a from scale-rotated-ellipse

Percentage Accurate: 2.4% → 16.2%

Time: 39.9s

Alternatives: 9

Speedup: 6.8×

• 29.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 16.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{a}{y-scale \cdot y-scale}\\ t_1 := \left(\left(-a\right) \cdot b\_m\right) \cdot b\_m\\ t_2 := \frac{1}{{y-scale}^{2}}\\ t_3 := 0.011111111111111112 \cdot \left(angle \cdot \pi\right)\\ t_4 := \cos t\_3\\ t_5 := \frac{t\_4}{{x-scale}^{2}}\\ t_6 := 0.5 \cdot \frac{t\_4}{{y-scale}^{2}}\\ t_7 := \frac{1}{{x-scale}^{2}}\\ \mathbf{if}\;b\_m \leq 5 \cdot 10^{+83}:\\ \;\;\;\;\left(\left(\left(x-scale \cdot y-scale\right) \cdot \frac{\frac{\frac{\sqrt{\left(\mathsf{fma}\left(t\_0, a, \mathsf{fma}\left(\frac{b\_m}{x-scale \cdot x-scale}, b\_m, \left|\frac{b\_m \cdot b\_m}{x-scale \cdot x-scale} - t\_0 \cdot a\right|\right)\right) \cdot \left(\left(\left(a \cdot b\_m\right) \cdot b\_m\right) \cdot \left(-a\right)\right)\right) \cdot \left(\left(t\_1 \cdot 8\right) \cdot a\right)}}{\left|y-scale \cdot x-scale\right|}}{t\_1 \cdot -4}}{a}\right) \cdot y-scale\right) \cdot x-scale\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x-scale \cdot y-scale\right) \cdot \left(0.25 \cdot \frac{b\_m \cdot \sqrt{8 \cdot \left(\left(\sqrt{\frac{{\sin t\_3}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}} + {\left(0.5 \cdot t\_2 - \mathsf{fma}\left(0.5, t\_7, \mathsf{fma}\left(0.5, t\_5, t\_6\right)\right)\right)}^{2}} + \mathsf{fma}\left(0.5, t\_7, \mathsf{fma}\left(0.5, t\_2, 0.5 \cdot t\_5\right)\right)\right) - t\_6\right)}}{\left|x-scale \cdot y-scale\right|}\right)\right) \cdot y-scale\right) \cdot x-scale\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale)))
        (t_1 (* (* (- a) b_m) b_m))
        (t_2 (/ 1.0 (pow y-scale 2.0)))
        (t_3 (* 0.011111111111111112 (* angle PI)))
        (t_4 (cos t_3))
        (t_5 (/ t_4 (pow x-scale 2.0)))
        (t_6 (* 0.5 (/ t_4 (pow y-scale 2.0))))
        (t_7 (/ 1.0 (pow x-scale 2.0))))
   (if (<= b_m 5e+83)
     (*
      (*
       (*
        (* x-scale y-scale)
        (/
         (/
          (/
           (sqrt
            (*
             (*
              (fma
               t_0
               a
               (fma
                (/ b_m (* x-scale x-scale))
                b_m
                (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) (* t_0 a)))))
              (* (* (* a b_m) b_m) (- a)))
             (* (* t_1 8.0) a)))
           (fabs (* y-scale x-scale)))
          (* t_1 -4.0))
         a))
       y-scale)
      x-scale)
     (*
      (*
       (*
        (* x-scale y-scale)
        (*
         0.25
         (/
          (*
           b_m
           (sqrt
            (*
             8.0
             (-
              (+
               (sqrt
                (+
                 (/
                  (pow (sin t_3) 2.0)
                  (* (pow x-scale 2.0) (pow y-scale 2.0)))
                 (pow (- (* 0.5 t_2) (fma 0.5 t_7 (fma 0.5 t_5 t_6))) 2.0)))
               (fma 0.5 t_7 (fma 0.5 t_2 (* 0.5 t_5))))
              t_6))))
          (fabs (* x-scale y-scale)))))
       y-scale)
      x-scale))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * y_45_scale);
	double t_1 = (-a * b_m) * b_m;
