^{2}} + \left(t\_3 + t\_8\right)\right)\right)}}{t\_6}}{t\_7}}{b \cdot a} \cdot t\_0\right) \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs y-scale) x-scale))
        (t_1 (cos (* (* 2.0 PI) (* angle 0.005555555555555556))))
        (t_2 (pow (fabs y-scale) 2.0))
        (t_3 (/ (pow a 2.0) t_2))
        (t_4 (* (* (* b a) b) (- a)))
        (t_5 (* t_4 8.0))
        (t_6 (fabs (* x-scale (fabs y-scale))))
        (t_7 (* 4.0 (* b a)))
        (t_8 (/ (pow b 2.0) (pow x-scale 2.0))))
   (if (<= (fabs y-scale) 4.2e+103)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            t_5
            (*
             t_4
             (+
              (/
               (fma (* (fma t_1 0.5 0.5) a) a (* (* (- 0.5 (* t_1 0.5)) b) b))
               (* (fabs y-scale) (fabs y-scale)))
              (/ (sqrt (pow a 4.0)) t_2)))))
          t_6)
         t_7)
        (* b a))
       t_0)
      t_0)
     (*
      (*
       (/
        (/
         (/
          (sqrt (* t_5 (* t_4 (+ (sqrt (pow (- t_3 t_8) 2.0)) (+ t_3 t_8)))))
          t_6)
         t_7)
        (* b a))
       t_0)
      t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(y_45_scale) * x_45_scale;
	double t_1 = cos(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556)));
	double t_2 = pow(fabs(y_45_scale), 2.0);
	double t_3 = pow(a, 2.0) / t_2;
	double t_4 = ((b * a) * b) * -a;
	double t_5 = t_4 * 8.0;
	double t_6 = fabs((x_45_scale * fabs(y_45_scale)));
	double t_7 = 4.0 * (b * a);
	double t_8 = pow(b, 2.0) / pow(x_45_scale, 2.0);
	double tmp;
	if (fabs(y_45_scale) <= 4.2e+103) {
		tmp = ((((sqrt((t_5 * (t_4 * ((fma((fma(t_1, 0.5, 0.5) * a), a, (((0.5 - (t_1 * 0.5)) * b) * b)) / (fabs(y_45_scale) * fabs(y_45_scale))) + (sqrt(pow(a, 4.0)) / t_2))))) / t_6) / t_7) / (b * a)) * t_0) * t_0;
	} else {
		tmp = ((((sqrt((t_5 * (t_4 * (sqrt(pow((t_3 - t_8), 2.0)) + (t_3 + t_8))))) / t_6) / t_7) / (b * a)) * t_0) * t_0;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	t_1 = cos(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556)))
	t_2 = abs(y_45_scale) ^ 2.0
	t_3 = Float64((a ^ 2.0) / t_2)
	t_4 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	t_5 = Float64(t_4 * 8.0)
	t_6 = abs(Float64(x_45_scale * abs(y_45_scale)))
	t_7 = Float64(4.0 * Float64(b * a))
	t_8 = Float64((b ^ 2.0) / (x_45_scale ^ 2.0))
	tmp = 0.0
	if (abs(y_45_scale) <= 4.2e+103)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_5 * Float64(t_4 * Float64(Float64(fma(Float64(fma(t_1, 0.5, 0.5) * a), a, Float64(Float64(Float64(0.5 - Float64(t_1 * 0.5)) * b) * b)) / Float64(abs(y_45_scale) * abs(y_45_scale))) + Float64(sqrt((a ^ 4.0)) / t_2))))) / t_6) / t_7) / Float64(b * a)) * t_0) * t_0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(t_5 * Float64(t_4 * Float64(sqrt((Float64(t_3 - t_8) ^ 2.0)) + Float64(t_3 + t_8))))) / t_6) / t_7) / Float64(b * a)) * t_0) * t_0);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[a, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * 8.0), $MachinePrecision]}, Block[{t$95$6 = N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[Power[b, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 4.2e+103], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$5 * N[(t$95$4 * N[(N[(N[(N[(N[(t$95$1 * 0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision] / t$95$7), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(t$95$5 * N[(t$95$4 * N[(N[Sqrt[N[Power[N[(t$95$3 - t$95$8), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(t$95$3 + t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision] / t$95$7), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.2000000000000003e103

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in y-scale around 0

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.2%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left({a}^{2}, 0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in angle around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-pow.f64 10.4%

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

   11. Applied rewrites 10.4%

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

   if 4.2000000000000003e103 < y-scale

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 13.1%

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

Alternative 7: 12.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|y-scale\right|\right)}^{2}\\ t_1 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ t_2 := \frac{{a}^{2}}{t\_0}\\ t_3 := \left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ t_4 := \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_5 := \left|y-scale\right| \cdot x-scale\\ t_6 := \frac{{b}^{2}}{{x-scale}^{2}}\\ \mathbf{if}\;\left|y-scale\right| \leq 5.5 \cdot 10^{+106}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_1 \cdot 8\right) \cdot \left(t\_1 \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_4, 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - t\_4 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{\left|y-scale\right| \cdot \left|y-scale\right|} + \frac{\sqrt{{a}^{4}}}{t\_0}\right)\right)}}{\left|x-scale \cdot \left|y-scale\right|\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot t\_5\right) \cdot t\_5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\sqrt{\left(8 \cdot t\_3\right) \cdot \left(t\_3 \cdot \left(\sqrt{{\left(t\_6 - t\_2\right)}^{2}} + \left(t\_2 + t\_6\right)\right)\right)}}{\left|t\_5\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot t\_5\right) \cdot t\_5\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs y-scale) 2.0))
        (t_1 (* (* (* b a) b) (- a)))
        (t_2 (/ (pow a 2.0) t_0))
        (t_3 (* (* (* (- a) b) b) a))
        (t_4 (cos (* (* 2.0 PI) (* angle 0.005555555555555556))))
        (t_5 (* (fabs y-scale) x-scale))
        (t_6 (/ (pow b 2.0) (pow x-scale 2.0))))
   (if (<= (fabs y-scale) 5.5e+106)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            (* t_1 8.0)
            (*
             t_1
             (+
              (/
               (fma (* (fma t_4 0.5 0.5) a) a (* (* (- 0.5 (* t_4 0.5)) b) b))
               (* (fabs y-scale) (fabs y-scale)))
              (/ (sqrt (pow a 4.0)) t_0)))))
          (fabs (* x-scale (fabs y-scale))))
         (* 4.0 (* b a)))
        (* b a))
       t_5)
      t_5)
     (*
      (*
       (/
        (/
         (sqrt
          (* (* 8.0 t_3) (* t_3 (+ (sqrt (pow (- t_6 t_2) 2.0)) (+ t_2 t_6)))))
         (fabs t_5))
        (* (* (* a b) 4.0) (* a b)))
       t_5)
      t_5))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(y_45_scale), 2.0);
	double t_1 = ((b * a) * b) * -a;
	double t_2 = pow(a, 2.0) / t_0;
	double t_3 = ((-a * b) * b) * a;
	double t_4 = cos(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556)));
	double t_5 = fabs(y_45_scale) * x_45_scale;
	double t_6 = pow(b, 2.0) / pow(x_45_scale, 2.0);
	double tmp;
	if (fabs(y_45_scale) <= 5.5e+106) {
		tmp = ((((sqrt(((t_1 * 8.0) * (t_1 * ((fma((fma(t_4, 0.5, 0.5) * a), a, (((0.5 - (t_4 * 0.5)) * b) * b)) / (fabs(y_45_scale) * fabs(y_45_scale))) + (sqrt(pow(a, 4.0)) / t_0))))) / fabs((x_45_scale * fabs(y_45_scale)))) / (4.0 * (b * a))) / (b * a)) * t_5) * t_5;
	} else {
		tmp = (((sqrt(((8.0 * t_3) * (t_3 * (sqrt(pow((t_6 - t_2), 2.0)) + (t_2 + t_6))))) / fabs(t_5)) / (((a * b) * 4.0) * (a * b))) * t_5) * t_5;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0
	t_1 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	t_2 = Float64((a ^ 2.0) / t_0)
	t_3 = Float64(Float64(Float64(Float64(-a) * b) * b) * a)
	t_4 = cos(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556)))
	t_5 = Float64(abs(y_45_scale) * x_45_scale)
	t_6 = Float64((b ^ 2.0) / (x_45_scale ^ 2.0))
	tmp = 0.0
	if (abs(y_45_scale) <= 5.5e+106)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_1 * 8.0) * Float64(t_1 * Float64(Float64(fma(Float64(fma(t_4, 0.5, 0.5) * a), a, Float64(Float64(Float64(0.5 - Float64(t_4 * 0.5)) * b) * b)) / Float64(abs(y_45_scale) * abs(y_45_scale))) + Float64(sqrt((a ^ 4.0)) / t_0))))) / abs(Float64(x_45_scale * abs(y_45_scale)))) / Float64(4.0 * Float64(b * a))) / Float64(b * a)) * t_5) * t_5);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * t_3) * Float64(t_3 * Float64(sqrt((Float64(t_6 - t_2) ^ 2.0)) + Float64(t_2 + t_6))))) / abs(t_5)) / Float64(Float64(Float64(a * b) * 4.0) * Float64(a * b))) * t_5) * t_5);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[a, 2.0], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[b, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 5.5e+106], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$1 * 8.0), $MachinePrecision] * N[(t$95$1 * N[(N[(N[(N[(N[(t$95$4 * 0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(0.5 - N[(t$95$4 * 0.5), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * t$95$5), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(N[(8.0 * t$95$3), $MachinePrecision] * N[(t$95$3 * N[(N[Sqrt[N[Power[N[(t$95$6 - t$95$2), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(t$95$2 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[t$95$5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * t$95$5), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 5.5e106

