a from scale-rotated-ellipse

Percentage Accurate: 2.6% → 16.7%

Time: 32.9s

Alternatives: 7

Speedup: 12.3×

• 69.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (/ angle 180) PI))
       (t_1 (sin t_0))
       (t_2 (cos t_0))
       (t_3
        (/
         (/ (+ (pow (* a t_2) 2) (pow (* b t_1) 2)) y-scale)
         y-scale))
       (t_4
        (/
         (/ (+ (pow (* a t_1) 2) (pow (* b t_2) 2)) x-scale)
         x-scale))
       (t_5 (* (* b a) (* b (- a))))
       (t_6 (/ (* 4 t_5) (pow (* x-scale y-scale) 2))))
  (/
   (-
    (sqrt
     (*
      (* (* 2 t_6) t_5)
      (+
       (+ t_4 t_3)
       (sqrt
        (+
         (pow (- t_4 t_3) 2)
         (pow
          (/
           (/ (* (* (* 2 (- (pow b 2) (pow a 2))) t_1) t_2) x-scale)
           y-scale)
          2)))))))
   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), $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], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2], $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], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2], $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 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2 * 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], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2 * N[(N[Power[b, 2], $MachinePrecision] - N[Power[a, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Initial Program: 2.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (/ angle 180) PI))
       (t_1 (sin t_0))
       (t_2 (cos t_0))
       (t_3
        (/
         (/ (+ (pow (* a t_2) 2) (pow (* b t_1) 2)) y-scale)
         y-scale))
       (t_4
        (/
         (/ (+ (pow (* a t_1) 2) (pow (* b t_2) 2)) x-scale)
         x-scale))
       (t_5 (* (* b a) (* b (- a))))
       (t_6 (/ (* 4 t_5) (pow (* x-scale y-scale) 2))))
  (/
   (-
    (sqrt
     (*
      (* (* 2 t_6) t_5)
      (+
       (+ t_4 t_3)
       (sqrt
        (+
         (pow (- t_4 t_3) 2)
         (pow
          (/
           (/ (* (* (* 2 (- (pow b 2) (pow a 2))) t_1) t_2) x-scale)
           y-scale)
          2)))))))
   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), $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], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2], $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], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2], $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 * t$95$5), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2 * 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], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2 * N[(N[Power[b, 2], $MachinePrecision] - N[Power[a, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision]]]]]]]]