	double t_2 = 1.0 / pow(y_45_scale, 2.0);
	double t_3 = 0.011111111111111112 * (angle * ((double) M_PI));
	double t_4 = cos(t_3);
	double t_5 = t_4 / pow(x_45_scale, 2.0);
	double t_6 = 0.5 * (t_4 / pow(y_45_scale, 2.0));
	double t_7 = 1.0 / pow(x_45_scale, 2.0);
	double tmp;
	if (b_m <= 5e+83) {
		tmp = (((x_45_scale * y_45_scale) * (((sqrt(((fma(t_0, a, fma((b_m / (x_45_scale * x_45_scale)), b_m, fabs((((b_m * b_m) / (x_45_scale * x_45_scale)) - (t_0 * a))))) * (((a * b_m) * b_m) * -a)) * ((t_1 * 8.0) * a))) / fabs((y_45_scale * x_45_scale))) / (t_1 * -4.0)) / a)) * y_45_scale) * x_45_scale;
	} else {
		tmp = (((x_45_scale * y_45_scale) * (0.25 * ((b_m * sqrt((8.0 * ((sqrt(((pow(sin(t_3), 2.0) / (pow(x_45_scale, 2.0) * pow(y_45_scale, 2.0))) + pow(((0.5 * t_2) - fma(0.5, t_7, fma(0.5, t_5, t_6))), 2.0))) + fma(0.5, t_7, fma(0.5, t_2, (0.5 * t_5)))) - t_6)))) / fabs((x_45_scale * y_45_scale))))) * y_45_scale) * x_45_scale;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(Float64(Float64(-a) * b_m) * b_m)
	t_2 = Float64(1.0 / (y_45_scale ^ 2.0))
	t_3 = Float64(0.011111111111111112 * Float64(angle * pi))
	t_4 = cos(t_3)
	t_5 = Float64(t_4 / (x_45_scale ^ 2.0))
	t_6 = Float64(0.5 * Float64(t_4 / (y_45_scale ^ 2.0)))
	t_7 = Float64(1.0 / (x_45_scale ^ 2.0))
	tmp = 0.0
	if (b_m <= 5e+83)
		tmp = Float64(Float64(Float64(Float64(x_45_scale * y_45_scale) * Float64(Float64(Float64(sqrt(Float64(Float64(fma(t_0, a, fma(Float64(b_m / Float64(x_45_scale * x_45_scale)), b_m, abs(Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale * x_45_scale)) - Float64(t_0 * a))))) * Float64(Float64(Float64(a * b_m) * b_m) * Float64(-a))) * Float64(Float64(t_1 * 8.0) * a))) / abs(Float64(y_45_scale * x_45_scale))) / Float64(t_1 * -4.0)) / a)) * y_45_scale) * x_45_scale);
	else
		tmp = Float64(Float64(Float64(Float64(x_45_scale * y_45_scale) * Float64(0.25 * Float64(Float64(b_m * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(Float64((sin(t_3) ^ 2.0) / Float64((x_45_scale ^ 2.0) * (y_45_scale ^ 2.0))) + (Float64(Float64(0.5 * t_2) - fma(0.5, t_7, fma(0.5, t_5, t_6))) ^ 2.0))) + fma(0.5, t_7, fma(0.5, t_2, Float64(0.5 * t_5)))) - t_6)))) / abs(Float64(x_45_scale * y_45_scale))))) * y_45_scale) * x_45_scale);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-a) * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(0.5 * N[(t$95$4 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 5e+83], N[(N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$0 * a + N[(N[(b$95$m / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b$95$m + N[Abs[N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * 8.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * -4.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision], N[(N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(0.25 * N[(N[(b$95$m * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(N[(N[Power[N[Sin[t$95$3], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * 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$7 + N[(0.5 * t$95$5 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(0.5 * t$95$7 + N[(0.5 * t$95$2 + N[(0.5 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$6), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 5.00000000000000029e83