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in y-scale around 0

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.2%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left({a}^{2}, 0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in angle around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-pow.f64 10.4%

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

   11. Applied rewrites 10.4%

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

   if 5.5e106 < y-scale

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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 angle around 0

      \[\leadsto \left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 8.4%

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

Alternative 8: 11.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot a\\ t_1 := \left(t\_0 \cdot \left|b\right|\right) \cdot \left(-a\right)\\ t_2 := \left|y-scale\right| \cdot \left|x-scale\right|\\ t_3 := \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ t_4 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|y-scale\right| \leq 1.85 \cdot 10^{+159}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_1 \cdot 8\right) \cdot \left(t\_1 \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(t\_3, 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - t\_3 \cdot 0.5\right) \cdot \left|b\right|\right) \cdot \left|b\right|\right)}{\left|y-scale\right| \cdot \left|y-scale\right|} + \frac{\sqrt{{a}^{4}}}{{\left(\left|y-scale\right|\right)}^{2}}\right)\right)}}{\left|\left|x-scale\right| \cdot \left|y-scale\right|\right|}}{4 \cdot t\_0}}{t\_0} \cdot t\_2\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{\frac{\left(-\sqrt{\left(8 \cdot {a}^{4}\right) \cdot \left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) + \sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right)} \cdot \left(t\_4 \cdot \left|b\right|\right)\right) \cdot \left|x-scale\right|}{a \cdot a}}{t\_4}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs b) a))
        (t_1 (* (* t_0 (fabs b)) (- a)))
        (t_2 (* (fabs y-scale) (fabs x-scale)))
        (t_3 (cos (* (* 2.0 PI) (* angle 0.005555555555555556))))
        (t_4 (* (fabs b) (fabs b))))
   (if (<= (fabs y-scale) 1.85e+159)
     (*
      (*
       (/
        (/
         (/
          (sqrt
           (*
            (* t_1 8.0)
            (*
             t_1
             (+
              (/
               (fma
                (* (fma t_3 0.5 0.5) a)
                a
                (* (* (- 0.5 (* t_3 0.5)) (fabs b)) (fabs b)))
               (* (fabs y-scale) (fabs y-scale)))
              (/ (sqrt (pow a 4.0)) (pow (fabs y-scale) 2.0))))))
          (fabs (* (fabs x-scale) (fabs y-scale))))
         (* 4.0 t_0))
        t_0)
       t_2)
      t_2)
     (*
      -0.25
      (/
       (/
        (*
         (-
          (*
           (sqrt
            (*
             (* 8.0 (pow a 4.0))
             (+
              (-
               0.5
               (* (cos (* (* angle (+ PI PI)) 0.005555555555555556)) 0.5))
              (sqrt (pow (sin (* (* angle PI) 0.005555555555555556)) 4.0)))))
           (* t_4 (fabs b))))
         (fabs x-scale))
        (* a a))
       t_4)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) * a;
	double t_1 = (t_0 * fabs(b)) * -a;
	double t_2 = fabs(y_45_scale) * fabs(x_45_scale);
	double t_3 = cos(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556)));
	double t_4 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(y_45_scale) <= 1.85e+159) {
		tmp = ((((sqrt(((t_1 * 8.0) * (t_1 * ((fma((fma(t_3, 0.5, 0.5) * a), a, (((0.5 - (t_3 * 0.5)) * fabs(b)) * fabs(b))) / (fabs(y_45_scale) * fabs(y_45_scale))) + (sqrt(pow(a, 4.0)) / pow(fabs(y_45_scale), 2.0)))))) / fabs((fabs(x_45_scale) * fabs(y_45_scale)))) / (4.0 * t_0)) / t_0) * t_2) * t_2;
	} else {
		tmp = -0.25 * (((-(sqrt(((8.0 * pow(a, 4.0)) * ((0.5 - (cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556)) * 0.5)) + sqrt(pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))))) * (t_4 * fabs(b))) * fabs(x_45_scale)) / (a * a)) / t_4);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) * a)
	t_1 = Float64(Float64(t_0 * abs(b)) * Float64(-a))
	t_2 = Float64(abs(y_45_scale) * abs(x_45_scale))
	t_3 = cos(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556)))
	t_4 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(y_45_scale) <= 1.85e+159)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_1 * 8.0) * Float64(t_1 * Float64(Float64(fma(Float64(fma(t_3, 0.5, 0.5) * a), a, Float64(Float64(Float64(0.5 - Float64(t_3 * 0.5)) * abs(b)) * abs(b))) / Float64(abs(y_45_scale) * abs(y_45_scale))) + Float64(sqrt((a ^ 4.0)) / (abs(y_45_scale) ^ 2.0)))))) / abs(Float64(abs(x_45_scale) * abs(y_45_scale)))) / Float64(4.0 * t_0)) / t_0) * t_2) * t_2);
	else
		tmp = Float64(-0.25 * Float64(Float64(Float64(Float64(-Float64(sqrt(Float64(Float64(8.0 * (a ^ 4.0)) * Float64(Float64(0.5 - Float64(cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))))) * Float64(t_4 * abs(b)))) * abs(x_45_scale)) / Float64(a * a)) / t_4));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 1.85e+159], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$1 * 8.0), $MachinePrecision] * N[(t$95$1 * N[(N[(N[(N[(N[(t$95$3 * 0.5 + 0.5), $MachinePrecision] * a), $MachinePrecision] * a + N[(N[(N[(0.5 - N[(t$95$3 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision] / N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision], N[(-0.25 * N[(N[(N[((-N[(N[Sqrt[N[(N[(8.0 * N[Power[a, 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$4 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 1.85e159

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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)} \]

   5. Applied rewrites 12.2%

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

   6. Taylor expanded in y-scale around 0

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left({a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + {b}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 10.2%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right) \cdot a, a, \left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale \cdot y-scale} + \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left({a}^{2}, 0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), {b}^{2} \cdot \left(0.5 - 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}^{2}}}{{y-scale}^{2}}}\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in angle around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-pow.f64 10.4%