Alternative 1: 16.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|b\right| \cdot a\right) \cdot \left(\left|b\right| \cdot \left(-a\right)\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ t_2 := \sin t\_1\\ t_3 := \frac{\left(a \cdot \left|b\right|\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot \left|b\right|}{y-scale \cdot x-scale}\\ t_4 := \cos t\_1\\ t_5 := \frac{\frac{{\left(a \cdot t\_4\right)}^{2} + {\left(\left|b\right| \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_6 := \left(\pi \cdot angle\right) \cdot \frac{1}{180}\\ t_7 := \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(\left|b\right| \cdot \left|b\right|\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(t\_6 \cdot 2\right)\right) \cdot a\right) \cdot a\\ t_8 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ t_9 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(\left|b\right| \cdot t\_4\right)}^{2}}{x-scale}}{x-scale}\\ \mathbf{if}\;\frac{-\sqrt{\left(\left(2 \cdot t\_8\right) \cdot t\_0\right) \cdot \left(\left(t\_9 + t\_5\right) + \sqrt{{\left(t\_9 - t\_5\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_4}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_8} \leq \infty:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot t\_0\right) \cdot \frac{\frac{\left|t\_7\right| + t\_7}{y-scale}}{y-scale}}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{4} \cdot \left(\frac{\left|b\right| \cdot \left(x-scale \cdot x-scale\right)}{a} \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(\frac{1}{2} - \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{1}{2}\right) + \sqrt{{\sin t\_6}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{a}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* (fabs b) a) (* (fabs b) (- a))))
       (t_1 (* (/ angle 180) PI))
       (t_2 (sin t_1))
       (t_3
        (*
         (/ (* (* a (fabs b)) 4) (* y-scale x-scale))
         (/ (* (- a) (fabs b)) (* y-scale x-scale))))
       (t_4 (cos t_1))
       (t_5
        (/
         (/ (+ (pow (* a t_4) 2) (pow (* (fabs b) t_2) 2)) y-scale)
         y-scale))
       (t_6 (* (* PI angle) 1/180))
       (t_7
        (+
         (*
          (* 1/32400 (* (pow angle 2) (pow PI 2)))
          (* (fabs b) (fabs b)))
         (* (* (- 1/2 (* -1/2 (cos (* t_6 2)))) a) a)))
       (t_8 (/ (* 4 t_0) (pow (* x-scale y-scale) 2)))
       (t_9
        (/
         (/ (+ (pow (* a t_2) 2) (pow (* (fabs b) t_4) 2)) x-scale)
         x-scale)))
  (if (<=
       (/
        (-
         (sqrt
          (*
           (* (* 2 t_8) t_0)
           (+
            (+ t_9 t_5)
            (sqrt
             (+
              (pow (- t_9 t_5) 2)
              (pow
               (/
                (/
                 (* (* (* 2 (- (pow (fabs b) 2) (pow a 2))) t_2) t_4)
                 x-scale)
                y-scale)
               2)))))))
        t_8)
       INFINITY)
    (/
     (-
      (sqrt
       (*
        (* (* 2 t_3) t_0)
        (/ (/ (+ (fabs t_7) t_7) y-scale) y-scale))))
     t_3)
    (*
     1/4
     (*
      (/ (* (fabs b) (* x-scale x-scale)) a)
      (/
       (/
        (sqrt
         (*
          8
          (*
           (+
            (- 1/2 (* (cos (* (* PI angle) 1/90)) 1/2))
            (sqrt (pow (sin t_6) 4)))
           (pow a 4))))
        (fabs x-scale))
       a))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (fabs(b) * a) * (fabs(b) * -a);
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double t_2 = sin(t_1);
	double t_3 = (((a * fabs(b)) * 4.0) / (y_45_scale * x_45_scale)) * ((-a * fabs(b)) / (y_45_scale * x_45_scale));
	double t_4 = cos(t_1);
	double t_5 = ((pow((a * t_4), 2.0) + pow((fabs(b) * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_6 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_7 = ((3.08641975308642e-5 * (pow(angle, 2.0) * pow(((double) M_PI), 2.0))) * (fabs(b) * fabs(b))) + (((0.5 - (-0.5 * cos((t_6 * 2.0)))) * a) * a);