   1. Initial program 2.4%

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

   3. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 7.0%

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

   10. Applied rewrites 9.9%

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

   if 5.00000000000000029e83 < b

   1. Initial program 2.4%

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

   3. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 2.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 13.1%

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

Alternative 2: 11.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{a}{y-scale \cdot y-scale}\\ t_1 := \left(-a\right) \cdot b\_m\\ t_2 := t\_1 \cdot b\_m\\ t_3 := \frac{\left(a \cdot b\_m\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{t\_1}{y-scale \cdot x-scale}\\ t_4 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_5 := \mathsf{fma}\left({a}^{2}, {\cos t\_4}^{2}, {b\_m}^{2} \cdot {\sin t\_4}^{2}\right)\\ \mathbf{if}\;b\_m \leq 4.6 \cdot 10^{+208}:\\ \;\;\;\;\left(\left(\left(x-scale \cdot y-scale\right) \cdot \frac{\frac{\frac{\sqrt{\left(\mathsf{fma}\left(t\_0, a, \mathsf{fma}\left(\frac{b\_m}{x-scale \cdot x-scale}, b\_m, \left|\frac{b\_m \cdot b\_m}{x-scale \cdot x-scale} - t\_0 \cdot a\right|\right)\right) \cdot \left(\left(\left(a \cdot b\_m\right) \cdot b\_m\right) \cdot \left(-a\right)\right)\right) \cdot \left(\left(t\_2 \cdot 8\right) \cdot a\right)}}{\left|y-scale \cdot x-scale\right|}}{t\_2 \cdot -4}}{a}\right) \cdot y-scale\right) \cdot x-scale\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot \left(\left(b\_m \cdot a\right) \cdot \left(b\_m \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\sqrt{{t\_5}^{2}} + t\_5}{{y-scale}^{2}}}}{t\_3}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale)))
        (t_1 (* (- a) b_m))
        (t_2 (* t_1 b_m))
        (t_3
         (*
          (/ (* (* a b_m) 4.0) (* y-scale x-scale))
          (/ t_1 (* y-scale x-scale))))
        (t_4 (* 0.005555555555555556 (* angle PI)))
        (t_5
         (fma
          (pow a 2.0)
          (pow (cos t_4) 2.0)
          (* (pow b_m 2.0) (pow (sin t_4) 2.0)))))
   (if (<= b_m 4.6e+208)
     (*
      (*
       (*
        (* x-scale y-scale)
        (/
         (/
          (/
           (sqrt
            (*
             (*
              (fma
               t_0
               a
               (fma
                (/ b_m (* x-scale x-scale))
                b_m
                (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) (* t_0 a)))))
              (* (* (* a b_m) b_m) (- a)))
             (* (* t_2 8.0) a)))
           (fabs (* y-scale x-scale)))
          (* t_2 -4.0))
         a))
       y-scale)
      x-scale)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_3) (* (* b_m a) (* b_m (- a))))
         (/ (+ (sqrt (pow t_5 2.0)) t_5) (pow y-scale 2.0)))))
      t_3))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * y_45_scale);
	double t_1 = -a * b_m;
	double t_2 = t_1 * b_m;
	double t_3 = (((a * b_m) * 4.0) / (y_45_scale * x_45_scale)) * (t_1 / (y_45_scale * x_45_scale));
	double t_4 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_5 = fma(pow(a, 2.0), pow(cos(t_4), 2.0), (pow(b_m, 2.0) * pow(sin(t_4), 2.0)));
	double tmp;
	if (b_m <= 4.6e+208) {
		tmp = (((x_45_scale * y_45_scale) * (((sqrt(((fma(t_0, a, fma((b_m / (x_45_scale * x_45_scale)), b_m, fabs((((b_m * b_m) / (x_45_scale * x_45_scale)) - (t_0 * a))))) * (((a * b_m) * b_m) * -a)) * ((t_2 * 8.0) * a))) / fabs((y_45_scale * x_45_scale))) / (t_2 * -4.0)) / a)) * y_45_scale) * x_45_scale;
	} else {
		tmp = -sqrt((((2.0 * t_3) * ((b_m * a) * (b_m * -a))) * ((sqrt(pow(t_5, 2.0)) + t_5) / pow(y_45_scale, 2.0)))) / t_3;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(Float64(-a) * b_m)
	t_2 = Float64(t_1 * b_m)
	t_3 = Float64(Float64(Float64(Float64(a * b_m) * 4.0) / Float64(y_45_scale * x_45_scale)) * Float64(t_1 / Float64(y_45_scale * x_45_scale)))
	t_4 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_5 = fma((a ^ 2.0), (cos(t_4) ^ 2.0), Float64((b_m ^ 2.0) * (sin(t_4) ^ 2.0)))
	tmp = 0.0
	if (b_m <= 4.6e+208)
		tmp = Float64(Float64(Float64(Float64(x_45_scale * y_45_scale) * Float64(Float64(Float64(sqrt(Float64(Float64(fma(t_0, a, fma(Float64(b_m / Float64(x_45_scale * x_45_scale)), b_m, abs(Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale * x_45_scale)) - Float64(t_0 * a))))) * Float64(Float64(Float64(a * b_m) * b_m) * Float64(-a))) * Float64(Float64(t_2 * 8.0) * a))) / abs(Float64(y_45_scale * x_45_scale))) / Float64(t_2 * -4.0)) / a)) * y_45_scale) * x_45_scale);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * Float64(Float64(b_m * a) * Float64(b_m * Float64(-a)))) * Float64(Float64(sqrt((t_5 ^ 2.0)) + t_5) / (y_45_scale ^ 2.0))))) / t_3);
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) * b$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * b$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(a * b$95$m), $MachinePrecision] * 4.0), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[N[Cos[t$95$4], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Power[b$95$m, 2.0], $MachinePrecision] * N[Power[N[Sin[t$95$4], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 4.6e+208], N[(N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$0 * a + N[(N[(b$95$m / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b$95$m + N[Abs[N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * 8.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * -4.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$3), $MachinePrecision] * N[(N[(b$95$m * a), $MachinePrecision] * N[(b$95$m * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[Power[t$95$5, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.60000000000000001e208