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

   11. Applied rewrites 10.4%

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

   if 1.85e159 < y-scale

   1. Initial program 2.5%

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

   2. Taylor expanded in x-scale around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto -0.25 \cdot \frac{x-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)\right)}}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in b around -inf

      \[\leadsto -0.25 \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

   10. Applied rewrites 1.2%

       \[\leadsto -0.25 \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

   12. Applied rewrites 1.6%

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

Alternative 9: 6.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \sqrt{{t\_0}^{4}} + {t\_0}^{2}\\ \mathbf{if}\;\left|a\right| \leq 1.15 \cdot 10^{+108}:\\ \;\;\;\;0.25 \cdot \frac{\left|b\right| \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\left(\left|a\right|\right)}^{4} \cdot t\_1}{{y-scale}^{2}}}\right)}{{\left(\left|a\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \frac{\left|a\right| \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{\left(\left|b\right|\right)}^{4} \cdot t\_1}{{x-scale}^{2}}}\right)}{{\left(\left|b\right|\right)}^{2}}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* angle PI))))
        (t_1 (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0))))
   (if (<= (fabs a) 1.15e+108)
     (*
      0.25
      (/
       (*
        (fabs b)
        (*
         (pow y-scale 2.0)
         (sqrt (* 8.0 (/ (* (pow (fabs a) 4.0) t_1) (pow y-scale 2.0))))))
       (pow (fabs a) 2.0)))
     (*
      0.25
      (/
       (*
        (fabs a)
        (*
         (pow x-scale 2.0)
         (sqrt (* 8.0 (/ (* (pow (fabs b) 4.0) t_1) (pow x-scale 2.0))))))
       (pow (fabs b) 2.0))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_1 = sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0);
	double tmp;
	if (fabs(a) <= 1.15e+108) {
		tmp = 0.25 * ((fabs(b) * (pow(y_45_scale, 2.0) * sqrt((8.0 * ((pow(fabs(a), 4.0) * t_1) / pow(y_45_scale, 2.0)))))) / pow(fabs(a), 2.0));
	} else {
		tmp = 0.25 * ((fabs(a) * (pow(x_45_scale, 2.0) * sqrt((8.0 * ((pow(fabs(b), 4.0) * t_1) / pow(x_45_scale, 2.0)))))) / pow(fabs(b), 2.0));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	double t_1 = Math.sqrt(Math.pow(t_0, 4.0)) + Math.pow(t_0, 2.0);
	double tmp;
	if (Math.abs(a) <= 1.15e+108) {
		tmp = 0.25 * ((Math.abs(b) * (Math.pow(y_45_scale, 2.0) * Math.sqrt((8.0 * ((Math.pow(Math.abs(a), 4.0) * t_1) / Math.pow(y_45_scale, 2.0)))))) / Math.pow(Math.abs(a), 2.0));
	} else {
		tmp = 0.25 * ((Math.abs(a) * (Math.pow(x_45_scale, 2.0) * Math.sqrt((8.0 * ((Math.pow(Math.abs(b), 4.0) * t_1) / Math.pow(x_45_scale, 2.0)))))) / Math.pow(Math.abs(b), 2.0));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.cos((0.005555555555555556 * (angle * math.pi)))
	t_1 = math.sqrt(math.pow(t_0, 4.0)) + math.pow(t_0, 2.0)
	tmp = 0
	if math.fabs(a) <= 1.15e+108:
		tmp = 0.25 * ((math.fabs(b) * (math.pow(y_45_scale, 2.0) * math.sqrt((8.0 * ((math.pow(math.fabs(a), 4.0) * t_1) / math.pow(y_45_scale, 2.0)))))) / math.pow(math.fabs(a), 2.0))
	else:
		tmp = 0.25 * ((math.fabs(a) * (math.pow(x_45_scale, 2.0) * math.sqrt((8.0 * ((math.pow(math.fabs(b), 4.0) * t_1) / math.pow(x_45_scale, 2.0)))))) / math.pow(math.fabs(b), 2.0))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_1 = Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0))
	tmp = 0.0
	if (abs(a) <= 1.15e+108)
		tmp = Float64(0.25 * Float64(Float64(abs(b) * Float64((y_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64((abs(a) ^ 4.0) * t_1) / (y_45_scale ^ 2.0)))))) / (abs(a) ^ 2.0)));
	else
		tmp = Float64(0.25 * Float64(Float64(abs(a) * Float64((x_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64((abs(b) ^ 4.0) * t_1) / (x_45_scale ^ 2.0)))))) / (abs(b) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = cos((0.005555555555555556 * (angle * pi)));
	t_1 = sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0);
	tmp = 0.0;
	if (abs(a) <= 1.15e+108)
		tmp = 0.25 * ((abs(b) * ((y_45_scale ^ 2.0) * sqrt((8.0 * (((abs(a) ^ 4.0) * t_1) / (y_45_scale ^ 2.0)))))) / (abs(a) ^ 2.0));
	else
		tmp = 0.25 * ((abs(a) * ((x_45_scale ^ 2.0) * sqrt((8.0 * (((abs(b) ^ 4.0) * t_1) / (x_45_scale ^ 2.0)))))) / (abs(b) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.15e+108], N[(0.25 * N[(N[(N[Abs[b], $MachinePrecision] * N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.1499999999999999e108

   1. Initial program 2.5%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{a}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

   7. Applied rewrites 2.9%

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

   if 1.1499999999999999e108 < a

   1. Initial program 2.5%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto 0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