	double t_8 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double t_9 = ((pow((a * t_2), 2.0) + pow((fabs(b) * t_4), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((-sqrt((((2.0 * t_8) * t_0) * ((t_9 + t_5) + sqrt((pow((t_9 - t_5), 2.0) + pow((((((2.0 * (pow(fabs(b), 2.0) - pow(a, 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale), 2.0)))))) / t_8) <= ((double) INFINITY)) {
		tmp = -sqrt((((2.0 * t_3) * t_0) * (((fabs(t_7) + t_7) / y_45_scale) / y_45_scale))) / t_3;
	} else {
		tmp = 0.25 * (((fabs(b) * (x_45_scale * x_45_scale)) / a) * ((sqrt((8.0 * (((0.5 - (cos(((((double) M_PI) * angle) * 0.011111111111111112)) * 0.5)) + sqrt(pow(sin(t_6), 4.0))) * pow(a, 4.0)))) / fabs(x_45_scale)) / a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.abs(b) * a) * (Math.abs(b) * -a);
	double t_1 = (angle / 180.0) * Math.PI;
	double t_2 = Math.sin(t_1);
	double t_3 = (((a * Math.abs(b)) * 4.0) / (y_45_scale * x_45_scale)) * ((-a * Math.abs(b)) / (y_45_scale * x_45_scale));
	double t_4 = Math.cos(t_1);
	double t_5 = ((Math.pow((a * t_4), 2.0) + Math.pow((Math.abs(b) * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_6 = (Math.PI * angle) * 0.005555555555555556;
	double t_7 = ((3.08641975308642e-5 * (Math.pow(angle, 2.0) * Math.pow(Math.PI, 2.0))) * (Math.abs(b) * Math.abs(b))) + (((0.5 - (-0.5 * Math.cos((t_6 * 2.0)))) * a) * a);
	double t_8 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 2.0);
	double t_9 = ((Math.pow((a * t_2), 2.0) + Math.pow((Math.abs(b) * t_4), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((-Math.sqrt((((2.0 * t_8) * t_0) * ((t_9 + t_5) + Math.sqrt((Math.pow((t_9 - t_5), 2.0) + Math.pow((((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(a, 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale), 2.0)))))) / t_8) <= Double.POSITIVE_INFINITY) {
		tmp = -Math.sqrt((((2.0 * t_3) * t_0) * (((Math.abs(t_7) + t_7) / y_45_scale) / y_45_scale))) / t_3;
	} else {
		tmp = 0.25 * (((Math.abs(b) * (x_45_scale * x_45_scale)) / a) * ((Math.sqrt((8.0 * (((0.5 - (Math.cos(((Math.PI * angle) * 0.011111111111111112)) * 0.5)) + Math.sqrt(Math.pow(Math.sin(t_6), 4.0))) * Math.pow(a, 4.0)))) / Math.abs(x_45_scale)) / a));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (math.fabs(b) * a) * (math.fabs(b) * -a)
	t_1 = (angle / 180.0) * math.pi
	t_2 = math.sin(t_1)
	t_3 = (((a * math.fabs(b)) * 4.0) / (y_45_scale * x_45_scale)) * ((-a * math.fabs(b)) / (y_45_scale * x_45_scale))
	t_4 = math.cos(t_1)
	t_5 = ((math.pow((a * t_4), 2.0) + math.pow((math.fabs(b) * t_2), 2.0)) / y_45_scale) / y_45_scale
	t_6 = (math.pi * angle) * 0.005555555555555556
	t_7 = ((3.08641975308642e-5 * (math.pow(angle, 2.0) * math.pow(math.pi, 2.0))) * (math.fabs(b) * math.fabs(b))) + (((0.5 - (-0.5 * math.cos((t_6 * 2.0)))) * a) * a)
	t_8 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 2.0)
	t_9 = ((math.pow((a * t_2), 2.0) + math.pow((math.fabs(b) * t_4), 2.0)) / x_45_scale) / x_45_scale
	tmp = 0
	if (-math.sqrt((((2.0 * t_8) * t_0) * ((t_9 + t_5) + math.sqrt((math.pow((t_9 - t_5), 2.0) + math.pow((((((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(a, 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale), 2.0)))))) / t_8) <= math.inf:
		tmp = -math.sqrt((((2.0 * t_3) * t_0) * (((math.fabs(t_7) + t_7) / y_45_scale) / y_45_scale))) / t_3
	else:
		tmp = 0.25 * (((math.fabs(b) * (x_45_scale * x_45_scale)) / a) * ((math.sqrt((8.0 * (((0.5 - (math.cos(((math.pi * angle) * 0.011111111111111112)) * 0.5)) + math.sqrt(math.pow(math.sin(t_6), 4.0))) * math.pow(a, 4.0)))) / math.fabs(x_45_scale)) / a))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(abs(b) * a) * Float64(abs(b) * Float64(-a)))
	t_1 = Float64(Float64(angle / 180.0) * pi)
	t_2 = sin(t_1)
	t_3 = Float64(Float64(Float64(Float64(a * abs(b)) * 4.0) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(Float64(-a) * abs(b)) / Float64(y_45_scale * x_45_scale)))
	t_4 = cos(t_1)
	t_5 = Float64(Float64(Float64((Float64(a * t_4) ^ 2.0) + (Float64(abs(b) * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_6 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_7 = Float64(Float64(Float64(3.08641975308642e-5 * Float64((angle ^ 2.0) * (pi ^ 2.0))) * Float64(abs(b) * abs(b))) + Float64(Float64(Float64(0.5 - Float64(-0.5 * cos(Float64(t_6 * 2.0)))) * a) * a))
