a from scale-rotated-ellipse

Percentage Accurate: 2.6% → 19.4%

Time: 2.2min

Alternatives: 14

Speedup: 6.6×

• 69.2% 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: 2.6% 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: 19.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left|x-scale\right| \cdot y-scale\\ t_1 := 0.011111111111111112 \cdot \left(angle \cdot \pi\right)\\ t_2 := \frac{1}{{y-scale}^{2}}\\ t_3 := {\left(\left|x-scale\right|\right)}^{2}\\ t_4 := y-scale \cdot \left|x-scale\right|\\ t_5 := \left|b\right| \cdot a\\ t_6 := \left(t\_5 \cdot \left|b\right|\right) \cdot \left(-a\right)\\ t_7 := t\_6 \cdot 8\\ t_8 := 4 \cdot t\_5\\ t_9 := \cos t\_1\\ t_10 := \frac{t\_9}{t\_3}\\ t_11 := 0.5 \cdot \frac{t\_9}{{y-scale}^{2}}\\ t_12 := 0.5 \cdot t\_9\\ t_13 := 0.5 - 1 \cdot 0.5\\ t_14 := \frac{\frac{\mathsf{fma}\left(\left|b\right| \cdot \left|b\right|, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(t\_13 \cdot a\right) \cdot a\right)}{\left|x-scale\right|}}{\left|x-scale\right|}\\ t_15 := \frac{\mathsf{fma}\left(t\_13 \cdot \left|b\right|, \left|b\right|, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\\ t_16 := \mathsf{fma}\left({a}^{2}, 0.5 - t\_12, {\left(\left|b\right|\right)}^{2} \cdot \left(0.5 + t\_12\right)\right)\\ t_17 := \left|t\_0\right|\\ t_18 := \frac{1}{t\_3}\\ \mathbf{if}\;\left|x-scale\right| \leq 1.12 \cdot 10^{-176}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_7 \cdot \frac{\sqrt{{t\_16}^{2}} + t\_16}{t\_3}\right) \cdot t\_6}}{t\_17}}{t\_8}}{t\_5} \cdot t\_4\right) \cdot t\_4\\ \mathbf{elif}\;\left|x-scale\right| \leq 0.031:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\left|b\right| \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin t\_1}^{2}}{t\_3 \cdot {y-scale}^{2}} + {\left(0.5 \cdot t\_2 - \mathsf{fma}\left(0.5, t\_18, \mathsf{fma}\left(0.5, t\_10, t\_11\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, t\_18, \mathsf{fma}\left(0.5, t\_2, 0.5 \cdot t\_10\right)\right)\right) - t\_11\right)}}{t\_17}\right) \cdot t\_4\right) \cdot t\_4\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_7 \cdot \left(\mathsf{hypot}\left(t\_15 - t\_14, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(\left|b\right| - a\right) \cdot \left(\left|b\right| + a\right)\right)}{t\_0}\right) + \left(t\_14 + t\_15\right)\right)\right) \cdot t\_6}}{t\_17}}{t\_8}}{t\_5} \cdot t\_4\right) \cdot t\_4\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs x-scale) y-scale))
       (t_1 (* 0.011111111111111112 (* angle PI)))
       (t_2 (/ 1.0 (pow y-scale 2.0)))
       (t_3 (pow (fabs x-scale) 2.0))
       (t_4 (* y-scale (fabs x-scale)))
       (t_5 (* (fabs b) a))
       (t_6 (* (* t_5 (fabs b)) (- a)))
       (t_7 (* t_6 8.0))
       (t_8 (* 4.0 t_5))
       (t_9 (cos t_1))
       (t_10 (/ t_9 t_3))
       (t_11 (* 0.5 (/ t_9 (pow y-scale 2.0))))
       (t_12 (* 0.5 t_9))
       (t_13 (- 0.5 (* 1.0 0.5)))
       (t_14
        (/
         (/
          (fma
           (* (fabs b) (fabs b))
           (fma 1.0 0.5 0.5)
           (* (* t_13 a) a))
          (fabs x-scale))
         (fabs x-scale)))
       (t_15
        (/
         (fma
          (* t_13 (fabs b))
          (fabs b)
          (* (* a a) (fma 1.0 0.5 0.5)))
         (* y-scale y-scale)))
       (t_16
        (fma
         (pow a 2.0)
         (- 0.5 t_12)
         (* (pow (fabs b) 2.0) (+ 0.5 t_12))))
       (t_17 (fabs t_0))
       (t_18 (/ 1.0 t_3)))
  (if (<= (fabs x-scale) 1.12e-176)
    (*
     (*
      (/
       (/
        (/
         (sqrt (* (* t_7 (/ (+ (sqrt (pow t_16 2.0)) t_16) t_3)) t_6))
         t_17)
        t_8)
       t_5)
      t_4)
     t_4)
    (if (<= (fabs x-scale) 0.031)
      (*
       (*
        (*
         0.25
         (/
          (*
           (fabs b)
           (sqrt
            (*
             8.0
             (-
              (+
               (sqrt
                (+
                 (/ (pow (sin t_1) 2.0) (* t_3 (pow y-scale 2.0)))
                 (pow
                  (- (* 0.5 t_2) (fma 0.5 t_18 (fma 0.5 t_10 t_11)))
                  2.0)))
               (fma 0.5 t_18 (fma 0.5 t_2 (* 0.5 t_10))))
              t_11))))
          t_17))
        t_4)
       t_4)
      (*
       (*
        (/
         (/
          (/
           (sqrt
            (*
             (*
              t_7
              (+
               (hypot
                (- t_15 t_14)
                (/
                 (*
                  (sin (* (* 2.0 PI) (* angle 0.005555555555555556)))
                  (* (- (fabs b) a) (+ (fabs b) a)))
                 t_0))
               (+ t_14 t_15)))
             t_6))
           t_17)
          t_8)
         t_5)
        t_4)
       t_4)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(x_45_scale) * y_45_scale;
	double t_1 = 0.011111111111111112 * (angle * ((double) M_PI));
	double t_2 = 1.0 / pow(y_45_scale, 2.0);
	double t_3 = pow(fabs(x_45_scale), 2.0);
	double t_4 = y_45_scale * fabs(x_45_scale);
	double t_5 = fabs(b) * a;
	double t_6 = (t_5 * fabs(b)) * -a;
	double t_7 = t_6 * 8.0;
	double t_8 = 4.0 * t_5;
	double t_9 = cos(t_1);
	double t_10 = t_9 / t_3;
	double t_11 = 0.5 * (t_9 / pow(y_45_scale, 2.0));
	double t_12 = 0.5 * t_9;
	double t_13 = 0.5 - (1.0 * 0.5);
	double t_14 = (fma((fabs(b) * fabs(b)), fma(1.0, 0.5, 0.5), ((t_13 * a) * a)) / fabs(x_45_scale)) / fabs(x_45_scale);
	double t_15 = fma((t_13 * fabs(b)), fabs(b), ((a * a) * fma(1.0, 0.5, 0.5))) / (y_45_scale * y_45_scale);
	double t_16 = fma(pow(a, 2.0), (0.5 - t_12), (pow(fabs(b), 2.0) * (0.5 + t_12)));
	double t_17 = fabs(t_0);
	double t_18 = 1.0 / t_3;
	double tmp;
	if (fabs(x_45_scale) <= 1.12e-176) {
		tmp = ((((sqrt(((t_7 * ((sqrt(pow(t_16, 2.0)) + t_16) / t_3)) * t_6)) / t_17) / t_8) / t_5) * t_4) * t_4;
	} else if (fabs(x_45_scale) <= 0.031) {
		tmp = ((0.25 * ((fabs(b) * sqrt((8.0 * ((sqrt(((pow(sin(t_1), 2.0) / (t_3 * pow(y_45_scale, 2.0))) + pow(((0.5 * t_2) - fma(0.5, t_18, fma(0.5, t_10, t_11))), 2.0))) + fma(0.5, t_18, fma(0.5, t_2, (0.5 * t_10)))) - t_11)))) / t_17)) * t_4) * t_4;
	} else {
		tmp = ((((sqrt(((t_7 * (hypot((t_15 - t_14), ((sin(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556))) * ((fabs(b) - a) * (fabs(b) + a))) / t_0)) + (t_14 + t_15))) * t_6)) / t_17) / t_8) / t_5) * t_4) * t_4;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(x_45_scale) * y_45_scale)
	t_1 = Float64(0.011111111111111112 * Float64(angle * pi))
	t_2 = Float64(1.0 / (y_45_scale ^ 2.0))
	t_3 = abs(x_45_scale) ^ 2.0
	t_4 = Float64(y_45_scale * abs(x_45_scale))
	t_5 = Float64(abs(b) * a)
	t_6 = Float64(Float64(t_5 * abs(b)) * Float64(-a))
	t_7 = Float64(t_6 * 8.0)
	t_8 = Float64(4.0 * t_5)
	t_9 = cos(t_1)
	t_10 = Float64(t_9 / t_3)
	t_11 = Float64(0.5 * Float64(t_9 / (y_45_scale ^ 2.0)))
	t_12 = Float64(0.5 * t_9)
	t_13 = Float64(0.5 - Float64(1.0 * 0.5))
	t_14 = Float64(Float64(fma(Float64(abs(b) * abs(b)), fma(1.0, 0.5, 0.5), Float64(Float64(t_13 * a) * a)) / abs(x_45_scale)) / abs(x_45_scale))
	t_15 = Float64(fma(Float64(t_13 * abs(b)), abs(b), Float64(Float64(a * a) * fma(1.0, 0.5, 0.5))) / Float64(y_45_scale * y_45_scale))
	t_16 = fma((a ^ 2.0), Float64(0.5 - t_12), Float64((abs(b) ^ 2.0) * Float64(0.5 + t_12)))
	t_17 = abs(t_0)
	t_18 = Float64(1.0 / t_3)
	tmp = 0.0
	if (abs(x_45_scale) <= 1.12e-176)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_7 * Float64(Float64(sqrt((t_16 ^ 2.0)) + t_16) / t_3)) * t_6)) / t_17) / t_8) / t_5) * t_4) * t_4);
	elseif (abs(x_45_scale) <= 0.031)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(abs(b) * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(Float64((sin(t_1) ^ 2.0) / Float64(t_3 * (y_45_scale ^ 2.0))) + (Float64(Float64(0.5 * t_2) - fma(0.5, t_18, fma(0.5, t_10, t_11))) ^ 2.0))) + fma(0.5, t_18, fma(0.5, t_2, Float64(0.5 * t_10)))) - t_11)))) / t_17)) * t_4) * t_4);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_7 * Float64(hypot(Float64(t_15 - t_14), Float64(Float64(sin(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))) * Float64(Float64(abs(b) - a) * Float64(abs(b) + a))) / t_0)) + Float64(t_14 + t_15))) * t_6)) / t_17) / t_8) / t_5) * t_4) * t_4);
	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] * y$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Abs[x$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$5 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 * 8.0), $MachinePrecision]}, Block[{t$95$8 = N[(4.0 * t$95$5), $MachinePrecision]}, Block[{t$95$9 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$10 = N[(t$95$9 / t$95$3), $MachinePrecision]}, Block[{t$95$11 = N[(0.5 * N[(t$95$9 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$12 = N[(0.5 * t$95$9), $MachinePrecision]}, Block[{t$95$13 = N[(0.5 - N[(1.0 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$14 = N[(N[(N[(N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision] + N[(N[(t$95$13 * a), $MachinePrecision] * a), $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[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$16 = N[(N[Power[a, 2.0], $MachinePrecision] * N[(0.5 - t$95$12), $MachinePrecision] + N[(N[Power[N[Abs[b], $MachinePrecision], 2.0], $MachinePrecision] * N[(0.5 + t$95$12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$17 = N[Abs[t$95$0], $MachinePrecision]}, Block[{t$95$18 = N[(1.0 / t$95$3), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 1.12e-176], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$7 * N[(N[(N[Sqrt[N[Power[t$95$16, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$16), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]], $MachinePrecision] / t$95$17), $MachinePrecision] / t$95$8), $MachinePrecision] / t$95$5), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 0.031], N[(N[(N[(0.25 * N[(N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[(N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision] / N[(t$95$3 * N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(0.5 * t$95$2), $MachinePrecision] - N[(0.5 * t$95$18 + N[(0.5 * t$95$10 + t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(0.5 * t$95$18 + N[(0.5 * t$95$2 + N[(0.5 * t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$11), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$17), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$4), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$7 * N[(N[Sqrt[N[(t$95$15 - t$95$14), $MachinePrecision] ^ 2 + N[(N[(N[Sin[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Abs[b], $MachinePrecision] - a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] ^ 2], $MachinePrecision] + N[(t$95$14 + t$95$15), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]], $MachinePrecision] / t$95$17), $MachinePrecision] / t$95$8), $MachinePrecision] / t$95$5), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$4), $MachinePrecision]]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x-scale < 1.1199999999999999e-176