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

   7. Applied rewrites 2.9%

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

Alternative 10: 5.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\left|y-scale\right|\right)}^{2}\\ t_1 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := \left|b\right| \cdot \left|b\right|\\ \mathbf{if}\;\left|y-scale\right| \leq 3.5 \cdot 10^{+130}:\\ \;\;\;\;0.25 \cdot \frac{\left|a\right| \cdot \left(t\_0 \cdot \sqrt{8 \cdot \frac{{\left(\left|b\right|\right)}^{4} \cdot \left(\sqrt{{t\_1}^{4}} + {t\_1}^{2}\right)}{t\_0}}\right)}{{\left(\left|b\right|\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \frac{\frac{\left(-\sqrt{\left(8 \cdot {\left(\left|a\right|\right)}^{4}\right) \cdot \left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) + \sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right)} \cdot \left(t\_2 \cdot \left|b\right|\right)\right) \cdot \left|x-scale\right|}{\left|a\right| \cdot \left|a\right|}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (pow (fabs y-scale) 2.0))
        (t_1 (sin (* 0.005555555555555556 (* angle PI))))
        (t_2 (* (fabs b) (fabs b))))
   (if (<= (fabs y-scale) 3.5e+130)
     (*
      0.25
      (/
       (*
        (fabs a)
        (*
         t_0
         (sqrt
          (*
           8.0
           (/
            (* (pow (fabs b) 4.0) (+ (sqrt (pow t_1 4.0)) (pow t_1 2.0)))
            t_0)))))
       (pow (fabs b) 2.0)))
     (*
      -0.25
      (/
       (/
        (*
         (-
          (*
           (sqrt
            (*
             (* 8.0 (pow (fabs a) 4.0))
             (+
              (-
               0.5
               (* (cos (* (* angle (+ PI PI)) 0.005555555555555556)) 0.5))
              (sqrt (pow (sin (* (* angle PI) 0.005555555555555556)) 4.0)))))
           (* t_2 (fabs b))))
         (fabs x-scale))
        (* (fabs a) (fabs a)))
       t_2)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = pow(fabs(y_45_scale), 2.0);
	double t_1 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_2 = fabs(b) * fabs(b);
	double tmp;
	if (fabs(y_45_scale) <= 3.5e+130) {
		tmp = 0.25 * ((fabs(a) * (t_0 * sqrt((8.0 * ((pow(fabs(b), 4.0) * (sqrt(pow(t_1, 4.0)) + pow(t_1, 2.0))) / t_0))))) / pow(fabs(b), 2.0));
	} else {
		tmp = -0.25 * (((-(sqrt(((8.0 * pow(fabs(a), 4.0)) * ((0.5 - (cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556)) * 0.5)) + sqrt(pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))))) * (t_2 * fabs(b))) * fabs(x_45_scale)) / (fabs(a) * fabs(a))) / t_2);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.pow(Math.abs(y_45_scale), 2.0);
	double t_1 = Math.sin((0.005555555555555556 * (angle * Math.PI)));
	double t_2 = Math.abs(b) * Math.abs(b);
	double tmp;
	if (Math.abs(y_45_scale) <= 3.5e+130) {
		tmp = 0.25 * ((Math.abs(a) * (t_0 * Math.sqrt((8.0 * ((Math.pow(Math.abs(b), 4.0) * (Math.sqrt(Math.pow(t_1, 4.0)) + Math.pow(t_1, 2.0))) / t_0))))) / Math.pow(Math.abs(b), 2.0));
	} else {
		tmp = -0.25 * (((-(Math.sqrt(((8.0 * Math.pow(Math.abs(a), 4.0)) * ((0.5 - (Math.cos(((angle * (Math.PI + Math.PI)) * 0.005555555555555556)) * 0.5)) + Math.sqrt(Math.pow(Math.sin(((angle * Math.PI) * 0.005555555555555556)), 4.0))))) * (t_2 * Math.abs(b))) * Math.abs(x_45_scale)) / (Math.abs(a) * Math.abs(a))) / t_2);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pow(math.fabs(y_45_scale), 2.0)
	t_1 = math.sin((0.005555555555555556 * (angle * math.pi)))
	t_2 = math.fabs(b) * math.fabs(b)
	tmp = 0
	if math.fabs(y_45_scale) <= 3.5e+130:
		tmp = 0.25 * ((math.fabs(a) * (t_0 * math.sqrt((8.0 * ((math.pow(math.fabs(b), 4.0) * (math.sqrt(math.pow(t_1, 4.0)) + math.pow(t_1, 2.0))) / t_0))))) / math.pow(math.fabs(b), 2.0))
	else:
		tmp = -0.25 * (((-(math.sqrt(((8.0 * math.pow(math.fabs(a), 4.0)) * ((0.5 - (math.cos(((angle * (math.pi + math.pi)) * 0.005555555555555556)) * 0.5)) + math.sqrt(math.pow(math.sin(((angle * math.pi) * 0.005555555555555556)), 4.0))))) * (t_2 * math.fabs(b))) * math.fabs(x_45_scale)) / (math.fabs(a) * math.fabs(a))) / t_2)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0
	t_1 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_2 = Float64(abs(b) * abs(b))
	tmp = 0.0
	if (abs(y_45_scale) <= 3.5e+130)
		tmp = Float64(0.25 * Float64(Float64(abs(a) * Float64(t_0 * sqrt(Float64(8.0 * Float64(Float64((abs(b) ^ 4.0) * Float64(sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / t_0))))) / (abs(b) ^ 2.0)));
	else
		tmp = Float64(-0.25 * Float64(Float64(Float64(Float64(-Float64(sqrt(Float64(Float64(8.0 * (abs(a) ^ 4.0)) * Float64(Float64(0.5 - Float64(cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))))) * Float64(t_2 * abs(b)))) * abs(x_45_scale)) / Float64(abs(a) * abs(a))) / t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(y_45_scale) ^ 2.0;
	t_1 = sin((0.005555555555555556 * (angle * pi)));
	t_2 = abs(b) * abs(b);
	tmp = 0.0;
	if (abs(y_45_scale) <= 3.5e+130)
		tmp = 0.25 * ((abs(a) * (t_0 * sqrt((8.0 * (((abs(b) ^ 4.0) * (sqrt((t_1 ^ 4.0)) + (t_1 ^ 2.0))) / t_0))))) / (abs(b) ^ 2.0));
	else
		tmp = -0.25 * (((-(sqrt(((8.0 * (abs(a) ^ 4.0)) * ((0.5 - (cos(((angle * (pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(((angle * pi) * 0.005555555555555556)) ^ 4.0))))) * (t_2 * abs(b))) * abs(x_45_scale)) / (abs(a) * abs(a))) / t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 3.5e+130], N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[(8.0 * N[(N[(N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$1, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(N[((-N[(N[Sqrt[N[(N[(8.0 * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$2 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 3.5000000000000001e130

   1. Initial program 2.5%

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

   2. Taylor expanded in a around inf

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

   4. Applied rewrites 0.8%

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

   5. Taylor expanded in x-scale around 0

      \[\leadsto 0.25 \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

   7. Applied rewrites 1.7%

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

   if 3.5000000000000001e130 < y-scale

   1. Initial program 2.5%

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

   2. Taylor expanded in x-scale around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto -0.25 \cdot \frac{x-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)\right)}}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in b around -inf

      \[\leadsto -0.25 \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

   10. Applied rewrites 1.2%

       \[\leadsto -0.25 \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

   12. Applied rewrites 1.6%

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

Alternative 11: 5.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ 0.25 \cdot \frac{\left|a\right| \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{{t\_0}^{4}} + {t\_0}^{2}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* 0.005555555555555556 (* angle PI)))))
   (*
    0.25
    (/
     (*
      (fabs a)
      (*
       (pow x-scale 2.0)
       (sqrt
        (*
         8.0
         (/
          (* (pow b 4.0) (+ (sqrt (pow t_0 4.0)) (pow t_0 2.0)))
          (pow x-scale 2.0))))))
     (pow b 2.0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos((0.005555555555555556 * (angle * ((double) M_PI))));
	return 0.25 * ((fabs(a) * (pow(x_45_scale, 2.0) * sqrt((8.0 * ((pow(b, 4.0) * (sqrt(pow(t_0, 4.0)) + pow(t_0, 2.0))) / pow(x_45_scale, 2.0)))))) / pow(b, 2.0));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.cos((0.005555555555555556 * (angle * Math.PI)));
	return 0.25 * ((Math.abs(a) * (Math.pow(x_45_scale, 2.0) * Math.sqrt((8.0 * ((Math.pow(b, 4.0) * (Math.sqrt(Math.pow(t_0, 4.0)) + Math.pow(t_0, 2.0))) / Math.pow(x_45_scale, 2.0)))))) / Math.pow(b, 2.0));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.cos((0.005555555555555556 * (angle * math.pi)))
	return 0.25 * ((math.fabs(a) * (math.pow(x_45_scale, 2.0) * math.sqrt((8.0 * ((math.pow(b, 4.0) * (math.sqrt(math.pow(t_0, 4.0)) + math.pow(t_0, 2.0))) / math.pow(x_45_scale, 2.0)))))) / math.pow(b, 2.0))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(0.005555555555555556 * Float64(angle * pi)))
	return Float64(0.25 * Float64(Float64(abs(a) * Float64((x_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64((b ^ 4.0) * Float64(sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0))) / (x_45_scale ^ 2.0)))))) / (b ^ 2.0)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = cos((0.005555555555555556 * (angle * pi)));
	tmp = 0.25 * ((abs(a) * ((x_45_scale ^ 2.0) * sqrt((8.0 * (((b ^ 4.0) * (sqrt((t_0 ^ 4.0)) + (t_0 ^ 2.0))) / (x_45_scale ^ 2.0)))))) / (b ^ 2.0));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(0.25 * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Power[b, 4.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$0, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.5%

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

2. Taylor expanded in a around inf

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

4. Applied rewrites 0.8%

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

5. Taylor expanded in y-scale around 0

   \[\leadsto 0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
6. Step-by-step derivation

   1. lower-sqrt.f64 N/A

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

7. Applied rewrites 2.9%

   \[\leadsto 0.25 \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
8. Add Preprocessing