   1. Initial program 2.4%

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

   3. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 7.0%

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

   10. Applied rewrites 9.9%

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

   if 4.60000000000000001e208 < b

   1. Initial program 2.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 3.1

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 3.1

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 3.1

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

   3. Applied rewrites 3.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 4.6

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 4.6

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lower-*.f64 4.6

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

   5. Applied rewrites 4.6%

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

   6. Taylor expanded in y-scale around 0

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

   8. Applied rewrites 6.3%

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

Alternative 3: 11.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 4 \cdot \left(b\_m \cdot a\right)\\ t_1 := \frac{a}{y-scale \cdot y-scale}\\ t_2 := \left(\left(-a\right) \cdot b\_m\right) \cdot b\_m\\ t_3 := \frac{b\_m}{x-scale \cdot x-scale}\\ t_4 := b\_m \cdot \left(-a\right)\\ \mathbf{if}\;b\_m \leq 2.7 \cdot 10^{+119}:\\ \;\;\;\;\left(\left(\left(x-scale \cdot y-scale\right) \cdot \frac{\frac{\frac{\sqrt{\left(\mathsf{fma}\left(t\_1, a, \mathsf{fma}\left(t\_3, b\_m, \left|\frac{b\_m \cdot b\_m}{x-scale \cdot x-scale} - t\_1 \cdot a\right|\right)\right) \cdot \left(\left(\left(a \cdot b\_m\right) \cdot b\_m\right) \cdot \left(-a\right)\right)\right) \cdot \left(\left(t\_2 \cdot 8\right) \cdot a\right)}}{\left|y-scale \cdot x-scale\right|}}{t\_2 \cdot -4}}{a}\right) \cdot y-scale\right) \cdot x-scale\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\mathsf{fma}\left(a, t\_1, \mathsf{fma}\left(b\_m, t\_3, \left|a \cdot t\_1 - b\_m \cdot t\_3\right|\right)\right) \cdot \left(\left(\left(\frac{t\_4}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot t\_0\right) \cdot 2\right) \cdot \left(\left(\left(b\_m \cdot a\right) \cdot b\_m\right) \cdot \left(-a\right)\right)\right)}}{t\_0}}{t\_4} \cdot \left(\left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 4.0 (* b_m a)))
        (t_1 (/ a (* y-scale y-scale)))
        (t_2 (* (* (- a) b_m) b_m))
        (t_3 (/ b_m (* x-scale x-scale)))
        (t_4 (* b_m (- a))))
   (if (<= b_m 2.7e+119)
     (*
      (*
       (*
        (* x-scale y-scale)
        (/
         (/
          (/
           (sqrt
            (*
             (*
              (fma
               t_1
               a
               (fma
                t_3
                b_m
                (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) (* t_1 a)))))
              (* (* (* a b_m) b_m) (- a)))
             (* (* t_2 8.0) a)))
           (fabs (* y-scale x-scale)))
          (* t_2 -4.0))
         a))
       y-scale)
      x-scale)
     (*