   1. Initial program 2.6%

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

   3. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

   8. Applied rewrites 12.9%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 12.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Taylor expanded in angle around 0

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

   14. Applied rewrites 12.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   15. Taylor expanded in angle around 0

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

   17. Applied rewrites 12.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   18. Taylor expanded in angle around 0

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

   20. Applied rewrites 13.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   21. Taylor expanded in angle around 0

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

   23. Applied rewrites 13.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   24. Taylor expanded in angle around 0

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

   26. Applied rewrites 14.5%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   27. Taylor expanded in angle around 0

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

   29. Applied rewrites 14.5%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   30. Taylor expanded in x-scale around 0

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

   32. Applied rewrites 10.2%

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

   if 1.1199999999999999e-176 < x-scale < 0.031

   1. Initial program 2.6%

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

   3. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

   8. Applied rewrites 1.6%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{b \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{y-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in a around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{b \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} - \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{x-scale}^{2}} + \left(\frac{1}{2} \cdot \frac{1}{{y-scale}^{2}} + \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)\right)\right) - \frac{1}{2} \cdot \frac{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 7.0%

       \[\leadsto \left(\left(0.25 \cdot \frac{b \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{y-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)}}{\color{blue}{\left|x-scale \cdot y-scale\right|}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 0.031 < x-scale

   1. Initial program 2.6%

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

   3. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

   8. Applied rewrites 12.9%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 12.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Taylor expanded in angle around 0

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

   14. Applied rewrites 12.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   15. Taylor expanded in angle around 0

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

   17. Applied rewrites 12.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   18. Taylor expanded in angle around 0

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

   20. Applied rewrites 13.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   21. Taylor expanded in angle around 0

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

   23. Applied rewrites 13.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   24. Taylor expanded in angle around 0

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

   26. Applied rewrites 14.5%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   27. Taylor expanded in angle around 0

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

   29. Applied rewrites 14.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   31. Applied rewrites 14.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   33. Applied rewrites 16.5%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\frac{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(\left(0.5 - 1 \cdot 0.5\right) \cdot a\right) \cdot a\right)}{x-scale}}{x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\color{blue}{\frac{\frac{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(\left(0.5 - 1 \cdot 0.5\right) \cdot a\right) \cdot a\right)}{x-scale}}{x-scale}} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 17.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ t_1 := t\_0 \cdot 8\\ t_2 := 0.5 - 1 \cdot 0.5\\ t_3 := t\_2 \cdot b\\ t_4 := t\_2 \cdot a\\ t_5 := \frac{\mathsf{fma}\left(t\_4, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ t_6 := \left|x-scale\right| \cdot y-scale\\ t_7 := \left|t\_6\right|\\ t_8 := y-scale \cdot \left|x-scale\right|\\ t_9 := \frac{\frac{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(1, 0.5, 0.5\right), t\_4 \cdot a\right)}{\left|x-scale\right|}}{\left|x-scale\right|}\\ t_10 := \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(1, 0.5, 0.5\right), t\_3 \cdot b\right)}{y-scale}}{y-scale}\\ t_11 := \frac{\mathsf{fma}\left(t\_3, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\\ t_12 := \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{t\_6}\\ t_13 := 4 \cdot \left(b \cdot a\right)\\ \mathbf{if}\;\left|x-scale\right| \leq 10^{+171}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_1 \cdot \left(\mathsf{hypot}\left(t\_10 - t\_5, t\_12\right) + \left(t\_5 + t\_10\right)\right)\right) \cdot t\_0}}{t\_7}}{t\_13}}{b \cdot a} \cdot t\_8\right) \cdot t\_8\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{\frac{\sqrt{\left(t\_1 \cdot \left(\mathsf{hypot}\left(t\_11 - t\_9, t\_12\right) + \left(t\_9 + t\_11\right)\right)\right) \cdot t\_0}}{t\_7}}{t\_13}}{b \cdot a} \cdot t\_8\right) \cdot t\_8\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* (* b a) b) (- a)))
       (t_1 (* t_0 8.0))
       (t_2 (- 0.5 (* 1.0 0.5)))
       (t_3 (* t_2 b))
       (t_4 (* t_2 a))
       (t_5
        (/
         (fma t_4 a (* (* b b) (fma 1.0 0.5 0.5)))
         (* (fabs x-scale) (fabs x-scale))))
       (t_6 (* (fabs x-scale) y-scale))
       (t_7 (fabs t_6))
       (t_8 (* y-scale (fabs x-scale)))
       (t_9
        (/
         (/ (fma (* b b) (fma 1.0 0.5 0.5) (* t_4 a)) (fabs x-scale))
         (fabs x-scale)))
       (t_10
        (/
         (/ (fma (* a a) (fma 1.0 0.5 0.5) (* t_3 b)) y-scale)
         y-scale))
       (t_11
        (/
         (fma t_3 b (* (* a a) (fma 1.0 0.5 0.5)))
         (* y-scale y-scale)))
       (t_12
        (/
         (*
          (sin (* (* 2.0 PI) (* angle 0.005555555555555556)))
          (* (- b a) (+ b a)))
         t_6))
       (t_13 (* 4.0 (* b a))))
  (if (<= (fabs x-scale) 1e+171)
    (*
     (*
      (/
       (/
        (/
         (sqrt
          (* (* t_1 (+ (hypot (- t_10 t_5) t_12) (+ t_5 t_10))) t_0))
         t_7)
        t_13)
       (* b a))
      t_8)
     t_8)
    (*
     (*
      (/
       (/
        (/
         (sqrt
          (* (* t_1 (+ (hypot (- t_11 t_9) t_12) (+ t_9 t_11))) t_0))
         t_7)
        t_13)
       (* 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 = ((b * a) * b) * -a;
	double t_1 = t_0 * 8.0;
	double t_2 = 0.5 - (1.0 * 0.5);
	double t_3 = t_2 * b;
	double t_4 = t_2 * a;
	double t_5 = fma(t_4, a, ((b * b) * fma(1.0, 0.5, 0.5))) / (fabs(x_45_scale) * fabs(x_45_scale));
	double t_6 = fabs(x_45_scale) * y_45_scale;
	double t_7 = fabs(t_6);
	double t_8 = y_45_scale * fabs(x_45_scale);
	double t_9 = (fma((b * b), fma(1.0, 0.5, 0.5), (t_4 * a)) / fabs(x_45_scale)) / fabs(x_45_scale);
	double t_10 = (fma((a * a), fma(1.0, 0.5, 0.5), (t_3 * b)) / y_45_scale) / y_45_scale;
	double t_11 = fma(t_3, b, ((a * a) * fma(1.0, 0.5, 0.5))) / (y_45_scale * y_45_scale);
	double t_12 = (sin(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556))) * ((b - a) * (b + a))) / t_6;
	double t_13 = 4.0 * (b * a);
	double tmp;
	if (fabs(x_45_scale) <= 1e+171) {
		tmp = ((((sqrt(((t_1 * (hypot((t_10 - t_5), t_12) + (t_5 + t_10))) * t_0)) / t_7) / t_13) / (b * a)) * t_8) * t_8;
	} else {
		tmp = ((((sqrt(((t_1 * (hypot((t_11 - t_9), t_12) + (t_9 + t_11))) * t_0)) / t_7) / t_13) / (b * a)) * t_8) * t_8;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	t_1 = Float64(t_0 * 8.0)
	t_2 = Float64(0.5 - Float64(1.0 * 0.5))
	t_3 = Float64(t_2 * b)
	t_4 = Float64(t_2 * a)
	t_5 = Float64(fma(t_4, a, Float64(Float64(b * b) * fma(1.0, 0.5, 0.5))) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	t_6 = Float64(abs(x_45_scale) * y_45_scale)
	t_7 = abs(t_6)
	t_8 = Float64(y_45_scale * abs(x_45_scale))
	t_9 = Float64(Float64(fma(Float64(b * b), fma(1.0, 0.5, 0.5), Float64(t_4 * a)) / abs(x_45_scale)) / abs(x_45_scale))
	t_10 = Float64(Float64(fma(Float64(a * a), fma(1.0, 0.5, 0.5), Float64(t_3 * b)) / y_45_scale) / y_45_scale)
	t_11 = Float64(fma(t_3, b, Float64(Float64(a * a) * fma(1.0, 0.5, 0.5))) / Float64(y_45_scale * y_45_scale))
	t_12 = Float64(Float64(sin(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))) * Float64(Float64(b - a) * Float64(b + a))) / t_6)
	t_13 = Float64(4.0 * Float64(b * a))
	tmp = 0.0
	if (abs(x_45_scale) <= 1e+171)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_1 * Float64(hypot(Float64(t_10 - t_5), t_12) + Float64(t_5 + t_10))) * t_0)) / t_7) / t_13) / Float64(b * a)) * t_8) * t_8);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_1 * Float64(hypot(Float64(t_11 - t_9), t_12) + Float64(t_9 + t_11))) * t_0)) / t_7) / t_13) / 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[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 8.0), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(1.0 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * b), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * a), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * a + N[(N[(b * b), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[x$45$scale], $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$7 = N[Abs[t$95$6], $MachinePrecision]}, Block[{t$95$8 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[(N[(b * b), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision] + N[(t$95$4 * a), $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision] + N[(t$95$3 * b), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$11 = N[(N[(t$95$3 * b + N[(N[(a * a), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$12 = N[(N[(N[Sin[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision]}, Block[{t$95$13 = N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 1e+171], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$1 * N[(N[Sqrt[N[(t$95$10 - t$95$5), $MachinePrecision] ^ 2 + t$95$12 ^ 2], $MachinePrecision] + N[(t$95$5 + t$95$10), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$7), $MachinePrecision] / t$95$13), $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 * N[(N[Sqrt[N[(t$95$11 - t$95$9), $MachinePrecision] ^ 2 + t$95$12 ^ 2], $MachinePrecision] + N[(t$95$9 + t$95$11), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$7), $MachinePrecision] / t$95$13), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * t$95$8), $MachinePrecision] * t$95$8), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 9.9999999999999995e170