Alternative 12: 4.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} t_0 := \left|x-scale\right| \cdot \left|x-scale\right|\\ t_1 := \frac{\left|b\right|}{t\_0}\\ t_2 := \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_3 := \left|b\right| \cdot \left|b\right|\\ t_4 := \left|a\right| \cdot \left|b\right|\\ t_5 := -\left|a\right|\\ t_6 := t\_5 \cdot \left|b\right|\\ \mathbf{if}\;\left|a\right| \leq 3.2 \cdot 10^{+153}:\\ \;\;\;\;-0.25 \cdot \frac{\frac{\left(-\sqrt{\left(8 \cdot {\left(\left|a\right|\right)}^{4}\right) \cdot \left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) + \sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right)} \cdot \left(t\_3 \cdot \left|b\right|\right)\right) \cdot \left|x-scale\right|}{\left|a\right| \cdot \left|a\right|}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left(\left(\left(\left(t\_4 \cdot \left|b\right|\right) \cdot \frac{t\_5}{\left(\left(y-scale \cdot y-scale\right) \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|}\right) \cdot 4\right) \cdot 2\right) \cdot \left(t\_6 \cdot t\_4\right)\right) \cdot \mathsf{fma}\left(\left|b\right|, t\_1, \mathsf{fma}\left(\left|a\right|, t\_2, \left|\left|a\right| \cdot t\_2 - \left|b\right| \cdot t\_1\right|\right)\right)}}{4 \cdot t\_4}}{t\_6} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs x-scale) (fabs x-scale)))
        (t_1 (/ (fabs b) t_0))
        (t_2 (/ (fabs a) (* y-scale y-scale)))
        (t_3 (* (fabs b) (fabs b)))
        (t_4 (* (fabs a) (fabs b)))
        (t_5 (- (fabs a)))
        (t_6 (* t_5 (fabs b))))
   (if (<= (fabs a) 3.2e+153)
     (*
      -0.25
      (/
       (/
        (*
         (-
          (*
           (sqrt
            (*
             (* 8.0 (pow (fabs a) 4.0))
             (+
              (-
               0.5
               (* (cos (* (* angle (+ PI PI)) 0.005555555555555556)) 0.5))
              (sqrt (pow (sin (* (* angle PI) 0.005555555555555556)) 4.0)))))
           (* t_3 (fabs b))))
         (fabs x-scale))
        (* (fabs a) (fabs a)))
       t_3))
     (*
      (/
       (/
        (-
         (sqrt
          (*
           (*
            (*
             (*
              (*
               (* t_4 (fabs b))
               (/
                t_5
                (* (* (* y-scale y-scale) (fabs x-scale)) (fabs x-scale))))
              4.0)
             2.0)
            (* t_6 t_4))
           (fma
            (fabs b)
            t_1
            (fma (fabs a) t_2 (fabs (- (* (fabs a) t_2) (* (fabs b) t_1))))))))
        (* 4.0 t_4))
       t_6)
      (* (* y-scale y-scale) t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(x_45_scale) * fabs(x_45_scale);
	double t_1 = fabs(b) / t_0;
	double t_2 = fabs(a) / (y_45_scale * y_45_scale);
	double t_3 = fabs(b) * fabs(b);
	double t_4 = fabs(a) * fabs(b);
	double t_5 = -fabs(a);
	double t_6 = t_5 * fabs(b);
	double tmp;
	if (fabs(a) <= 3.2e+153) {
		tmp = -0.25 * (((-(sqrt(((8.0 * pow(fabs(a), 4.0)) * ((0.5 - (cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556)) * 0.5)) + sqrt(pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))))) * (t_3 * fabs(b))) * fabs(x_45_scale)) / (fabs(a) * fabs(a))) / t_3);
	} else {
		tmp = ((-sqrt(((((((t_4 * fabs(b)) * (t_5 / (((y_45_scale * y_45_scale) * fabs(x_45_scale)) * fabs(x_45_scale)))) * 4.0) * 2.0) * (t_6 * t_4)) * fma(fabs(b), t_1, fma(fabs(a), t_2, fabs(((fabs(a) * t_2) - (fabs(b) * t_1))))))) / (4.0 * t_4)) / t_6) * ((y_45_scale * y_45_scale) * t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(x_45_scale) * abs(x_45_scale))
	t_1 = Float64(abs(b) / t_0)
	t_2 = Float64(abs(a) / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(abs(b) * abs(b))
	t_4 = Float64(abs(a) * abs(b))
	t_5 = Float64(-abs(a))
	t_6 = Float64(t_5 * abs(b))
	tmp = 0.0
	if (abs(a) <= 3.2e+153)
		tmp = Float64(-0.25 * Float64(Float64(Float64(Float64(-Float64(sqrt(Float64(Float64(8.0 * (abs(a) ^ 4.0)) * Float64(Float64(0.5 - Float64(cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))))) * Float64(t_3 * abs(b)))) * abs(x_45_scale)) / Float64(abs(a) * abs(a))) / t_3));
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(t_4 * abs(b)) * Float64(t_5 / Float64(Float64(Float64(y_45_scale * y_45_scale) * abs(x_45_scale)) * abs(x_45_scale)))) * 4.0) * 2.0) * Float64(t_6 * t_4)) * fma(abs(b), t_1, fma(abs(a), t_2, abs(Float64(Float64(abs(a) * t_2) - Float64(abs(b) * t_1)))))))) / Float64(4.0 * t_4)) / t_6) * Float64(Float64(y_45_scale * y_45_scale) * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[a], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$6 = N[(t$95$5 * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 3.2e+153], N[(-0.25 * N[(N[(N[((-N[(N[Sqrt[N[(N[(8.0 * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$3 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[(t$95$4 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(t$95$5 / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$6 * t$95$4), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * t$95$1 + N[(N[Abs[a], $MachinePrecision] * t$95$2 + N[Abs[N[(N[(N[Abs[a], $MachinePrecision] * t$95$2), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(4.0 * t$95$4), $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision] * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.2000000000000001e153

   1. Initial program 2.5%

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

   2. Taylor expanded in x-scale around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto -0.25 \cdot \frac{x-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)\right)}}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in b around -inf

      \[\leadsto -0.25 \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

   10. Applied rewrites 1.2%

       \[\leadsto -0.25 \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

   12. Applied rewrites 1.6%

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

   if 3.2000000000000001e153 < a

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 4.2%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Applied rewrites 1.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]

   8. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{x-scale \cdot x-scale}, \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \left|a \cdot \frac{a}{y-scale \cdot y-scale} - b \cdot \frac{b}{x-scale \cdot x-scale}\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{\left(-a\right) \cdot b}} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 3.8% accurate, 4.4× speedup