      (/
       (/
        (-
         (sqrt
          (*
           (fma a t_1 (fma b_m t_3 (fabs (- (* a t_1) (* b_m t_3)))))
           (*
            (* (* (/ t_4 (* (* (* y-scale x-scale) x-scale) y-scale)) t_0) 2.0)
            (* (* (* b_m a) b_m) (- a))))))
        t_0)
       t_4)
      (* (* (* x-scale y-scale) x-scale) y-scale)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 4.0 * (b_m * a);
	double t_1 = a / (y_45_scale * y_45_scale);
	double t_2 = (-a * b_m) * b_m;
	double t_3 = b_m / (x_45_scale * x_45_scale);
	double t_4 = b_m * -a;
	double tmp;
	if (b_m <= 2.7e+119) {
		tmp = (((x_45_scale * y_45_scale) * (((sqrt(((fma(t_1, a, fma(t_3, b_m, fabs((((b_m * b_m) / (x_45_scale * x_45_scale)) - (t_1 * a))))) * (((a * b_m) * b_m) * -a)) * ((t_2 * 8.0) * a))) / fabs((y_45_scale * x_45_scale))) / (t_2 * -4.0)) / a)) * y_45_scale) * x_45_scale;
	} else {
		tmp = ((-sqrt((fma(a, t_1, fma(b_m, t_3, fabs(((a * t_1) - (b_m * t_3))))) * ((((t_4 / (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * t_0) * 2.0) * (((b_m * a) * b_m) * -a)))) / t_0) / t_4) * (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(4.0 * Float64(b_m * a))
	t_1 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_2 = Float64(Float64(Float64(-a) * b_m) * b_m)
	t_3 = Float64(b_m / Float64(x_45_scale * x_45_scale))
	t_4 = Float64(b_m * Float64(-a))
	tmp = 0.0
	if (b_m <= 2.7e+119)
		tmp = Float64(Float64(Float64(Float64(x_45_scale * y_45_scale) * Float64(Float64(Float64(sqrt(Float64(Float64(fma(t_1, a, fma(t_3, b_m, abs(Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale * x_45_scale)) - Float64(t_1 * a))))) * Float64(Float64(Float64(a * b_m) * b_m) * Float64(-a))) * Float64(Float64(t_2 * 8.0) * a))) / abs(Float64(y_45_scale * x_45_scale))) / Float64(t_2 * -4.0)) / a)) * y_45_scale) * x_45_scale);
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(fma(a, t_1, fma(b_m, t_3, abs(Float64(Float64(a * t_1) - Float64(b_m * t_3))))) * Float64(Float64(Float64(Float64(t_4 / Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)) * t_0) * 2.0) * Float64(Float64(Float64(b_m * a) * b_m) * Float64(-a)))))) / t_0) / t_4) * Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(4.0 * N[(b$95$m * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[((-a) * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(b$95$m / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(b$95$m * (-a)), $MachinePrecision]}, If[LessEqual[b$95$m, 2.7e+119], N[(N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$1 * a + N[(t$95$3 * b$95$m + N[Abs[N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * 8.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * -4.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(a * t$95$1 + N[(b$95$m * t$95$3 + N[Abs[N[(N[(a * t$95$1), $MachinePrecision] - N[(b$95$m * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$4 / N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(b$95$m * a), $MachinePrecision] * b$95$m), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision] / t$95$4), $MachinePrecision] * N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6999999999999998e119