   1. Initial program 2.6%

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

   3. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

   8. Applied rewrites 12.9%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 12.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Taylor expanded in angle around 0

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

   14. Applied rewrites 12.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   15. Taylor expanded in angle around 0

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

   17. Applied rewrites 12.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   18. Taylor expanded in angle around 0

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

   20. Applied rewrites 13.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   21. Taylor expanded in angle around 0

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

   23. Applied rewrites 13.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   24. Taylor expanded in angle around 0

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

   26. Applied rewrites 14.5%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   27. Taylor expanded in angle around 0

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

   29. Applied rewrites 14.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   31. Applied rewrites 14.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   33. Applied rewrites 16.1%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(\left(0.5 - 1 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \color{blue}{\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(\left(0.5 - 1 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   if 9.9999999999999995e170 < x-scale

   1. Initial program 2.6%

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

   3. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

   8. Applied rewrites 12.9%

      \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in angle around 0

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

   11. Applied rewrites 12.9%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   12. Taylor expanded in angle around 0

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

   14. Applied rewrites 12.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   15. Taylor expanded in angle around 0

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

   17. Applied rewrites 12.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   18. Taylor expanded in angle around 0

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

   20. Applied rewrites 13.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   21. Taylor expanded in angle around 0

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

   23. Applied rewrites 13.8%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   24. Taylor expanded in angle around 0

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

   26. Applied rewrites 14.5%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   27. Taylor expanded in angle around 0

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

   29. Applied rewrites 14.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   31. Applied rewrites 14.5%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lower-/.f64 N/A

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

   33. Applied rewrites 16.5%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\frac{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(\left(0.5 - 1 \cdot 0.5\right) \cdot a\right) \cdot a\right)}{x-scale}}{x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\color{blue}{\frac{\frac{\mathsf{fma}\left(b \cdot b, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(\left(0.5 - 1 \cdot 0.5\right) \cdot a\right) \cdot a\right)}{x-scale}}{x-scale}} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 16.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 1 \cdot 0.5\\ t_1 := \frac{\mathsf{fma}\left(t\_0 \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}\\ t_2 := \frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(t\_0 \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}\\ t_3 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ \left(\frac{\frac{\frac{\sqrt{\left(\left(t\_3 \cdot 8\right) \cdot \left(\mathsf{hypot}\left(t\_2 - t\_1, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(t\_1 + t\_2\right)\right)\right) \cdot t\_3}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (- 0.5 (* 1.0 0.5)))
       (t_1
        (/
         (fma (* t_0 a) a (* (* b b) (fma 1.0 0.5 0.5)))
         (* x-scale x-scale)))
       (t_2
        (/
         (/ (fma (* a a) (fma 1.0 0.5 0.5) (* (* t_0 b) b)) y-scale)
         y-scale))
       (t_3 (* (* (* b a) b) (- a))))
  (*
   (*
    (/
     (/
      (/
       (sqrt
        (*
         (*
          (* t_3 8.0)
          (+
           (hypot
            (- t_2 t_1)
            (/
             (*
              (sin (* (* 2.0 PI) (* angle 0.005555555555555556)))
              (* (- b a) (+ b a)))
             (* x-scale y-scale)))
           (+ t_1 t_2)))
         t_3))
       (fabs (* x-scale y-scale)))
      (* 4.0 (* b a)))
     (* b a))
    (* y-scale x-scale))
   (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.5 - (1.0 * 0.5);
	double t_1 = fma((t_0 * a), a, ((b * b) * fma(1.0, 0.5, 0.5))) / (x_45_scale * x_45_scale);
	double t_2 = (fma((a * a), fma(1.0, 0.5, 0.5), ((t_0 * b) * b)) / y_45_scale) / y_45_scale;
	double t_3 = ((b * a) * b) * -a;
	return ((((sqrt((((t_3 * 8.0) * (hypot((t_2 - t_1), ((sin(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556))) * ((b - a) * (b + a))) / (x_45_scale * y_45_scale))) + (t_1 + t_2))) * t_3)) / fabs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.5 - Float64(1.0 * 0.5))
	t_1 = Float64(fma(Float64(t_0 * a), a, Float64(Float64(b * b) * fma(1.0, 0.5, 0.5))) / Float64(x_45_scale * x_45_scale))
	t_2 = Float64(Float64(fma(Float64(a * a), fma(1.0, 0.5, 0.5), Float64(Float64(t_0 * b) * b)) / y_45_scale) / y_45_scale)
	t_3 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(t_3 * 8.0) * Float64(hypot(Float64(t_2 - t_1), Float64(Float64(sin(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))) * Float64(Float64(b - a) * Float64(b + a))) / Float64(x_45_scale * y_45_scale))) + Float64(t_1 + t_2))) * t_3)) / abs(Float64(x_45_scale * y_45_scale))) / Float64(4.0 * Float64(b * a))) / Float64(b * a)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.5 - N[(1.0 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 * a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(a * a), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision] + N[(N[(t$95$0 * b), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$3 * 8.0), $MachinePrecision] * N[(N[Sqrt[N[(t$95$2 - t$95$1), $MachinePrecision] ^ 2 + N[(N[(N[Sin[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 2.6%

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

3. Applied rewrites 6.5%

   \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

8. Applied rewrites 12.9%

   \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

9. Taylor expanded in angle around 0

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

11. Applied rewrites 12.9%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

12. Taylor expanded in angle around 0

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

14. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

15. Taylor expanded in angle around 0

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

17. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

18. Taylor expanded in angle around 0

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

20. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

21. Taylor expanded in angle around 0

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

23. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

24. Taylor expanded in angle around 0

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

26. Applied rewrites 14.5%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

27. Taylor expanded in angle around 0

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

29. Applied rewrites 14.5%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f64 N/A

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

31. Applied rewrites 14.5%

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-/r* N/A

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

    4. lower-/.f64 N/A

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

33. Applied rewrites 16.1%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(\left(0.5 - 1 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \color{blue}{\frac{\frac{\mathsf{fma}\left(a \cdot a, \mathsf{fma}\left(1, 0.5, 0.5\right), \left(\left(0.5 - 1 \cdot 0.5\right) \cdot b\right) \cdot b\right)}{y-scale}}{y-scale}}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
34. Add Preprocessing