Math

\[\begin{array}{l} t_0 := \left|x-scale\right| \cdot \left|x-scale\right|\\ t_1 := \frac{\left|b\right|}{t\_0}\\ t_2 := \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_3 := \left|a\right| \cdot \left|b\right|\\ t_4 := \left|b\right| \cdot \left|b\right|\\ t_5 := -\left|a\right|\\ t_6 := t\_5 \cdot \left|b\right|\\ \mathbf{if}\;\left|a\right| \leq 1.48 \cdot 10^{+134}:\\ \;\;\;\;-0.25 \cdot \left(\frac{\left|x-scale\right|}{\left|a\right| \cdot \left|a\right|} \cdot \frac{-\sqrt{\left(8 \cdot {\left(\left|a\right|\right)}^{4}\right) \cdot \left(\left(0.5 - \cos \left(\left(angle \cdot \left(\pi + \pi\right)\right) \cdot 0.005555555555555556\right) \cdot 0.5\right) + \sqrt{{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{4}}\right)} \cdot \left(t\_4 \cdot \left|b\right|\right)}{t\_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left(\left(\left(\left(t\_3 \cdot \left|b\right|\right) \cdot \frac{t\_5}{\left(\left(y-scale \cdot y-scale\right) \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|}\right) \cdot 4\right) \cdot 2\right) \cdot \left(t\_6 \cdot t\_3\right)\right) \cdot \mathsf{fma}\left(\left|b\right|, t\_1, \mathsf{fma}\left(\left|a\right|, t\_2, \left|\left|a\right| \cdot t\_2 - \left|b\right| \cdot t\_1\right|\right)\right)}}{4 \cdot t\_3}}{t\_6} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot t\_0\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs x-scale) (fabs x-scale)))
        (t_1 (/ (fabs b) t_0))
        (t_2 (/ (fabs a) (* y-scale y-scale)))
        (t_3 (* (fabs a) (fabs b)))
        (t_4 (* (fabs b) (fabs b)))
        (t_5 (- (fabs a)))
        (t_6 (* t_5 (fabs b))))
   (if (<= (fabs a) 1.48e+134)
     (*
      -0.25
      (*
       (/ (fabs x-scale) (* (fabs a) (fabs a)))
       (/
        (-
         (*
          (sqrt
           (*
            (* 8.0 (pow (fabs a) 4.0))
            (+
             (- 0.5 (* (cos (* (* angle (+ PI PI)) 0.005555555555555556)) 0.5))
             (sqrt (pow (sin (* (* angle PI) 0.005555555555555556)) 4.0)))))
          (* t_4 (fabs b))))
        t_4)))
     (*
      (/
       (/
        (-
         (sqrt
          (*
           (*
            (*
             (*
              (*
               (* t_3 (fabs b))
               (/
                t_5
                (* (* (* y-scale y-scale) (fabs x-scale)) (fabs x-scale))))
              4.0)
             2.0)
            (* t_6 t_3))
           (fma
            (fabs b)
            t_1
            (fma (fabs a) t_2 (fabs (- (* (fabs a) t_2) (* (fabs b) t_1))))))))
        (* 4.0 t_3))
       t_6)
      (* (* y-scale y-scale) t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(x_45_scale) * fabs(x_45_scale);
	double t_1 = fabs(b) / t_0;
	double t_2 = fabs(a) / (y_45_scale * y_45_scale);
	double t_3 = fabs(a) * fabs(b);
	double t_4 = fabs(b) * fabs(b);
	double t_5 = -fabs(a);
	double t_6 = t_5 * fabs(b);
	double tmp;
	if (fabs(a) <= 1.48e+134) {
		tmp = -0.25 * ((fabs(x_45_scale) / (fabs(a) * fabs(a))) * (-(sqrt(((8.0 * pow(fabs(a), 4.0)) * ((0.5 - (cos(((angle * (((double) M_PI) + ((double) M_PI))) * 0.005555555555555556)) * 0.5)) + sqrt(pow(sin(((angle * ((double) M_PI)) * 0.005555555555555556)), 4.0))))) * (t_4 * fabs(b))) / t_4));
	} else {
		tmp = ((-sqrt(((((((t_3 * fabs(b)) * (t_5 / (((y_45_scale * y_45_scale) * fabs(x_45_scale)) * fabs(x_45_scale)))) * 4.0) * 2.0) * (t_6 * t_3)) * fma(fabs(b), t_1, fma(fabs(a), t_2, fabs(((fabs(a) * t_2) - (fabs(b) * t_1))))))) / (4.0 * t_3)) / t_6) * ((y_45_scale * y_45_scale) * t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(x_45_scale) * abs(x_45_scale))
	t_1 = Float64(abs(b) / t_0)
	t_2 = Float64(abs(a) / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(abs(a) * abs(b))
	t_4 = Float64(abs(b) * abs(b))
	t_5 = Float64(-abs(a))
	t_6 = Float64(t_5 * abs(b))
	tmp = 0.0
	if (abs(a) <= 1.48e+134)
		tmp = Float64(-0.25 * Float64(Float64(abs(x_45_scale) / Float64(abs(a) * abs(a))) * Float64(Float64(-Float64(sqrt(Float64(Float64(8.0 * (abs(a) ^ 4.0)) * Float64(Float64(0.5 - Float64(cos(Float64(Float64(angle * Float64(pi + pi)) * 0.005555555555555556)) * 0.5)) + sqrt((sin(Float64(Float64(angle * pi) * 0.005555555555555556)) ^ 4.0))))) * Float64(t_4 * abs(b)))) / t_4)));
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(t_3 * abs(b)) * Float64(t_5 / Float64(Float64(Float64(y_45_scale * y_45_scale) * abs(x_45_scale)) * abs(x_45_scale)))) * 4.0) * 2.0) * Float64(t_6 * t_3)) * fma(abs(b), t_1, fma(abs(a), t_2, abs(Float64(Float64(abs(a) * t_2) - Float64(abs(b) * t_1)))))))) / Float64(4.0 * t_3)) / t_6) * Float64(Float64(y_45_scale * y_45_scale) * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[a], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$6 = N[(t$95$5 * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.48e+134], N[(-0.25 * N[(N[(N[Abs[x$45$scale], $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-N[(N[Sqrt[N[(N[(8.0 * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 - N[(N[Cos[N[(N[(angle * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(angle * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$4 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[(t$95$3 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(t$95$5 / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$6 * t$95$3), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * t$95$1 + N[(N[Abs[a], $MachinePrecision] * t$95$2 + N[Abs[N[(N[(N[Abs[a], $MachinePrecision] * t$95$2), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(4.0 * t$95$3), $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision] * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4800000000000001e134

   1. Initial program 2.5%

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

   2. Taylor expanded in x-scale around -inf

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

   4. Applied rewrites 0.5%

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

   5. Taylor expanded in y-scale around 0

      \[\leadsto -0.25 \cdot \frac{x-scale \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}^{2}} + \left({a}^{2} \cdot {\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2} + {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)\right)}}{{a}^{\color{blue}{2}} \cdot {b}^{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in b around -inf

      \[\leadsto -0.25 \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{-1}{4} \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

   10. Applied rewrites 1.2%

       \[\leadsto -0.25 \cdot \frac{x-scale \cdot \left(-1 \cdot \left({b}^{3} \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\sqrt{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}} + {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}\right)\right)}{{a}^{2} \cdot {b}^{2}} \]

   12. Applied rewrites 1.4%

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

   if 1.4800000000000001e134 < a

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 4.2%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Applied rewrites 1.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]