	t_8 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	t_9 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(abs(b) * t_4) ^ 2.0)) / x_45_scale) / x_45_scale)
	tmp = 0.0
	if (Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_8) * t_0) * Float64(Float64(t_9 + t_5) + sqrt(Float64((Float64(t_9 - t_5) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (a ^ 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_8) <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * t_0) * Float64(Float64(Float64(abs(t_7) + t_7) / y_45_scale) / y_45_scale)))) / t_3);
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(abs(b) * Float64(x_45_scale * x_45_scale)) / a) * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) * 0.5)) + sqrt((sin(t_6) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale)) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (abs(b) * a) * (abs(b) * -a);
	t_1 = (angle / 180.0) * pi;
	t_2 = sin(t_1);
	t_3 = (((a * abs(b)) * 4.0) / (y_45_scale * x_45_scale)) * ((-a * abs(b)) / (y_45_scale * x_45_scale));
	t_4 = cos(t_1);
	t_5 = ((((a * t_4) ^ 2.0) + ((abs(b) * t_2) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_6 = (pi * angle) * 0.005555555555555556;
	t_7 = ((3.08641975308642e-5 * ((angle ^ 2.0) * (pi ^ 2.0))) * (abs(b) * abs(b))) + (((0.5 - (-0.5 * cos((t_6 * 2.0)))) * a) * a);
	t_8 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 2.0);
	t_9 = ((((a * t_2) ^ 2.0) + ((abs(b) * t_4) ^ 2.0)) / x_45_scale) / x_45_scale;
	tmp = 0.0;
	if ((-sqrt((((2.0 * t_8) * t_0) * ((t_9 + t_5) + sqrt((((t_9 - t_5) ^ 2.0) + ((((((2.0 * ((abs(b) ^ 2.0) - (a ^ 2.0))) * t_2) * t_4) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_8) <= Inf)
		tmp = -sqrt((((2.0 * t_3) * t_0) * (((abs(t_7) + t_7) / y_45_scale) / y_45_scale))) / t_3;
	else
		tmp = 0.25 * (((abs(b) * (x_45_scale * x_45_scale)) / a) * ((sqrt((8.0 * (((0.5 - (cos(((pi * angle) * 0.011111111111111112)) * 0.5)) + sqrt((sin(t_6) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale)) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision] * 4), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[((-a) * N[Abs[b], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(a * t$95$4), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(N[Abs[b], $MachinePrecision] * t$95$2), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$6 = N[(N[(Pi * angle), $MachinePrecision] * 1/180), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(1/32400 * N[(N[Power[angle, 2], $MachinePrecision] * N[Power[Pi, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1/2 - N[(-1/2 * N[Cos[N[(t$95$6 * 2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[(4 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(N[Abs[b], $MachinePrecision] * t$95$4), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[N[((-N[Sqrt[N[(N[(N[(2 * t$95$8), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(t$95$9 + t$95$5), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$9 - t$95$5), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[a, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$4), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$8), $MachinePrecision], Infinity], N[((-N[Sqrt[N[(N[(N[(2 * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[Abs[t$95$7], $MachinePrecision] + t$95$7), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], N[(1/4 * N[(N[(N[(N[Abs[b], $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(N[(N[Sqrt[N[(8 * N[(N[(N[(1/2 - N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[t$95$6], $MachinePrecision], 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[a, 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64)))) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (+.f64 (+.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64)))))))) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64)))) < +inf.0