   1. Initial program 2.4%

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

   3. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 7.0%

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

   10. Applied rewrites 9.9%

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

   if 2.6999999999999998e119 < b

   1. Initial program 2.4%

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

   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 4.0%

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

   6. Applied rewrites 3.8%

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

   8. Applied rewrites 7.5%

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

Alternative 4: 10.5% accurate, 6.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{a}{y-scale \cdot y-scale}\\ t_1 := \left(\left(-a\right) \cdot b\_m\right) \cdot b\_m\\ t_2 := \frac{b\_m}{x-scale \cdot x-scale}\\ t_3 := b\_m \cdot \left(-a\right)\\ t_4 := \left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\\ \mathbf{if}\;b\_m \leq 3.7 \cdot 10^{+171}:\\ \;\;\;\;\left(\left(\left(x-scale \cdot y-scale\right) \cdot \frac{\frac{\frac{\sqrt{\left(\mathsf{fma}\left(t\_0, a, \mathsf{fma}\left(t\_2, b\_m, \left|\frac{b\_m \cdot b\_m}{x-scale \cdot x-scale} - t\_0 \cdot a\right|\right)\right) \cdot \left(\left(\left(a \cdot b\_m\right) \cdot b\_m\right) \cdot \left(-a\right)\right)\right) \cdot \left(\left(t\_1 \cdot 8\right) \cdot a\right)}}{\left|y-scale \cdot x-scale\right|}}{t\_1 \cdot -4}}{a}\right) \cdot y-scale\right) \cdot x-scale\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(a, t\_0, \mathsf{fma}\left(b\_m, t\_2, \left|a \cdot t\_0 - b\_m \cdot t\_2\right|\right)\right) \cdot \left(\left(\left(\frac{t\_3}{t\_4} \cdot \left(4 \cdot \left(b\_m \cdot a\right)\right)\right) \cdot 2\right) \cdot \left(\left(\left(b\_m \cdot a\right) \cdot b\_m\right) \cdot \left(-a\right)\right)\right)}}{b\_m \cdot a} \cdot \frac{t\_4}{4 \cdot t\_3}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale)))
        (t_1 (* (* (- a) b_m) b_m))
        (t_2 (/ b_m (* x-scale x-scale)))
        (t_3 (* b_m (- a)))
        (t_4 (* (* (* y-scale x-scale) x-scale) y-scale)))
   (if (<= b_m 3.7e+171)
     (*
      (*
       (*
        (* x-scale y-scale)
        (/
         (/
          (/
           (sqrt
            (*
             (*
              (fma
               t_0
               a
               (fma
                t_2
                b_m
                (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) (* t_0 a)))))
              (* (* (* a b_m) b_m) (- a)))
             (* (* t_1 8.0) a)))
           (fabs (* y-scale x-scale)))
          (* t_1 -4.0))
         a))
       y-scale)
      x-scale)
     (*
      (/
       (-
        (sqrt
         (*
          (fma a t_0 (fma b_m t_2 (fabs (- (* a t_0) (* b_m t_2)))))
          (*
           (* (* (/ t_3 t_4) (* 4.0 (* b_m a))) 2.0)
           (* (* (* b_m a) b_m) (- a))))))
       (* b_m a))
      (/ t_4 (* 4.0 t_3))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * y_45_scale);
	double t_1 = (-a * b_m) * b_m;
	double t_2 = b_m / (x_45_scale * x_45_scale);
	double t_3 = b_m * -a;
	double t_4 = ((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale;
	double tmp;
	if (b_m <= 3.7e+171) {
		tmp = (((x_45_scale * y_45_scale) * (((sqrt(((fma(t_0, a, fma(t_2, b_m, fabs((((b_m * b_m) / (x_45_scale * x_45_scale)) - (t_0 * a))))) * (((a * b_m) * b_m) * -a)) * ((t_1 * 8.0) * a))) / fabs((y_45_scale * x_45_scale))) / (t_1 * -4.0)) / a)) * y_45_scale) * x_45_scale;
	} else {
		tmp = (-sqrt((fma(a, t_0, fma(b_m, t_2, fabs(((a * t_0) - (b_m * t_2))))) * ((((t_3 / t_4) * (4.0 * (b_m * a))) * 2.0) * (((b_m * a) * b_m) * -a)))) / (b_m * a)) * (t_4 / (4.0 * t_3));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(Float64(Float64(-a) * b_m) * b_m)