   8. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{x-scale \cdot x-scale}, \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \left|a \cdot \frac{a}{y-scale \cdot y-scale} - b \cdot \frac{b}{x-scale \cdot x-scale}\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{\left(-a\right) \cdot b}} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 3.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} t_0 := \left|y-scale\right| \cdot x-scale\\ t_1 := \left(-a\right) \cdot b\\ t_2 := \frac{b}{x-scale \cdot x-scale}\\ t_3 := \left|y-scale\right| \cdot \left|y-scale\right|\\ t_4 := \frac{a}{t\_3}\\ \mathbf{if}\;\left|y-scale\right| \leq 9 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(t\_3 \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(t\_1 \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(b, t\_2, \mathsf{fma}\left(a, t\_4, \left|a \cdot t\_4 - b \cdot t\_2\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{t\_1} \cdot \left(t\_3 \cdot \left(x-scale \cdot x-scale\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{{a}^{2} \cdot \sqrt{8 \cdot \left({b}^{4} \cdot \left(\sqrt{\frac{{b}^{4}}{{x-scale}^{4}}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}{\left|t\_0\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot t\_0\right) \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs y-scale) x-scale))
        (t_1 (* (- a) b))
        (t_2 (/ b (* x-scale x-scale)))
        (t_3 (* (fabs y-scale) (fabs y-scale)))
        (t_4 (/ a t_3)))
   (if (<= (fabs y-scale) 9e+137)
     (*
      (/
       (/
        (-
         (sqrt
          (*
           (*
            (*
             (* (* (* (* a b) b) (/ (- a) (* (* t_3 x-scale) x-scale))) 4.0)
             2.0)
            (* t_1 (* a b)))
           (fma b t_2 (fma a t_4 (fabs (- (* a t_4) (* b t_2))))))))
        (* 4.0 (* a b)))
       t_1)
      (* t_3 (* x-scale x-scale)))
     (*
      (*
       (/
        (/
         (*
          (pow a 2.0)
          (sqrt
           (*
            8.0
            (*
             (pow b 4.0)
             (+
              (sqrt (/ (pow b 4.0) (pow x-scale 4.0)))
              (/ (pow b 2.0) (pow x-scale 2.0)))))))
         (fabs t_0))
        (* (* (* a b) 4.0) (* a b)))
       t_0)
      t_0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(y_45_scale) * x_45_scale;
	double t_1 = -a * b;
	double t_2 = b / (x_45_scale * x_45_scale);
	double t_3 = fabs(y_45_scale) * fabs(y_45_scale);
	double t_4 = a / t_3;
	double tmp;
	if (fabs(y_45_scale) <= 9e+137) {
		tmp = ((-sqrt((((((((a * b) * b) * (-a / ((t_3 * x_45_scale) * x_45_scale))) * 4.0) * 2.0) * (t_1 * (a * b))) * fma(b, t_2, fma(a, t_4, fabs(((a * t_4) - (b * t_2))))))) / (4.0 * (a * b))) / t_1) * (t_3 * (x_45_scale * x_45_scale));
	} else {
		tmp = ((((pow(a, 2.0) * sqrt((8.0 * (pow(b, 4.0) * (sqrt((pow(b, 4.0) / pow(x_45_scale, 4.0))) + (pow(b, 2.0) / pow(x_45_scale, 2.0))))))) / fabs(t_0)) / (((a * b) * 4.0) * (a * b))) * t_0) * t_0;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(y_45_scale) * x_45_scale)
	t_1 = Float64(Float64(-a) * b)
	t_2 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_3 = Float64(abs(y_45_scale) * abs(y_45_scale))
	t_4 = Float64(a / t_3)
	tmp = 0.0
	if (abs(y_45_scale) <= 9e+137)
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * b) * b) * Float64(Float64(-a) / Float64(Float64(t_3 * x_45_scale) * x_45_scale))) * 4.0) * 2.0) * Float64(t_1 * Float64(a * b))) * fma(b, t_2, fma(a, t_4, abs(Float64(Float64(a * t_4) - Float64(b * t_2)))))))) / Float64(4.0 * Float64(a * b))) / t_1) * Float64(t_3 * Float64(x_45_scale * x_45_scale)));
	else
		tmp = Float64(Float64(Float64(Float64(Float64((a ^ 2.0) * sqrt(Float64(8.0 * Float64((b ^ 4.0) * Float64(sqrt(Float64((b ^ 4.0) / (x_45_scale ^ 4.0))) + Float64((b ^ 2.0) / (x_45_scale ^ 2.0))))))) / abs(t_0)) / Float64(Float64(Float64(a * b) * 4.0) * Float64(a * b))) * t_0) * t_0);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[y$45$scale], $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[((-a) * b), $MachinePrecision]}, Block[{t$95$2 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(a / t$95$3), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 9e+137], N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * N[((-a) / N[(N[(t$95$3 * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$1 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * t$95$2 + N[(a * t$95$4 + N[Abs[N[(N[(a * t$95$4), $MachinePrecision] - N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t$95$3 * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[(N[Sqrt[N[(N[Power[b, 4.0], $MachinePrecision] / N[Power[x$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Power[b, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 9.0000000000000003e137

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 4.2%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   6. Applied rewrites 1.9%

      \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]

   8. Applied rewrites 3.7%

      \[\leadsto \color{blue}{\frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{x-scale \cdot x-scale}, \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \left|a \cdot \frac{a}{y-scale \cdot y-scale} - b \cdot \frac{b}{x-scale \cdot x-scale}\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{\left(-a\right) \cdot b}} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]

   if 9.0000000000000003e137 < y-scale

   1. Initial program 2.5%

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

   3. Applied rewrites 5.9%

      \[\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 a around 0

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

   6. Applied rewrites 1.0%

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

   7. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 2.4%

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

   9. Applied rewrites 2.4%

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

Alternative 15: 3.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{a}{y-scale \cdot y-scale}\\ t_1 := \left(-a\right) \cdot b\\ t_2 := \frac{b}{x-scale \cdot x-scale}\\ \frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(t\_1 \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(b, t\_2, \mathsf{fma}\left(a, t\_0, \left|a \cdot t\_0 - b \cdot t\_2\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{t\_1} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale)))
        (t_1 (* (- a) b))
        (t_2 (/ b (* x-scale x-scale))))
   (*
    (/
     (/
      (-
       (sqrt
        (*
         (*
          (*
           (*
            (*
             (* (* a b) b)
             (/ (- a) (* (* (* y-scale y-scale) x-scale) x-scale)))
            4.0)
           2.0)
          (* t_1 (* a b)))
         (fma b t_2 (fma a t_0 (fabs (- (* a t_0) (* b t_2))))))))
      (* 4.0 (* a b)))
     t_1)
    (* (* y-scale y-scale) (* x-scale x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * y_45_scale);
	double t_1 = -a * b;
	double t_2 = b / (x_45_scale * x_45_scale);
	return ((-sqrt((((((((a * b) * b) * (-a / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 4.0) * 2.0) * (t_1 * (a * b))) * fma(b, t_2, fma(a, t_0, fabs(((a * t_0) - (b * t_2))))))) / (4.0 * (a * b))) / t_1) * ((y_45_scale * y_45_scale) * (x_45_scale * x_45_scale));
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(Float64(-a) * b)
	t_2 = Float64(b / Float64(x_45_scale * x_45_scale))
	return Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(a * b) * b) * Float64(Float64(-a) / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 4.0) * 2.0) * Float64(t_1 * Float64(a * b))) * fma(b, t_2, fma(a, t_0, abs(Float64(Float64(a * t_0) - Float64(b * t_2)))))))) / Float64(4.0 * Float64(a * b))) / t_1) * Float64(Float64(y_45_scale * y_45_scale) * Float64(x_45_scale * x_45_scale)))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) * b), $MachinePrecision]}, Block[{t$95$2 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * N[((-a) / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$1 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * t$95$2 + N[(a * t$95$0 + N[Abs[N[(N[(a * t$95$0), $MachinePrecision] - N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 2.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Step-by-step derivation

4. Applied rewrites 4.2%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

6. Applied rewrites 1.9%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]

8. Applied rewrites 3.7%

   \[\leadsto \color{blue}{\frac{\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{x-scale \cdot x-scale}, \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \left|a \cdot \frac{a}{y-scale \cdot y-scale} - b \cdot \frac{b}{x-scale \cdot x-scale}\right|\right)\right)}}{4 \cdot \left(a \cdot b\right)}}{\left(-a\right) \cdot b}} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right) \]
9. Add Preprocessing

Alternative 16: 3.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ \left(\frac{\frac{-\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(-a\right) \cdot b\right)\right) \cdot \left(a \cdot b\right)\right) \cdot \mathsf{fma}\left(t\_0, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|t\_0 \cdot b - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(4 \cdot \left(a \cdot b\right)\right) \cdot b}}{-a} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale))))
   (*
    (*
     (/
      (/
       (-
        (sqrt
         (*
          (*
           (*
            (*
             (*
              (/ (- a) (* (* y-scale (* x-scale y-scale)) x-scale))
              (* (* a b) b))
             8.0)
            (* (- a) b))
           (* a b))
          (fma
           t_0
           b
           (fma
            (/ a (* y-scale y-scale))
            a
            (fabs (- (* t_0 b) (/ (* a a) (* y-scale y-scale)))))))))
       (* (* 4.0 (* a b)) b))
      (- a))
     (* y-scale y-scale))
    (* x-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * x_45_scale);
	return (((-sqrt(((((((-a / ((y_45_scale * (x_45_scale * y_45_scale)) * x_45_scale)) * ((a * b) * b)) * 8.0) * (-a * b)) * (a * b)) * fma(t_0, b, fma((a / (y_45_scale * y_45_scale)), a, fabs(((t_0 * b) - ((a * a) / (y_45_scale * y_45_scale)))))))) / ((4.0 * (a * b)) * b)) / -a) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	return Float64(Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-a) / Float64(Float64(y_45_scale * Float64(x_45_scale * y_45_scale)) * x_45_scale)) * Float64(Float64(a * b) * b)) * 8.0) * Float64(Float64(-a) * b)) * Float64(a * b)) * fma(t_0, b, fma(Float64(a / Float64(y_45_scale * y_45_scale)), a, abs(Float64(Float64(t_0 * b) - Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale))))))))) / Float64(Float64(4.0 * Float64(a * b)) * b)) / Float64(-a)) * Float64(y_45_scale * y_45_scale)) * Float64(x_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[((-a) / N[(N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision] * N[((-a) * b), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * b + N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a + N[Abs[N[(N[(t$95$0 * b), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / (-a)), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Step-by-step derivation