Alternative 4: 14.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot \left|b\right|\\ t_1 := \frac{\left|b\right|}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ t_2 := -\left|a\right|\\ t_3 := \left(t\_2 \cdot \left|b\right|\right) \cdot t\_0\\ t_4 := \left|x-scale\right| \cdot y-scale\\ t_5 := t\_4 \cdot \left|x-scale\right|\\ t_6 := y-scale \cdot \left|x-scale\right|\\ t_7 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_8 := 4 \cdot t\_0\\ t_9 := t\_8 \cdot \left|b\right|\\ t_10 := \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_11 := \mathsf{fma}\left(\left|a\right|, t\_10, \mathsf{fma}\left(\left|b\right|, t\_1, \left|\left|a\right| \cdot t\_10 - \left|b\right| \cdot t\_1\right|\right)\right)\\ \mathbf{if}\;\left|a\right| \leq 3 \cdot 10^{-123}:\\ \;\;\;\;\left(\frac{\sqrt{\left(\left(\frac{t\_9 \cdot t\_2}{t\_5 \cdot y-scale} \cdot 2\right) \cdot t\_3\right) \cdot t\_11}}{t\_0 \cdot t\_8} \cdot t\_5\right) \cdot y-scale\\ \mathbf{elif}\;\left|a\right| \leq 8.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\left|b\right| \cdot \frac{\sqrt{8 \cdot \left({\left(\left|a\right|\right)}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_7\right)}^{2}} + t\_7\right)\right)\right)}}{\left|x-scale\right|}}{{\left(\left|a\right|\right)}^{2} \cdot \left|t\_4\right|}\right) \cdot t\_6\right) \cdot t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\left(\left(\frac{\left(t\_8 \cdot \frac{\left|b\right|}{t\_4}\right) \cdot t\_2}{t\_4} \cdot 2\right) \cdot t\_3\right) \cdot t\_11}}{t\_9 \cdot \left|a\right|} \cdot t\_5\right) \cdot y-scale\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs a) (fabs b)))
       (t_1 (/ (fabs b) (* (fabs x-scale) (fabs x-scale))))
       (t_2 (- (fabs a)))
       (t_3 (* (* t_2 (fabs b)) t_0))
       (t_4 (* (fabs x-scale) y-scale))
       (t_5 (* t_4 (fabs x-scale)))
       (t_6 (* y-scale (fabs x-scale)))
       (t_7 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
       (t_8 (* 4.0 t_0))
       (t_9 (* t_8 (fabs b)))
       (t_10 (/ (fabs a) (* y-scale y-scale)))
       (t_11
        (fma
         (fabs a)
         t_10
         (fma
          (fabs b)
          t_1
          (fabs (- (* (fabs a) t_10) (* (fabs b) t_1)))))))
  (if (<= (fabs a) 3e-123)
    (*
     (*
      (/
       (sqrt (* (* (* (/ (* t_9 t_2) (* t_5 y-scale)) 2.0) t_3) t_11))
       (* t_0 t_8))
      t_5)
     y-scale)
    (if (<= (fabs a) 8.5e+33)
      (*
       (*
        (*
         0.25
         (/
          (*
           (fabs b)
           (/
            (sqrt
             (*
              8.0
              (*
               (pow (fabs a) 4.0)
               (+ 0.5 (+ (sqrt (pow (+ 0.5 t_7) 2.0)) t_7)))))
            (fabs x-scale)))
          (* (pow (fabs a) 2.0) (fabs t_4))))
        t_6)
       t_6)
      (*
       (*
        (/
         (sqrt
          (*
           (* (* (/ (* (* t_8 (/ (fabs b) t_4)) t_2) t_4) 2.0) t_3)
           t_11))
         (* t_9 (fabs a)))
        t_5)
       y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(a) * fabs(b);
	double t_1 = fabs(b) / (fabs(x_45_scale) * fabs(x_45_scale));
	double t_2 = -fabs(a);
	double t_3 = (t_2 * fabs(b)) * t_0;
	double t_4 = fabs(x_45_scale) * y_45_scale;
	double t_5 = t_4 * fabs(x_45_scale);
	double t_6 = y_45_scale * fabs(x_45_scale);
	double t_7 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_8 = 4.0 * t_0;
	double t_9 = t_8 * fabs(b);
	double t_10 = fabs(a) / (y_45_scale * y_45_scale);
	double t_11 = fma(fabs(a), t_10, fma(fabs(b), t_1, fabs(((fabs(a) * t_10) - (fabs(b) * t_1)))));
	double tmp;
	if (fabs(a) <= 3e-123) {
		tmp = ((sqrt((((((t_9 * t_2) / (t_5 * y_45_scale)) * 2.0) * t_3) * t_11)) / (t_0 * t_8)) * t_5) * y_45_scale;
	} else if (fabs(a) <= 8.5e+33) {
		tmp = ((0.25 * ((fabs(b) * (sqrt((8.0 * (pow(fabs(a), 4.0) * (0.5 + (sqrt(pow((0.5 + t_7), 2.0)) + t_7))))) / fabs(x_45_scale))) / (pow(fabs(a), 2.0) * fabs(t_4)))) * t_6) * t_6;
	} else {
		tmp = ((sqrt(((((((t_8 * (fabs(b) / t_4)) * t_2) / t_4) * 2.0) * t_3) * t_11)) / (t_9 * fabs(a))) * t_5) * y_45_scale;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) * abs(b))
	t_1 = Float64(abs(b) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	t_2 = Float64(-abs(a))
	t_3 = Float64(Float64(t_2 * abs(b)) * t_0)
	t_4 = Float64(abs(x_45_scale) * y_45_scale)
	t_5 = Float64(t_4 * abs(x_45_scale))
	t_6 = Float64(y_45_scale * abs(x_45_scale))
	t_7 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_8 = Float64(4.0 * t_0)
	t_9 = Float64(t_8 * abs(b))
	t_10 = Float64(abs(a) / Float64(y_45_scale * y_45_scale))
	t_11 = fma(abs(a), t_10, fma(abs(b), t_1, abs(Float64(Float64(abs(a) * t_10) - Float64(abs(b) * t_1)))))
	tmp = 0.0
	if (abs(a) <= 3e-123)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(t_9 * t_2) / Float64(t_5 * y_45_scale)) * 2.0) * t_3) * t_11)) / Float64(t_0 * t_8)) * t_5) * y_45_scale);
	elseif (abs(a) <= 8.5e+33)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(abs(b) * Float64(sqrt(Float64(8.0 * Float64((abs(a) ^ 4.0) * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_7) ^ 2.0)) + t_7))))) / abs(x_45_scale))) / Float64((abs(a) ^ 2.0) * abs(t_4)))) * t_6) * t_6);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(t_8 * Float64(abs(b) / t_4)) * t_2) / t_4) * 2.0) * t_3) * t_11)) / Float64(t_9 * abs(a))) * t_5) * y_45_scale);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$3 = N[(N[(t$95$2 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[x$45$scale], $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(4.0 * t$95$0), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$8 * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[(N[Abs[a], $MachinePrecision] * t$95$10 + N[(N[Abs[b], $MachinePrecision] * t$95$1 + N[Abs[N[(N[(N[Abs[a], $MachinePrecision] * t$95$10), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 3e-123], N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(t$95$9 * t$95$2), $MachinePrecision] / N[(t$95$5 * y$45$scale), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$11), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$8), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * y$45$scale), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 8.5e+33], N[(N[(N[(0.25 * N[(N[(N[Abs[b], $MachinePrecision] * N[(N[Sqrt[N[(8.0 * N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$7), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] * N[Abs[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision] * t$95$6), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[(t$95$8 * N[(N[Abs[b], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$4), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$11), $MachinePrecision]], $MachinePrecision] / N[(t$95$9 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * y$45$scale), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.9999999999999998e-123

   1. Initial program 2.6%

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

   2. 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 3.7%

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

   8. Applied rewrites 4.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 5.1%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 5.1%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 5.1%

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

   10. Applied rewrites 5.1%

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

   if 2.9999999999999998e-123 < a < 8.4999999999999998e33

   1. Initial program 2.6%

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

   3. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

   8. Applied rewrites 1.6%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{b \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{y-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in x-scale around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{b \cdot \frac{\sqrt{8 \cdot \left({a}^{4} \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}}{x-scale}}{{a}^{\color{blue}{2}} \cdot \left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