   1. Initial program 2.6%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 4.2%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 4.2%

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 4.2%

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

   6. Applied rewrites 4.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 6.4%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 6.4%

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 6.4%

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

   8. Applied rewrites 6.4%

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

   10. Applied rewrites 8.4%

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

   11. Taylor expanded in angle around 0

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{2} - \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{2} - \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{2} - \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       3. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{2} - \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       4. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{2} - \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       5. lower-PI.f64 8.2%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{2} - \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   13. Applied rewrites 8.2%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{2} - \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   14. Taylor expanded in angle around 0

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
   15. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       3. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       4. lower-pow.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

       5. lower-PI.f64 9.0%

          \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   16. Applied rewrites 9.0%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(\left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right) \cdot 2\right)\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64)))) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (+.f64 (+.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64)))))))) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64))))

   1. Initial program 2.6%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 8.3%

      \[\leadsto \frac{1}{4} \cdot \left(\frac{b \cdot \left(x-scale \cdot x-scale\right)}{a} \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(\frac{1}{2} - \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{1}{2}\right) + \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{\color{blue}{a}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 15.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|b\right| \cdot a\right) \cdot \left(\left|b\right| \cdot \left(-a\right)\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ t_2 := \sin t\_1\\ t_3 := \left(\pi \cdot angle\right) \cdot \frac{1}{180}\\ t_4 := \sin t\_3\\ t_5 := {\left(t\_4 \cdot \left|b\right|\right)}^{2} + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot \cos \left(t\_3 \cdot 2\right)\right) \cdot a\right) \cdot a\\ t_6 := \frac{\left(a \cdot \left|b\right|\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot \left|b\right|}{y-scale \cdot x-scale}\\ t_7 := \cos t\_1\\ t_8 := \frac{\frac{{\left(a \cdot t\_7\right)}^{2} + {\left(\left|b\right| \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_9 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ t_10 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(\left|b\right| \cdot t\_7\right)}^{2}}{x-scale}}{x-scale}\\ \mathbf{if}\;\frac{-\sqrt{\left(\left(2 \cdot t\_9\right) \cdot t\_0\right) \cdot \left(\left(t\_10 + t\_8\right) + \sqrt{{\left(t\_10 - t\_8\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_7}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_9} \leq \infty:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_0\right) \cdot \frac{\frac{\left|t\_5\right| + t\_5}{y-scale}}{y-scale}}}{t\_6}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{4} \cdot \left(\frac{\left|b\right| \cdot \left(x-scale \cdot x-scale\right)}{a} \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(\frac{1}{2} - \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{1}{2}\right) + \sqrt{{t\_4}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{a}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* (fabs b) a) (* (fabs b) (- a))))