4. Applied rewrites 4.2%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

6. Applied rewrites 1.9%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]

8. Applied rewrites 1.8%

   \[\leadsto \color{blue}{\left(\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{x-scale \cdot x-scale}, \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \left|a \cdot \frac{a}{y-scale \cdot y-scale} - b \cdot \frac{b}{x-scale \cdot x-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot \left(-a\right)} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)} \]

10. Applied rewrites 2.6%

    \[\leadsto \left(\color{blue}{\frac{\frac{-\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(-a\right) \cdot b\right)\right) \cdot \left(a \cdot b\right)\right) \cdot \mathsf{fma}\left(\frac{b}{x-scale \cdot x-scale}, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|\frac{b}{x-scale \cdot x-scale} \cdot b - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(4 \cdot \left(a \cdot b\right)\right) \cdot b}}{-a}} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right) \]
11. Add Preprocessing

Alternative 17: 2.6% accurate, 6.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ \left(\left(\frac{\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(-a\right) \cdot b\right)\right) \cdot \left(a \cdot b\right)\right) \cdot \mathsf{fma}\left(t\_0, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|t\_0 \cdot b - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot y-scale\right) \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale))))
   (*
    (*
     (*
      (/
       (sqrt
        (*
         (*
          (*
           (*
            (*
             (/ (- a) (* (* y-scale (* x-scale y-scale)) x-scale))
             (* (* a b) b))
            8.0)
           (* (- a) b))
          (* a b))
         (fma
          t_0
          b
          (fma
           (/ a (* y-scale y-scale))
           a
           (fabs (- (* t_0 b) (/ (* a a) (* y-scale y-scale))))))))
       (* (* (* 4.0 (* a b)) b) a))
      y-scale)
     y-scale)
    (* x-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * x_45_scale);
	return (((sqrt(((((((-a / ((y_45_scale * (x_45_scale * y_45_scale)) * x_45_scale)) * ((a * b) * b)) * 8.0) * (-a * b)) * (a * b)) * fma(t_0, b, fma((a / (y_45_scale * y_45_scale)), a, fabs(((t_0 * b) - ((a * a) / (y_45_scale * y_45_scale)))))))) / (((4.0 * (a * b)) * b) * a)) * y_45_scale) * y_45_scale) * (x_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	return Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-a) / Float64(Float64(y_45_scale * Float64(x_45_scale * y_45_scale)) * x_45_scale)) * Float64(Float64(a * b) * b)) * 8.0) * Float64(Float64(-a) * b)) * Float64(a * b)) * fma(t_0, b, fma(Float64(a / Float64(y_45_scale * y_45_scale)), a, abs(Float64(Float64(t_0 * b) - Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale)))))))) / Float64(Float64(Float64(4.0 * Float64(a * b)) * b) * a)) * y_45_scale) * y_45_scale) * Float64(x_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[((-a) / N[(N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision] * N[((-a) * b), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * b + N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a + N[Abs[N[(N[(t$95$0 * b), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Step-by-step derivation

4. Applied rewrites 4.2%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

6. Applied rewrites 1.9%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]

8. Applied rewrites 1.8%

   \[\leadsto \color{blue}{\left(\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{x-scale \cdot x-scale}, \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \left|a \cdot \frac{a}{y-scale \cdot y-scale} - b \cdot \frac{b}{x-scale \cdot x-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot \left(-a\right)} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)} \]

10. Applied rewrites 3.3%

    \[\leadsto \color{blue}{\left(\left(\frac{\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(-a\right) \cdot b\right)\right) \cdot \left(a \cdot b\right)\right) \cdot \mathsf{fma}\left(\frac{b}{x-scale \cdot x-scale}, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|\frac{b}{x-scale \cdot x-scale} \cdot b - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot y-scale\right) \cdot y-scale\right)} \cdot \left(x-scale \cdot x-scale\right) \]
11. Add Preprocessing

Alternative 18: 1.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ \left(\frac{\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(-a\right) \cdot b\right)\right) \cdot \left(a \cdot b\right)\right) \cdot \mathsf{fma}\left(t\_0, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|t\_0 \cdot b - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale))))
   (*
    (*
     (/
      (sqrt
       (*
        (*
         (*
          (*
           (*
            (/ (- a) (* (* y-scale (* x-scale y-scale)) x-scale))
            (* (* a b) b))
           8.0)
          (* (- a) b))
         (* a b))
        (fma
         t_0
         b
         (fma
          (/ a (* y-scale y-scale))
          a
          (fabs (- (* t_0 b) (/ (* a a) (* y-scale y-scale))))))))
      (* (* (* 4.0 (* a b)) b) a))
     (* y-scale y-scale))
    (* x-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * x_45_scale);
	return ((sqrt(((((((-a / ((y_45_scale * (x_45_scale * y_45_scale)) * x_45_scale)) * ((a * b) * b)) * 8.0) * (-a * b)) * (a * b)) * fma(t_0, b, fma((a / (y_45_scale * y_45_scale)), a, fabs(((t_0 * b) - ((a * a) / (y_45_scale * y_45_scale)))))))) / (((4.0 * (a * b)) * b) * a)) * (y_45_scale * y_45_scale)) * (x_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	return Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-a) / Float64(Float64(y_45_scale * Float64(x_45_scale * y_45_scale)) * x_45_scale)) * Float64(Float64(a * b) * b)) * 8.0) * Float64(Float64(-a) * b)) * Float64(a * b)) * fma(t_0, b, fma(Float64(a / Float64(y_45_scale * y_45_scale)), a, abs(Float64(Float64(t_0 * b) - Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale)))))))) / Float64(Float64(Float64(4.0 * Float64(a * b)) * b) * a)) * Float64(y_45_scale * y_45_scale)) * Float64(x_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[((-a) / N[(N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision] * N[((-a) * b), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * b + N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a + N[Abs[N[(N[(t$95$0 * b), $MachinePrecision] - N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 2.5%

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

2. Taylor expanded in angle around 0

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Step-by-step derivation

4. Applied rewrites 4.2%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \color{blue}{\left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

6. Applied rewrites 1.9%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right) \cdot \left(\left(\left(4 \cdot \frac{\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)}\right) \cdot 2\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(4 \cdot \left(b \cdot a\right)\right) \cdot \left(b \cdot \left(-a\right)\right)} \cdot \left(\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)\right)} \]

8. Applied rewrites 1.8%

   \[\leadsto \color{blue}{\left(\frac{-\sqrt{\left(\left(\left(\left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \frac{-a}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(b, \frac{b}{x-scale \cdot x-scale}, \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \left|a \cdot \frac{a}{y-scale \cdot y-scale} - b \cdot \frac{b}{x-scale \cdot x-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot \left(-a\right)} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)} \]

10. Applied rewrites 1.9%

    \[\leadsto \color{blue}{\left(\frac{\sqrt{\left(\left(\left(\left(\frac{-a}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(\left(a \cdot b\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(-a\right) \cdot b\right)\right) \cdot \left(a \cdot b\right)\right) \cdot \mathsf{fma}\left(\frac{b}{x-scale \cdot x-scale}, b, \mathsf{fma}\left(\frac{a}{y-scale \cdot y-scale}, a, \left|\frac{b}{x-scale \cdot x-scale} \cdot b - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)\right)}}{\left(\left(4 \cdot \left(a \cdot b\right)\right) \cdot b\right) \cdot a} \cdot \left(y-scale \cdot y-scale\right)\right)} \cdot \left(x-scale \cdot x-scale\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025195 
(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))))