	t_2 = Float64(b_m / Float64(x_45_scale * x_45_scale))
	t_3 = Float64(b_m * Float64(-a))
	t_4 = Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)
	tmp = 0.0
	if (b_m <= 3.7e+171)
		tmp = Float64(Float64(Float64(Float64(x_45_scale * y_45_scale) * Float64(Float64(Float64(sqrt(Float64(Float64(fma(t_0, a, fma(t_2, b_m, abs(Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale * x_45_scale)) - Float64(t_0 * a))))) * Float64(Float64(Float64(a * b_m) * b_m) * Float64(-a))) * Float64(Float64(t_1 * 8.0) * a))) / abs(Float64(y_45_scale * x_45_scale))) / Float64(t_1 * -4.0)) / a)) * y_45_scale) * x_45_scale);
	else
		tmp = Float64(Float64(Float64(-sqrt(Float64(fma(a, t_0, fma(b_m, t_2, abs(Float64(Float64(a * t_0) - Float64(b_m * t_2))))) * Float64(Float64(Float64(Float64(t_3 / t_4) * Float64(4.0 * Float64(b_m * a))) * 2.0) * Float64(Float64(Float64(b_m * a) * b_m) * Float64(-a)))))) / Float64(b_m * a)) * Float64(t_4 / Float64(4.0 * t_3)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-a) * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]}, Block[{t$95$2 = N[(b$95$m / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b$95$m * (-a)), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, If[LessEqual[b$95$m, 3.7e+171], N[(N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(N[(N[(t$95$0 * a + N[(t$95$2 * b$95$m + N[Abs[N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(a * b$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * 8.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * -4.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision], N[(N[((-N[Sqrt[N[(N[(a * t$95$0 + N[(b$95$m * t$95$2 + N[Abs[N[(N[(a * t$95$0), $MachinePrecision] - N[(b$95$m * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$3 / t$95$4), $MachinePrecision] * N[(4.0 * N[(b$95$m * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(N[(b$95$m * a), $MachinePrecision] * b$95$m), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(b$95$m * a), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 / N[(4.0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 3.69999999999999998e171

   1. Initial program 2.4%

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

   3. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in angle around 0

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

   8. Applied rewrites 7.0%

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

   10. Applied rewrites 9.9%

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

   if 3.69999999999999998e171 < b

   1. Initial program 2.4%

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

   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 4.0%

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

   6. Applied rewrites 3.8%

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

   8. Applied rewrites 6.7%

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

Alternative 5: 9.9% accurate, 6.8× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale))) (t_1 (* (* (- a) b_m) b_m)))
   (*
    (*
     (*
      (* x-scale y-scale)
      (/
       (/
        (/
         (sqrt
          (*
           (*
            (fma
             t_0
             a
             (fma
              (/ b_m (* x-scale x-scale))
              b_m
              (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) (* t_0 a)))))
            (* (* (* a b_m) b_m) (- a)))
           (* (* t_1 8.0) a)))
         (fabs (* y-scale x-scale)))
        (* t_1 -4.0))
       a))
     y-scale)
    x-scale)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.4%