   11. Applied rewrites 3.2%

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

   if 8.4999999999999998e33 < a

   1. Initial program 2.6%

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

   2. 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 3.7%

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

   8. Applied rewrites 4.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

   10. Applied rewrites 5.5%

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

Alternative 5: 14.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} t_0 := \left|a\right| \cdot \left|b\right|\\ t_1 := \frac{\left|b\right|}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ t_2 := -\left|a\right|\\ t_3 := \left(t\_2 \cdot \left|b\right|\right) \cdot t\_0\\ t_4 := \left|x-scale\right| \cdot y-scale\\ t_5 := t\_4 \cdot \left|x-scale\right|\\ t_6 := y-scale \cdot \left|x-scale\right|\\ t_7 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_8 := 4 \cdot t\_0\\ t_9 := t\_8 \cdot \left|b\right|\\ t_10 := \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_11 := \mathsf{fma}\left(\left|a\right|, t\_10, \mathsf{fma}\left(\left|b\right|, t\_1, \left|\left|a\right| \cdot t\_10 - \left|b\right| \cdot t\_1\right|\right)\right)\\ \mathbf{if}\;\left|a\right| \leq 3 \cdot 10^{-123}:\\ \;\;\;\;\left(\frac{\sqrt{\left(\left(\frac{t\_9 \cdot t\_2}{t\_5 \cdot y-scale} \cdot 2\right) \cdot t\_3\right) \cdot t\_11}}{t\_0 \cdot t\_8} \cdot t\_5\right) \cdot y-scale\\ \mathbf{elif}\;\left|a\right| \leq 1.5 \cdot 10^{+34}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\frac{\left|b\right| \cdot \sqrt{8 \cdot \left({\left(\left|a\right|\right)}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_7\right)}^{2}} + t\_7\right)\right)\right)}}{\left|x-scale\right|}}{{\left(\left|a\right|\right)}^{2} \cdot \left|t\_4\right|}\right) \cdot t\_6\right) \cdot t\_6\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\left(\left(\frac{\left(t\_8 \cdot \frac{\left|b\right|}{t\_4}\right) \cdot t\_2}{t\_4} \cdot 2\right) \cdot t\_3\right) \cdot t\_11}}{t\_9 \cdot \left|a\right|} \cdot t\_5\right) \cdot y-scale\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs a) (fabs b)))
       (t_1 (/ (fabs b) (* (fabs x-scale) (fabs x-scale))))
       (t_2 (- (fabs a)))
       (t_3 (* (* t_2 (fabs b)) t_0))
       (t_4 (* (fabs x-scale) y-scale))
       (t_5 (* t_4 (fabs x-scale)))
       (t_6 (* y-scale (fabs x-scale)))
       (t_7 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
       (t_8 (* 4.0 t_0))
       (t_9 (* t_8 (fabs b)))
       (t_10 (/ (fabs a) (* y-scale y-scale)))
       (t_11
        (fma
         (fabs a)
         t_10
         (fma
          (fabs b)
          t_1
          (fabs (- (* (fabs a) t_10) (* (fabs b) t_1)))))))
  (if (<= (fabs a) 3e-123)
    (*
     (*
      (/
       (sqrt (* (* (* (/ (* t_9 t_2) (* t_5 y-scale)) 2.0) t_3) t_11))
       (* t_0 t_8))
      t_5)
     y-scale)
    (if (<= (fabs a) 1.5e+34)
      (*
       (*
        (*
         0.25
         (/
          (/
           (*
            (fabs b)
            (sqrt
             (*
              8.0
              (*
               (pow (fabs a) 4.0)
               (+ 0.5 (+ (sqrt (pow (+ 0.5 t_7) 2.0)) t_7))))))
           (fabs x-scale))
          (* (pow (fabs a) 2.0) (fabs t_4))))
        t_6)
       t_6)
      (*
       (*
        (/
         (sqrt
          (*
           (* (* (/ (* (* t_8 (/ (fabs b) t_4)) t_2) t_4) 2.0) t_3)
           t_11))
         (* t_9 (fabs a)))
        t_5)
       y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(a) * fabs(b);
	double t_1 = fabs(b) / (fabs(x_45_scale) * fabs(x_45_scale));
	double t_2 = -fabs(a);
	double t_3 = (t_2 * fabs(b)) * t_0;
	double t_4 = fabs(x_45_scale) * y_45_scale;
	double t_5 = t_4 * fabs(x_45_scale);
	double t_6 = y_45_scale * fabs(x_45_scale);
	double t_7 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_8 = 4.0 * t_0;
	double t_9 = t_8 * fabs(b);
	double t_10 = fabs(a) / (y_45_scale * y_45_scale);
	double t_11 = fma(fabs(a), t_10, fma(fabs(b), t_1, fabs(((fabs(a) * t_10) - (fabs(b) * t_1)))));
	double tmp;
	if (fabs(a) <= 3e-123) {
		tmp = ((sqrt((((((t_9 * t_2) / (t_5 * y_45_scale)) * 2.0) * t_3) * t_11)) / (t_0 * t_8)) * t_5) * y_45_scale;
	} else if (fabs(a) <= 1.5e+34) {
		tmp = ((0.25 * (((fabs(b) * sqrt((8.0 * (pow(fabs(a), 4.0) * (0.5 + (sqrt(pow((0.5 + t_7), 2.0)) + t_7)))))) / fabs(x_45_scale)) / (pow(fabs(a), 2.0) * fabs(t_4)))) * t_6) * t_6;
	} else {
		tmp = ((sqrt(((((((t_8 * (fabs(b) / t_4)) * t_2) / t_4) * 2.0) * t_3) * t_11)) / (t_9 * fabs(a))) * t_5) * y_45_scale;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) * abs(b))
	t_1 = Float64(abs(b) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	t_2 = Float64(-abs(a))
	t_3 = Float64(Float64(t_2 * abs(b)) * t_0)
	t_4 = Float64(abs(x_45_scale) * y_45_scale)
	t_5 = Float64(t_4 * abs(x_45_scale))
	t_6 = Float64(y_45_scale * abs(x_45_scale))
	t_7 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_8 = Float64(4.0 * t_0)
	t_9 = Float64(t_8 * abs(b))
	t_10 = Float64(abs(a) / Float64(y_45_scale * y_45_scale))
	t_11 = fma(abs(a), t_10, fma(abs(b), t_1, abs(Float64(Float64(abs(a) * t_10) - Float64(abs(b) * t_1)))))
	tmp = 0.0
	if (abs(a) <= 3e-123)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(t_9 * t_2) / Float64(t_5 * y_45_scale)) * 2.0) * t_3) * t_11)) / Float64(t_0 * t_8)) * t_5) * y_45_scale);
	elseif (abs(a) <= 1.5e+34)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(Float64(abs(b) * sqrt(Float64(8.0 * Float64((abs(a) ^ 4.0) * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_7) ^ 2.0)) + t_7)))))) / abs(x_45_scale)) / Float64((abs(a) ^ 2.0) * abs(t_4)))) * t_6) * t_6);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(t_8 * Float64(abs(b) / t_4)) * t_2) / t_4) * 2.0) * t_3) * t_11)) / Float64(t_9 * abs(a))) * t_5) * y_45_scale);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$3 = N[(N[(t$95$2 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[x$45$scale], $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(4.0 * t$95$0), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$8 * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[(N[Abs[a], $MachinePrecision] * t$95$10 + N[(N[Abs[b], $MachinePrecision] * t$95$1 + N[Abs[N[(N[(N[Abs[a], $MachinePrecision] * t$95$10), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 3e-123], N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(t$95$9 * t$95$2), $MachinePrecision] / N[(t$95$5 * y$45$scale), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$11), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 * t$95$8), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * y$45$scale), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 1.5e+34], N[(N[(N[(0.25 * N[(N[(N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$7), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] * N[Abs[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision] * t$95$6), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[(t$95$8 * N[(N[Abs[b], $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$4), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$11), $MachinePrecision]], $MachinePrecision] / N[(t$95$9 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * y$45$scale), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 2.9999999999999998e-123

   1. Initial program 2.6%

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

   2. 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 3.7%

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

   8. Applied rewrites 4.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 5.1%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 5.1%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 5.1%

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

   10. Applied rewrites 5.1%

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

   if 2.9999999999999998e-123 < a < 1.5000000000000001e34

   1. Initial program 2.6%

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

   3. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

   8. Applied rewrites 1.6%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{b \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{y-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in x-scale around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{\frac{b \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}}{x-scale}}{\color{blue}{{a}^{2}} \cdot \left|x-scale \cdot y-scale\right|}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

   11. Applied rewrites 3.3%

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

   if 1.5000000000000001e34 < a

   1. Initial program 2.6%

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

   2. 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 3.7%

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

   8. Applied rewrites 4.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

   10. Applied rewrites 5.5%

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

Alternative 6: 14.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := \left|a\right| \cdot \left|b\right|\\ t_2 := \frac{\left|b\right|}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ t_3 := -\left|a\right|\\ t_4 := \left(t\_3 \cdot \left|b\right|\right) \cdot t\_1\\ t_5 := \left|x-scale\right| \cdot y-scale\\ t_6 := t\_5 \cdot \left|x-scale\right|\\ t_7 := y-scale \cdot \left|x-scale\right|\\ t_8 := 4 \cdot t\_1\\ t_9 := t\_8 \cdot \left|b\right|\\ t_10 := \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_11 := \mathsf{fma}\left(\left|a\right|, t\_10, \mathsf{fma}\left(\left|b\right|, t\_2, \left|\left|a\right| \cdot t\_10 - \left|b\right| \cdot t\_2\right|\right)\right)\\ \mathbf{if}\;\left|a\right| \leq 1.85 \cdot 10^{-117}:\\ \;\;\;\;\left(\frac{\sqrt{\left(\left(\frac{t\_9 \cdot t\_3}{t\_6 \cdot y-scale} \cdot 2\right) \cdot t\_4\right) \cdot t\_11}}{t\_1 \cdot t\_8} \cdot t\_6\right) \cdot y-scale\\ \mathbf{elif}\;\left|a\right| \leq 8.5 \cdot 10^{+33}:\\ \;\;\;\;\left(\left(0.25 \cdot \frac{\left|b\right| \cdot \sqrt{8 \cdot \left({\left(\left|a\right|\right)}^{4} \cdot \left(0.5 + \left(\sqrt{{\left(0.5 + t\_0\right)}^{2}} + t\_0\right)\right)\right)}}{{\left(\left|a\right|\right)}^{2} \cdot \left(\left|x-scale\right| \cdot \left|t\_5\right|\right)}\right) \cdot t\_7\right) \cdot t\_7\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\sqrt{\left(\left(\frac{\left(t\_8 \cdot \frac{\left|b\right|}{t\_5}\right) \cdot t\_3}{t\_5} \cdot 2\right) \cdot t\_4\right) \cdot t\_11}}{t\_9 \cdot \left|a\right|} \cdot t\_6\right) \cdot y-scale\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
       (t_1 (* (fabs a) (fabs b)))
       (t_2 (/ (fabs b) (* (fabs x-scale) (fabs x-scale))))
       (t_3 (- (fabs a)))
       (t_4 (* (* t_3 (fabs b)) t_1))
       (t_5 (* (fabs x-scale) y-scale))
       (t_6 (* t_5 (fabs x-scale)))
       (t_7 (* y-scale (fabs x-scale)))
       (t_8 (* 4.0 t_1))
       (t_9 (* t_8 (fabs b)))
       (t_10 (/ (fabs a) (* y-scale y-scale)))
       (t_11
        (fma
         (fabs a)
         t_10
         (fma
          (fabs b)
          t_2
          (fabs (- (* (fabs a) t_10) (* (fabs b) t_2)))))))
  (if (<= (fabs a) 1.85e-117)
    (*
     (*
      (/
       (sqrt (* (* (* (/ (* t_9 t_3) (* t_6 y-scale)) 2.0) t_4) t_11))
       (* t_1 t_8))
      t_6)
     y-scale)
    (if (<= (fabs a) 8.5e+33)
      (*
       (*
        (*
         0.25
         (/
          (*
           (fabs b)
           (sqrt
            (*
             8.0
             (*
              (pow (fabs a) 4.0)
              (+ 0.5 (+ (sqrt (pow (+ 0.5 t_0) 2.0)) t_0))))))
          (* (pow (fabs a) 2.0) (* (fabs x-scale) (fabs t_5)))))
        t_7)
       t_7)
      (*
       (*
        (/
         (sqrt
          (*
           (* (* (/ (* (* t_8 (/ (fabs b) t_5)) t_3) t_5) 2.0) t_4)
           t_11))
         (* t_9 (fabs a)))
        t_6)
       y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_1 = fabs(a) * fabs(b);
	double t_2 = fabs(b) / (fabs(x_45_scale) * fabs(x_45_scale));
	double t_3 = -fabs(a);
	double t_4 = (t_3 * fabs(b)) * t_1;
	double t_5 = fabs(x_45_scale) * y_45_scale;
	double t_6 = t_5 * fabs(x_45_scale);
	double t_7 = y_45_scale * fabs(x_45_scale);
	double t_8 = 4.0 * t_1;
	double t_9 = t_8 * fabs(b);
	double t_10 = fabs(a) / (y_45_scale * y_45_scale);
	double t_11 = fma(fabs(a), t_10, fma(fabs(b), t_2, fabs(((fabs(a) * t_10) - (fabs(b) * t_2)))));
	double tmp;
	if (fabs(a) <= 1.85e-117) {
		tmp = ((sqrt((((((t_9 * t_3) / (t_6 * y_45_scale)) * 2.0) * t_4) * t_11)) / (t_1 * t_8)) * t_6) * y_45_scale;
	} else if (fabs(a) <= 8.5e+33) {
		tmp = ((0.25 * ((fabs(b) * sqrt((8.0 * (pow(fabs(a), 4.0) * (0.5 + (sqrt(pow((0.5 + t_0), 2.0)) + t_0)))))) / (pow(fabs(a), 2.0) * (fabs(x_45_scale) * fabs(t_5))))) * t_7) * t_7;
	} else {
		tmp = ((sqrt(((((((t_8 * (fabs(b) / t_5)) * t_3) / t_5) * 2.0) * t_4) * t_11)) / (t_9 * fabs(a))) * t_6) * y_45_scale;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_1 = Float64(abs(a) * abs(b))
	t_2 = Float64(abs(b) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	t_3 = Float64(-abs(a))
	t_4 = Float64(Float64(t_3 * abs(b)) * t_1)
	t_5 = Float64(abs(x_45_scale) * y_45_scale)
	t_6 = Float64(t_5 * abs(x_45_scale))
	t_7 = Float64(y_45_scale * abs(x_45_scale))
	t_8 = Float64(4.0 * t_1)
	t_9 = Float64(t_8 * abs(b))
	t_10 = Float64(abs(a) / Float64(y_45_scale * y_45_scale))
	t_11 = fma(abs(a), t_10, fma(abs(b), t_2, abs(Float64(Float64(abs(a) * t_10) - Float64(abs(b) * t_2)))))
	tmp = 0.0
	if (abs(a) <= 1.85e-117)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(t_9 * t_3) / Float64(t_6 * y_45_scale)) * 2.0) * t_4) * t_11)) / Float64(t_1 * t_8)) * t_6) * y_45_scale);
	elseif (abs(a) <= 8.5e+33)
		tmp = Float64(Float64(Float64(0.25 * Float64(Float64(abs(b) * sqrt(Float64(8.0 * Float64((abs(a) ^ 4.0) * Float64(0.5 + Float64(sqrt((Float64(0.5 + t_0) ^ 2.0)) + t_0)))))) / Float64((abs(a) ^ 2.0) * Float64(abs(x_45_scale) * abs(t_5))))) * t_7) * t_7);
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(t_8 * Float64(abs(b) / t_5)) * t_3) / t_5) * 2.0) * t_4) * t_11)) / Float64(t_9 * abs(a))) * t_6) * y_45_scale);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[a], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$4 = N[(N[(t$95$3 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[x$45$scale], $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(4.0 * t$95$1), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$8 * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$11 = N[(N[Abs[a], $MachinePrecision] * t$95$10 + N[(N[Abs[b], $MachinePrecision] * t$95$2 + N[Abs[N[(N[(N[Abs[a], $MachinePrecision] * t$95$10), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.85e-117], N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(t$95$9 * t$95$3), $MachinePrecision] / N[(t$95$6 * y$45$scale), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$11), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 * t$95$8), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision] * y$45$scale), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 8.5e+33], N[(N[(N[(0.25 * N[(N[(N[Abs[b], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision] * N[(0.5 + N[(N[Sqrt[N[Power[N[(0.5 + t$95$0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$7), $MachinePrecision] * t$95$7), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[(t$95$8 * N[(N[Abs[b], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$5), $MachinePrecision] * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$11), $MachinePrecision]], $MachinePrecision] / N[(t$95$9 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision] * y$45$scale), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.8500000000000001e-117