       (t_1 (* (/ angle 180) PI))
       (t_2 (sin t_1))
       (t_3 (* (* PI angle) 1/180))
       (t_4 (sin t_3))
       (t_5
        (+
         (pow (* t_4 (fabs b)) 2)
         (* (* (- 1/2 (* -1/2 (cos (* t_3 2)))) a) a)))
       (t_6
        (*
         (/ (* (* a (fabs b)) 4) (* y-scale x-scale))
         (/ (* (- a) (fabs b)) (* y-scale x-scale))))
       (t_7 (cos t_1))
       (t_8
        (/
         (/ (+ (pow (* a t_7) 2) (pow (* (fabs b) t_2) 2)) y-scale)
         y-scale))
       (t_9 (/ (* 4 t_0) (pow (* x-scale y-scale) 2)))
       (t_10
        (/
         (/ (+ (pow (* a t_2) 2) (pow (* (fabs b) t_7) 2)) x-scale)
         x-scale)))
  (if (<=
       (/
        (-
         (sqrt
          (*
           (* (* 2 t_9) t_0)
           (+
            (+ t_10 t_8)
            (sqrt
             (+
              (pow (- t_10 t_8) 2)
              (pow
               (/
                (/
                 (* (* (* 2 (- (pow (fabs b) 2) (pow a 2))) t_2) t_7)
                 x-scale)
                y-scale)
               2)))))))
        t_9)
       INFINITY)
    (/
     (-
      (sqrt
       (*
        (* (* 2 t_6) t_0)
        (/ (/ (+ (fabs t_5) t_5) y-scale) y-scale))))
     t_6)
    (*
     1/4
     (*
      (/ (* (fabs b) (* x-scale x-scale)) a)
      (/
       (/
        (sqrt
         (*
          8
          (*
           (+
            (- 1/2 (* (cos (* (* PI angle) 1/90)) 1/2))
            (sqrt (pow t_4 4)))
           (pow a 4))))
        (fabs x-scale))
       a))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (fabs(b) * a) * (fabs(b) * -a);
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double t_2 = sin(t_1);
	double t_3 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_4 = sin(t_3);
	double t_5 = pow((t_4 * fabs(b)), 2.0) + (((0.5 - (-0.5 * cos((t_3 * 2.0)))) * a) * a);
	double t_6 = (((a * fabs(b)) * 4.0) / (y_45_scale * x_45_scale)) * ((-a * fabs(b)) / (y_45_scale * x_45_scale));
	double t_7 = cos(t_1);
	double t_8 = ((pow((a * t_7), 2.0) + pow((fabs(b) * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_9 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double t_10 = ((pow((a * t_2), 2.0) + pow((fabs(b) * t_7), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((-sqrt((((2.0 * t_9) * t_0) * ((t_10 + t_8) + sqrt((pow((t_10 - t_8), 2.0) + pow((((((2.0 * (pow(fabs(b), 2.0) - pow(a, 2.0))) * t_2) * t_7) / x_45_scale) / y_45_scale), 2.0)))))) / t_9) <= ((double) INFINITY)) {
		tmp = -sqrt((((2.0 * t_6) * t_0) * (((fabs(t_5) + t_5) / y_45_scale) / y_45_scale))) / t_6;
	} else {
		tmp = 0.25 * (((fabs(b) * (x_45_scale * x_45_scale)) / a) * ((sqrt((8.0 * (((0.5 - (cos(((((double) M_PI) * angle) * 0.011111111111111112)) * 0.5)) + sqrt(pow(t_4, 4.0))) * pow(a, 4.0)))) / fabs(x_45_scale)) / a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.abs(b) * a) * (Math.abs(b) * -a);
	double t_1 = (angle / 180.0) * Math.PI;
	double t_2 = Math.sin(t_1);
	double t_3 = (Math.PI * angle) * 0.005555555555555556;
	double t_4 = Math.sin(t_3);
	double t_5 = Math.pow((t_4 * Math.abs(b)), 2.0) + (((0.5 - (-0.5 * Math.cos((t_3 * 2.0)))) * a) * a);
	double t_6 = (((a * Math.abs(b)) * 4.0) / (y_45_scale * x_45_scale)) * ((-a * Math.abs(b)) / (y_45_scale * x_45_scale));
	double t_7 = Math.cos(t_1);
	double t_8 = ((Math.pow((a * t_7), 2.0) + Math.pow((Math.abs(b) * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_9 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 2.0);
	double t_10 = ((Math.pow((a * t_2), 2.0) + Math.pow((Math.abs(b) * t_7), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((-Math.sqrt((((2.0 * t_9) * t_0) * ((t_10 + t_8) + Math.sqrt((Math.pow((t_10 - t_8), 2.0) + Math.pow((((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(a, 2.0))) * t_2) * t_7) / x_45_scale) / y_45_scale), 2.0)))))) / t_9) <= Double.POSITIVE_INFINITY) {