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

3. Applied rewrites 3.2%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

5. Applied rewrites 7.0%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 7.0%

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

10. Applied rewrites 9.9%

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

Alternative 6: 7.7% accurate, 6.8× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (- a) b_m) b_m)) (t_1 (/ a (* y-scale y-scale))))
   (*
    (*
     (*
      (* x-scale y-scale)
      (/
       (/
        (sqrt
         (*
          (* (* t_0 8.0) a)
          (*
           (fma
            t_1
            a
            (fma
             (/ b_m (* x-scale x-scale))
             b_m
             (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) (* t_1 a)))))
           (* (* (* a b_m) b_m) (- a)))))
        (fabs (* x-scale y-scale)))
       (* (* -4.0 t_0) a)))
     y-scale)
    x-scale)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.4%

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

3. Applied rewrites 3.2%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

5. Applied rewrites 7.0%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 7.0%

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

10. Applied rewrites 7.7%

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

Alternative 7: 6.4% accurate, 6.8× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale))) (t_1 (* (* (- a) b_m) b_m)))
   (*
    (*
     (*
      x-scale
      (*
       y-scale
       (/
        (sqrt
         (*
          (*
           (fma
            t_0
            a
            (fma
             (/ b_m (* x-scale x-scale))
             b_m
             (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) (* t_0 a)))))
           (* (* (* a b_m) b_m) (- a)))
          (* (* t_1 8.0) a)))
        (* (fabs (* y-scale x-scale)) (* (* t_1 -4.0) a)))))
     y-scale)
    x-scale)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.4%

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

3. Applied rewrites 3.2%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

5. Applied rewrites 7.0%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 7.0%

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

10. Applied rewrites 6.4%

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

Alternative 8: 6.3% accurate, 6.8× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (- a) b_m) b_m)) (t_1 (/ a (* y-scale y-scale))))
   (*
    (*
     (*
      (/
       (sqrt
        (*
         (*
          (fma
           t_1
           a
           (fma
            (/ b_m (* x-scale x-scale))
            b_m
            (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) (* t_1 a)))))
          (* (* (* a b_m) b_m) (- a)))
         (* (* t_0 8.0) a)))
       (* (fabs (* y-scale x-scale)) (* (* t_0 -4.0) a)))
      (* y-scale x-scale))
     y-scale)
    x-scale)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.4%

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

3. Applied rewrites 3.2%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

5. Applied rewrites 7.0%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 7.0%

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

10. Applied rewrites 6.3%

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

Alternative 9: 6.3% accurate, 6.8× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (- a) b_m) b_m)) (t_1 (/ a (* y-scale y-scale))))
   (*
    (*
     (/
      (sqrt
       (*
        (*
         (fma
          t_1
          a
          (fma
           (/ b_m (* x-scale x-scale))
           b_m
           (fabs (- (/ (* b_m b_m) (* x-scale x-scale)) (* t_1 a)))))
         (* (* (* a b_m) b_m) (- a)))
        (* (* t_0 8.0) a)))
      (* (fabs (* y-scale x-scale)) (* (* t_0 -4.0) a)))
     (* y-scale x-scale))
    (* y-scale x-scale))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.4%

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

3. Applied rewrites 3.2%

   \[\leadsto \color{blue}{\left(\frac{\sqrt{\frac{\left(8 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)\right) \cdot \left(\left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right) \cdot \left(\mathsf{hypot}\left(\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} - \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{\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}\right) + \left(\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} + \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)\right)\right)}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale}}}{-4 \cdot \left(\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\right)} \cdot \left(y-scale \cdot x-scale\right)\right) \cdot \left(y-scale \cdot x-scale\right)} \]

5. Applied rewrites 7.0%

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

6. Taylor expanded in angle around 0

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

8. Applied rewrites 7.0%

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

10. Applied rewrites 6.3%

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

Reproduce

herbie shell --seed 2025149 
(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))))