   1. Initial program 2.6%

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

   2. 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 3.7%

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

   8. Applied rewrites 4.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-*.f64 5.1%

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 5.1%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 5.1%

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

   10. Applied rewrites 5.1%

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

   if 1.8500000000000001e-117 < a < 8.4999999999999998e33

   1. Initial program 2.6%

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

   3. Applied rewrites 6.5%

      \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

   8. Applied rewrites 1.6%

      \[\leadsto \left(\color{blue}{\left(0.25 \cdot \frac{b \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\left(\sqrt{\frac{{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot \frac{1}{{y-scale}^{2}} - \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, \frac{1}{{x-scale}^{2}}, \mathsf{fma}\left(0.5, \frac{1}{{y-scale}^{2}}, 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)\right)\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}\right)\right)}}{{a}^{2} \cdot \left|x-scale \cdot y-scale\right|}\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   9. Taylor expanded in x-scale around 0

      \[\leadsto \left(\left(0.25 \cdot \frac{b \cdot \sqrt{8 \cdot \left({a}^{4} \cdot \left(\frac{1}{2} + \left(\sqrt{{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)}}{\color{blue}{{a}^{2} \cdot \left(x-scale \cdot \left|x-scale \cdot y-scale\right|\right)}}\right) \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

   11. Applied rewrites 2.7%

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

   if 8.4999999999999998e33 < a

   1. Initial program 2.6%

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

   2. 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 3.7%

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

   8. Applied rewrites 4.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. frac-times N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*l* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*r/ N/A

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

   10. Applied rewrites 5.5%

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

Alternative 7: 13.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 1 \cdot 0.5\\ t_1 := \frac{\mathsf{fma}\left(t\_0 \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}\\ t_2 := \frac{\mathsf{fma}\left(t\_0 \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\\ t_3 := \left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\\ \left(\frac{\frac{\frac{\sqrt{\left(\left(t\_3 \cdot 8\right) \cdot \left(\mathsf{hypot}\left(t\_2 - t\_1, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(t\_1 + t\_2\right)\right)\right) \cdot t\_3}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (- 0.5 (* 1.0 0.5)))
       (t_1
        (/
         (fma (* t_0 a) a (* (* b b) (fma 1.0 0.5 0.5)))
         (* x-scale x-scale)))
       (t_2
        (/
         (fma (* t_0 b) b (* (* a a) (fma 1.0 0.5 0.5)))
         (* y-scale y-scale)))
       (t_3 (* (* (* (- a) b) a) b)))
  (*
   (*
    (/
     (/
      (/
       (sqrt
        (*
         (*
          (* t_3 8.0)
          (+
           (hypot
            (- t_2 t_1)
            (/
             (*
              (sin (* (* 2.0 PI) (* angle 0.005555555555555556)))
              (* (- b a) (+ b a)))
             (* x-scale y-scale)))
           (+ t_1 t_2)))
         t_3))
       (fabs (* x-scale y-scale)))
      (* 4.0 (* b a)))
     (* b a))
    (* y-scale x-scale))
   (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.5 - (1.0 * 0.5);
	double t_1 = fma((t_0 * a), a, ((b * b) * fma(1.0, 0.5, 0.5))) / (x_45_scale * x_45_scale);
	double t_2 = fma((t_0 * b), b, ((a * a) * fma(1.0, 0.5, 0.5))) / (y_45_scale * y_45_scale);
	double t_3 = ((-a * b) * a) * b;
	return ((((sqrt((((t_3 * 8.0) * (hypot((t_2 - t_1), ((sin(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556))) * ((b - a) * (b + a))) / (x_45_scale * y_45_scale))) + (t_1 + t_2))) * t_3)) / fabs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.5 - Float64(1.0 * 0.5))
	t_1 = Float64(fma(Float64(t_0 * a), a, Float64(Float64(b * b) * fma(1.0, 0.5, 0.5))) / Float64(x_45_scale * x_45_scale))
	t_2 = Float64(fma(Float64(t_0 * b), b, Float64(Float64(a * a) * fma(1.0, 0.5, 0.5))) / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(Float64(Float64(Float64(-a) * b) * a) * b)
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(t_3 * 8.0) * Float64(hypot(Float64(t_2 - t_1), Float64(Float64(sin(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))) * Float64(Float64(b - a) * Float64(b + a))) / Float64(x_45_scale * y_45_scale))) + Float64(t_1 + t_2))) * t_3)) / abs(Float64(x_45_scale * y_45_scale))) / Float64(4.0 * Float64(b * a))) / Float64(b * a)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.5 - N[(1.0 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 * a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[((-a) * b), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$3 * 8.0), $MachinePrecision] * N[(N[Sqrt[N[(t$95$2 - t$95$1), $MachinePrecision] ^ 2 + N[(N[(N[Sin[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 2.6%

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

3. Applied rewrites 6.5%

   \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

8. Applied rewrites 12.9%

   \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

9. Taylor expanded in angle around 0

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

11. Applied rewrites 12.9%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

12. Taylor expanded in angle around 0

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

14. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

15. Taylor expanded in angle around 0

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

17. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

18. Taylor expanded in angle around 0

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

20. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

21. Taylor expanded in angle around 0

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

23. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

24. Taylor expanded in angle around 0

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

26. Applied rewrites 14.5%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

27. Taylor expanded in angle around 0

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

29. Applied rewrites 14.5%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*l* N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*l* N/A

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

    8. lift-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. associate-*r* N/A

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

    11. lower-*.f64 N/A

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

    12. lift-*.f64 N/A

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

    13. *-commutative N/A

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

    14. lift-*.f64 N/A

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

    15. lower-*.f64 14.1%

        \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\color{blue}{\left(\left(b \cdot \left(-a\right)\right) \cdot a\right)} \cdot b\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

    16. lift-*.f64 N/A

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

    17. *-commutative N/A

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

    18. lift-*.f64 14.1%

        \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\color{blue}{\left(\left(-a\right) \cdot b\right)} \cdot a\right) \cdot b\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

31. Applied rewrites 14.1%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. associate-*l* N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*l* N/A

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

    8. lift-*.f64 N/A

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

    9. lift-*.f64 N/A

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

    10. associate-*r* N/A

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

    11. lower-*.f64 N/A

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

    12. lift-*.f64 N/A

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

    13. *-commutative N/A

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

    14. lift-*.f64 N/A

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

    15. lower-*.f64 14.7%

        \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\color{blue}{\left(\left(b \cdot \left(-a\right)\right) \cdot a\right)} \cdot b\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

    16. lift-*.f64 N/A

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

    17. *-commutative N/A

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

    18. lift-*.f64 14.7%

        \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\color{blue}{\left(\left(-a\right) \cdot b\right)} \cdot a\right) \cdot b\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

33. Applied rewrites 14.7%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \color{blue}{\left(\left(\left(\left(-a\right) \cdot b\right) \cdot a\right) \cdot b\right)}}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
34. Add Preprocessing

Alternative 8: 7.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 1 \cdot 0.5\\ t_1 := \frac{\mathsf{fma}\left(t\_0 \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}\\ t_2 := \frac{\mathsf{fma}\left(t\_0 \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\\ t_3 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ \left(\frac{\frac{\frac{\sqrt{\left(\left(t\_3 \cdot 8\right) \cdot \left(\mathsf{hypot}\left(t\_2 - t\_1, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(t\_1 + t\_2\right)\right)\right) \cdot t\_3}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (- 0.5 (* 1.0 0.5)))
       (t_1
        (/
         (fma (* t_0 a) a (* (* b b) (fma 1.0 0.5 0.5)))
         (* x-scale x-scale)))
       (t_2
        (/
         (fma (* t_0 b) b (* (* a a) (fma 1.0 0.5 0.5)))
         (* y-scale y-scale)))
       (t_3 (* (* (* b a) b) (- a))))
  (*
   (*
    (/
     (/
      (/
       (sqrt
        (*
         (*
          (* t_3 8.0)
          (+
           (hypot
            (- t_2 t_1)
            (/
             (*
              (sin (* (* 2.0 PI) (* angle 0.005555555555555556)))
              (* (- b a) (+ b a)))
             (* x-scale y-scale)))
           (+ t_1 t_2)))
         t_3))
       (fabs (* x-scale y-scale)))
      (* 4.0 (* b a)))
     (* b a))
    (* y-scale x-scale))
   (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.5 - (1.0 * 0.5);
	double t_1 = fma((t_0 * a), a, ((b * b) * fma(1.0, 0.5, 0.5))) / (x_45_scale * x_45_scale);
	double t_2 = fma((t_0 * b), b, ((a * a) * fma(1.0, 0.5, 0.5))) / (y_45_scale * y_45_scale);
	double t_3 = ((b * a) * b) * -a;
	return ((((sqrt((((t_3 * 8.0) * (hypot((t_2 - t_1), ((sin(((2.0 * ((double) M_PI)) * (angle * 0.005555555555555556))) * ((b - a) * (b + a))) / (x_45_scale * y_45_scale))) + (t_1 + t_2))) * t_3)) / fabs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.5 - Float64(1.0 * 0.5))
	t_1 = Float64(fma(Float64(t_0 * a), a, Float64(Float64(b * b) * fma(1.0, 0.5, 0.5))) / Float64(x_45_scale * x_45_scale))
	t_2 = Float64(fma(Float64(t_0 * b), b, Float64(Float64(a * a) * fma(1.0, 0.5, 0.5))) / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(t_3 * 8.0) * Float64(hypot(Float64(t_2 - t_1), Float64(Float64(sin(Float64(Float64(2.0 * pi) * Float64(angle * 0.005555555555555556))) * Float64(Float64(b - a) * Float64(b + a))) / Float64(x_45_scale * y_45_scale))) + Float64(t_1 + t_2))) * t_3)) / abs(Float64(x_45_scale * y_45_scale))) / Float64(4.0 * Float64(b * a))) / Float64(b * a)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.5 - N[(1.0 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 * a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$3 * 8.0), $MachinePrecision] * N[(N[Sqrt[N[(t$95$2 - t$95$1), $MachinePrecision] ^ 2 + N[(N[(N[Sin[N[(N[(2.0 * Pi), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 2.6%

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

3. Applied rewrites 6.5%

   \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

8. Applied rewrites 12.9%

   \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

9. Taylor expanded in angle around 0

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

11. Applied rewrites 12.9%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

12. Taylor expanded in angle around 0

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

14. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

15. Taylor expanded in angle around 0

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

17. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

18. Taylor expanded in angle around 0

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

20. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

21. Taylor expanded in angle around 0

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

23. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

24. Taylor expanded in angle around 0

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

26. Applied rewrites 14.5%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

27. Taylor expanded in angle around 0

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

29. Applied rewrites 14.5%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
30. Add Preprocessing

Alternative 9: 7.7% accurate, 3.3× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 1 \cdot 0.5\\ t_1 := \frac{\mathsf{fma}\left(t\_0 \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}\\ t_2 := \frac{\mathsf{fma}\left(t\_0 \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\\ t_3 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ \left(\frac{\frac{\frac{\sqrt{\left(\left(t\_3 \cdot 8\right) \cdot \left(\mathsf{hypot}\left(t\_2 - t\_1, 0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}{x-scale \cdot y-scale}\right) + \left(t\_1 + t\_2\right)\right)\right) \cdot t\_3}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (- 0.5 (* 1.0 0.5)))
       (t_1
        (/
         (fma (* t_0 a) a (* (* b b) (fma 1.0 0.5 0.5)))
         (* x-scale x-scale)))
       (t_2
        (/
         (fma (* t_0 b) b (* (* a a) (fma 1.0 0.5 0.5)))
         (* y-scale y-scale)))
       (t_3 (* (* (* b a) b) (- a))))
  (*
   (*
    (/
     (/
      (/
       (sqrt
        (*
         (*
          (* t_3 8.0)
          (+
           (hypot
            (- t_2 t_1)
            (*
             0.011111111111111112
             (/
              (* angle (* PI (* (+ a b) (- b a))))
              (* x-scale y-scale))))
           (+ t_1 t_2)))
         t_3))
       (fabs (* x-scale y-scale)))
      (* 4.0 (* b a)))
     (* b a))
    (* y-scale x-scale))
   (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.5 - (1.0 * 0.5);
	double t_1 = fma((t_0 * a), a, ((b * b) * fma(1.0, 0.5, 0.5))) / (x_45_scale * x_45_scale);
	double t_2 = fma((t_0 * b), b, ((a * a) * fma(1.0, 0.5, 0.5))) / (y_45_scale * y_45_scale);
	double t_3 = ((b * a) * b) * -a;
	return ((((sqrt((((t_3 * 8.0) * (hypot((t_2 - t_1), (0.011111111111111112 * ((angle * (((double) M_PI) * ((a + b) * (b - a)))) / (x_45_scale * y_45_scale)))) + (t_1 + t_2))) * t_3)) / fabs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.5 - Float64(1.0 * 0.5))
	t_1 = Float64(fma(Float64(t_0 * a), a, Float64(Float64(b * b) * fma(1.0, 0.5, 0.5))) / Float64(x_45_scale * x_45_scale))
	t_2 = Float64(fma(Float64(t_0 * b), b, Float64(Float64(a * a) * fma(1.0, 0.5, 0.5))) / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(t_3 * 8.0) * Float64(hypot(Float64(t_2 - t_1), Float64(0.011111111111111112 * Float64(Float64(angle * Float64(pi * Float64(Float64(a + b) * Float64(b - a)))) / Float64(x_45_scale * y_45_scale)))) + Float64(t_1 + t_2))) * t_3)) / abs(Float64(x_45_scale * y_45_scale))) / Float64(4.0 * Float64(b * a))) / Float64(b * a)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.5 - N[(1.0 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 * a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$3 * 8.0), $MachinePrecision] * N[(N[Sqrt[N[(t$95$2 - t$95$1), $MachinePrecision] ^ 2 + N[(0.011111111111111112 * N[(N[(angle * N[(Pi * N[(N[(a + b), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] + N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 2.6%

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

3. Applied rewrites 6.5%

   \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

8. Applied rewrites 12.9%

   \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

9. Taylor expanded in angle around 0

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

11. Applied rewrites 12.9%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

12. Taylor expanded in angle around 0

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

14. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

15. Taylor expanded in angle around 0

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

17. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

18. Taylor expanded in angle around 0

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

20. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

21. Taylor expanded in angle around 0

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

23. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

24. Taylor expanded in angle around 0

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

26. Applied rewrites 14.5%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

27. Taylor expanded in angle around 0

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

29. Applied rewrites 14.5%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

30. Taylor expanded in angle around 0

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-PI.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lower-+.f64 N/A

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

    8. lower--.f64 N/A

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

    9. lower-*.f64 14.7%

       \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, 0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}{x-scale \cdot \color{blue}{y-scale}}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

32. Applied rewrites 14.7%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \color{blue}{0.011111111111111112 \cdot \frac{angle \cdot \left(\pi \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}{x-scale \cdot y-scale}}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
33. Add Preprocessing

Alternative 10: 7.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 1 \cdot 0.5\\ t_1 := \left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\\ \left(\frac{\frac{\frac{\sqrt{\left(\left(t\_1 \cdot 8\right) \cdot \left(\sqrt{{\left(\frac{{a}^{2}}{{y-scale}^{2}} - \frac{{b}^{2}}{{x-scale}^{2}}\right)}^{2}} + \left(\frac{\mathsf{fma}\left(t\_0 \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(t\_0 \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot t\_1}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (- 0.5 (* 1.0 0.5))) (t_1 (* (* (* b a) b) (- a))))
  (*
   (*
    (/
     (/
      (/
       (sqrt
        (*
         (*
          (* t_1 8.0)
          (+
           (sqrt
            (pow
             (-
              (/ (pow a 2.0) (pow y-scale 2.0))
              (/ (pow b 2.0) (pow x-scale 2.0)))
             2.0))
           (+
            (/
             (fma (* t_0 a) a (* (* b b) (fma 1.0 0.5 0.5)))
             (* x-scale x-scale))
            (/
             (fma (* t_0 b) b (* (* a a) (fma 1.0 0.5 0.5)))
             (* y-scale y-scale)))))
         t_1))
       (fabs (* x-scale y-scale)))
      (* 4.0 (* b a)))
     (* b a))
    (* y-scale x-scale))
   (* y-scale x-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.5 - (1.0 * 0.5);
	double t_1 = ((b * a) * b) * -a;
	return ((((sqrt((((t_1 * 8.0) * (sqrt(pow(((pow(a, 2.0) / pow(y_45_scale, 2.0)) - (pow(b, 2.0) / pow(x_45_scale, 2.0))), 2.0)) + ((fma((t_0 * a), a, ((b * b) * fma(1.0, 0.5, 0.5))) / (x_45_scale * x_45_scale)) + (fma((t_0 * b), b, ((a * a) * fma(1.0, 0.5, 0.5))) / (y_45_scale * y_45_scale))))) * t_1)) / fabs((x_45_scale * y_45_scale))) / (4.0 * (b * a))) / (b * a)) * (y_45_scale * x_45_scale)) * (y_45_scale * x_45_scale);
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.5 - Float64(1.0 * 0.5))
	t_1 = Float64(Float64(Float64(b * a) * b) * Float64(-a))
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(Float64(t_1 * 8.0) * Float64(sqrt((Float64(Float64((a ^ 2.0) / (y_45_scale ^ 2.0)) - Float64((b ^ 2.0) / (x_45_scale ^ 2.0))) ^ 2.0)) + Float64(Float64(fma(Float64(t_0 * a), a, Float64(Float64(b * b) * fma(1.0, 0.5, 0.5))) / Float64(x_45_scale * x_45_scale)) + Float64(fma(Float64(t_0 * b), b, Float64(Float64(a * a) * fma(1.0, 0.5, 0.5))) / Float64(y_45_scale * y_45_scale))))) * t_1)) / abs(Float64(x_45_scale * y_45_scale))) / Float64(4.0 * Float64(b * a))) / Float64(b * a)) * Float64(y_45_scale * x_45_scale)) * Float64(y_45_scale * x_45_scale))
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.5 - N[(1.0 * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$1 * 8.0), $MachinePrecision] * N[(N[Sqrt[N[Power[N[(N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Power[b, 2.0], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[(t$95$0 * a), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 * b), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(1.0 * 0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 2.6%

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

3. Applied rewrites 6.5%

   \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\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) \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\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 13.0%

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

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

8. Applied rewrites 12.9%

   \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

9. Taylor expanded in angle around 0

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

11. Applied rewrites 12.9%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

12. Taylor expanded in angle around 0

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

14. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

15. Taylor expanded in angle around 0

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

17. Applied rewrites 12.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

18. Taylor expanded in angle around 0

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

20. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

21. Taylor expanded in angle around 0

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

23. Applied rewrites 13.8%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

24. Taylor expanded in angle around 0

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

26. Applied rewrites 14.5%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - \color{blue}{1} \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\cos \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right), 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

27. Taylor expanded in angle around 0

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

29. Applied rewrites 14.5%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\mathsf{hypot}\left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale}, \frac{\sin \left(\left(2 \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}{x-scale \cdot y-scale}\right) + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\color{blue}{1}, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]

30. Taylor expanded in angle around 0

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

32. Applied rewrites 13.4%

    \[\leadsto \left(\frac{\frac{\frac{\sqrt{\left(\left(\left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right) \cdot 8\right) \cdot \left(\color{blue}{\sqrt{{\left(\frac{{a}^{2}}{{y-scale}^{2}} - \frac{{b}^{2}}{{x-scale}^{2}}\right)}^{2}}} + \left(\frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot a, a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{x-scale \cdot x-scale} + \frac{\mathsf{fma}\left(\left(0.5 - 1 \cdot 0.5\right) \cdot b, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(1, 0.5, 0.5\right)\right)}{y-scale \cdot y-scale}\right)\right)\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)}}{\left|x-scale \cdot y-scale\right|}}{4 \cdot \left(b \cdot a\right)}}{b \cdot a} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right) \]
33. Add Preprocessing

Alternative 11: 7.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := \left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale\\ t_2 := \frac{a}{y-scale \cdot y-scale}\\ t_3 := 4 \cdot \left(a \cdot b\right)\\ \frac{\sqrt{\left(\left(\frac{\left(t\_3 \cdot b\right) \cdot \left(-a\right)}{t\_1} \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(a, t\_2, \mathsf{fma}\left(b, t\_0, \left|a \cdot t\_2 - b \cdot t\_0\right|\right)\right)}}{t\_3} \cdot \frac{t\_1}{a \cdot b} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (/ b (* x-scale x-scale)))
       (t_1 (* (* (* x-scale y-scale) x-scale) y-scale))
       (t_2 (/ a (* y-scale y-scale)))
       (t_3 (* 4.0 (* a b))))
  (*
   (/
    (sqrt
     (*
      (* (* (/ (* (* t_3 b) (- a)) t_1) 2.0) (* (* (- a) b) (* a b)))
      (fma a t_2 (fma b t_0 (fabs (- (* a t_2) (* b t_0)))))))
    t_3)
   (/ t_1 (* a b)))))

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);
	double t_1 = ((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale;
	double t_2 = a / (y_45_scale * y_45_scale);
	double t_3 = 4.0 * (a * b);
	return (sqrt(((((((t_3 * b) * -a) / t_1) * 2.0) * ((-a * b) * (a * b))) * fma(a, t_2, fma(b, t_0, fabs(((a * t_2) - (b * t_0))))))) / t_3) * (t_1 / (a * b));
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)
	t_2 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(4.0 * Float64(a * b))
	return Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(t_3 * b) * Float64(-a)) / t_1) * 2.0) * Float64(Float64(Float64(-a) * b) * Float64(a * b))) * fma(a, t_2, fma(b, t_0, abs(Float64(Float64(a * t_2) - Float64(b * t_0))))))) / t_3) * Float64(t_1 / Float64(a * b)))
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]}, Block[{t$95$1 = N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[(t$95$3 * b), $MachinePrecision] * (-a)), $MachinePrecision] / t$95$1), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * t$95$2 + N[(b * t$95$0 + N[Abs[N[(N[(a * t$95$2), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * N[(t$95$1 / N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 2.6%

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

2. 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 3.7%

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

8. Applied rewrites 7.4%

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

Alternative 12: 5.0% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (/ b (* x-scale x-scale)))
       (t_1 (/ a (* y-scale y-scale)))
       (t_2 (* (* 4.0 (* a b)) b)))
  (*
   (*
    (/
     (sqrt
      (*
       (*
        (*
         (/ (* t_2 (- a)) (* (* x-scale y-scale) (* x-scale y-scale)))
         2.0)
        (* (* (- a) b) (* a b)))
       (fma a t_1 (fma b t_0 (fabs (- (* a t_1) (* b t_0)))))))
     (* t_2 a))
    (* (* x-scale y-scale) x-scale))
   y-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);
	double t_1 = a / (y_45_scale * y_45_scale);
	double t_2 = (4.0 * (a * b)) * b;
	return ((sqrt((((((t_2 * -a) / ((x_45_scale * y_45_scale) * (x_45_scale * y_45_scale))) * 2.0) * ((-a * b) * (a * b))) * fma(a, t_1, fma(b, t_0, fabs(((a * t_1) - (b * t_0))))))) / (t_2 * a)) * ((x_45_scale * y_45_scale) * x_45_scale)) * y_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))
	t_1 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_2 = Float64(Float64(4.0 * Float64(a * b)) * b)
	return Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(t_2 * Float64(-a)) / Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))) * 2.0) * Float64(Float64(Float64(-a) * b) * Float64(a * b))) * fma(a, t_1, fma(b, t_0, abs(Float64(Float64(a * t_1) - Float64(b * t_0))))))) / Float64(t_2 * a)) * Float64(Float64(x_45_scale * y_45_scale) * x_45_scale)) * y_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]}, Block[{t$95$1 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, N[(N[(N[(N[Sqrt[N[(N[(N[(N[(N[(t$95$2 * (-a)), $MachinePrecision] / N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * t$95$1 + N[(b * t$95$0 + N[Abs[N[(N[(a * t$95$1), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 * a), $MachinePrecision]), $MachinePrecision] * N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision]]]]

Derivation

1. Initial program 2.6%

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

2. 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 3.7%

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

8. Applied rewrites 4.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 5.0%

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

10. Applied rewrites 5.0%

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

Alternative 13: 4.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := \left(x-scale \cdot y-scale\right) \cdot x-scale\\ t_2 := \left(4 \cdot \left(a \cdot b\right)\right) \cdot b\\ t_3 := \frac{a}{y-scale \cdot y-scale}\\ \left(\frac{\sqrt{\left(\left(\left(t\_2 \cdot \frac{-a}{t\_1 \cdot y-scale}\right) \cdot 2\right) \cdot \left(\left(\left(-a\right) \cdot b\right) \cdot \left(a \cdot b\right)\right)\right) \cdot \mathsf{fma}\left(a, t\_3, \mathsf{fma}\left(b, t\_0, \left|a \cdot t\_3 - b \cdot t\_0\right|\right)\right)}}{t\_2 \cdot a} \cdot t\_1\right) \cdot y-scale \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (/ b (* x-scale x-scale)))
       (t_1 (* (* x-scale y-scale) x-scale))
       (t_2 (* (* 4.0 (* a b)) b))
       (t_3 (/ a (* y-scale y-scale))))
  (*
   (*
    (/
     (sqrt
      (*
       (*
        (* (* t_2 (/ (- a) (* t_1 y-scale))) 2.0)
        (* (* (- a) b) (* a b)))
       (fma a t_3 (fma b t_0 (fabs (- (* a t_3) (* b t_0)))))))
     (* t_2 a))
    t_1)
   y-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);
	double t_1 = (x_45_scale * y_45_scale) * x_45_scale;
	double t_2 = (4.0 * (a * b)) * b;
	double t_3 = a / (y_45_scale * y_45_scale);
	return ((sqrt(((((t_2 * (-a / (t_1 * y_45_scale))) * 2.0) * ((-a * b) * (a * b))) * fma(a, t_3, fma(b, t_0, fabs(((a * t_3) - (b * t_0))))))) / (t_2 * a)) * t_1) * y_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))
	t_1 = Float64(Float64(x_45_scale * y_45_scale) * x_45_scale)
	t_2 = Float64(Float64(4.0 * Float64(a * b)) * b)
	t_3 = Float64(a / Float64(y_45_scale * y_45_scale))
	return Float64(Float64(Float64(sqrt(Float64(Float64(Float64(Float64(t_2 * Float64(Float64(-a) / Float64(t_1 * y_45_scale))) * 2.0) * Float64(Float64(Float64(-a) * b) * Float64(a * b))) * fma(a, t_3, fma(b, t_0, abs(Float64(Float64(a * t_3) - Float64(b * t_0))))))) / Float64(t_2 * a)) * t_1) * y_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]}, Block[{t$95$1 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$3 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[Sqrt[N[(N[(N[(N[(t$95$2 * N[((-a) / N[(t$95$1 * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * t$95$3 + N[(b * t$95$0 + N[Abs[N[(N[(a * t$95$3), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 * a), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * y$45$scale), $MachinePrecision]]]]]

Derivation

1. Initial program 2.6%

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

2. 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 3.7%

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

8. Applied rewrites 4.4%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 4.0%

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

10. Applied rewrites 4.0%

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

Alternative 14: 4.0% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.6%

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

2. 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 3.7%

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

8. Applied rewrites 4.4%

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

10. Applied rewrites 4.2%

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

Reproduce

herbie shell --seed 2025213 
(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))))