		tmp = -Math.sqrt((((2.0 * t_6) * t_0) * (((Math.abs(t_5) + t_5) / y_45_scale) / y_45_scale))) / t_6;
	} else {
		tmp = 0.25 * (((Math.abs(b) * (x_45_scale * x_45_scale)) / a) * ((Math.sqrt((8.0 * (((0.5 - (Math.cos(((Math.PI * angle) * 0.011111111111111112)) * 0.5)) + Math.sqrt(Math.pow(t_4, 4.0))) * Math.pow(a, 4.0)))) / Math.abs(x_45_scale)) / a));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (math.fabs(b) * a) * (math.fabs(b) * -a)
	t_1 = (angle / 180.0) * math.pi
	t_2 = math.sin(t_1)
	t_3 = (math.pi * angle) * 0.005555555555555556
	t_4 = math.sin(t_3)
	t_5 = math.pow((t_4 * math.fabs(b)), 2.0) + (((0.5 - (-0.5 * math.cos((t_3 * 2.0)))) * a) * a)
	t_6 = (((a * math.fabs(b)) * 4.0) / (y_45_scale * x_45_scale)) * ((-a * math.fabs(b)) / (y_45_scale * x_45_scale))
	t_7 = math.cos(t_1)
	t_8 = ((math.pow((a * t_7), 2.0) + math.pow((math.fabs(b) * t_2), 2.0)) / y_45_scale) / y_45_scale
	t_9 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 2.0)
	t_10 = ((math.pow((a * t_2), 2.0) + math.pow((math.fabs(b) * t_7), 2.0)) / x_45_scale) / x_45_scale
	tmp = 0
	if (-math.sqrt((((2.0 * t_9) * t_0) * ((t_10 + t_8) + math.sqrt((math.pow((t_10 - t_8), 2.0) + math.pow((((((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(a, 2.0))) * t_2) * t_7) / x_45_scale) / y_45_scale), 2.0)))))) / t_9) <= math.inf:
		tmp = -math.sqrt((((2.0 * t_6) * t_0) * (((math.fabs(t_5) + t_5) / y_45_scale) / y_45_scale))) / t_6
	else:
		tmp = 0.25 * (((math.fabs(b) * (x_45_scale * x_45_scale)) / a) * ((math.sqrt((8.0 * (((0.5 - (math.cos(((math.pi * angle) * 0.011111111111111112)) * 0.5)) + math.sqrt(math.pow(t_4, 4.0))) * math.pow(a, 4.0)))) / math.fabs(x_45_scale)) / a))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(abs(b) * a) * Float64(abs(b) * Float64(-a)))
	t_1 = Float64(Float64(angle / 180.0) * pi)
	t_2 = sin(t_1)
	t_3 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_4 = sin(t_3)
	t_5 = Float64((Float64(t_4 * abs(b)) ^ 2.0) + Float64(Float64(Float64(0.5 - Float64(-0.5 * cos(Float64(t_3 * 2.0)))) * a) * a))
	t_6 = Float64(Float64(Float64(Float64(a * abs(b)) * 4.0) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(Float64(-a) * abs(b)) / Float64(y_45_scale * x_45_scale)))
	t_7 = cos(t_1)
	t_8 = Float64(Float64(Float64((Float64(a * t_7) ^ 2.0) + (Float64(abs(b) * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_9 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	t_10 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(abs(b) * t_7) ^ 2.0)) / x_45_scale) / x_45_scale)
	tmp = 0.0
	if (Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_9) * t_0) * Float64(Float64(t_10 + t_8) + sqrt(Float64((Float64(t_10 - t_8) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (a ^ 2.0))) * t_2) * t_7) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_9) <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_0) * Float64(Float64(Float64(abs(t_5) + t_5) / y_45_scale) / y_45_scale)))) / t_6);
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(abs(b) * Float64(x_45_scale * x_45_scale)) / a) * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) * 0.5)) + sqrt((t_4 ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale)) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (abs(b) * a) * (abs(b) * -a);
	t_1 = (angle / 180.0) * pi;
	t_2 = sin(t_1);
	t_3 = (pi * angle) * 0.005555555555555556;
	t_4 = sin(t_3);
	t_5 = ((t_4 * abs(b)) ^ 2.0) + (((0.5 - (-0.5 * cos((t_3 * 2.0)))) * a) * a);
	t_6 = (((a * abs(b)) * 4.0) / (y_45_scale * x_45_scale)) * ((-a * abs(b)) / (y_45_scale * x_45_scale));
	t_7 = cos(t_1);
	t_8 = ((((a * t_7) ^ 2.0) + ((abs(b) * t_2) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_9 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 2.0);
	t_10 = ((((a * t_2) ^ 2.0) + ((abs(b) * t_7) ^ 2.0)) / x_45_scale) / x_45_scale;
	tmp = 0.0;
	if ((-sqrt((((2.0 * t_9) * t_0) * ((t_10 + t_8) + sqrt((((t_10 - t_8) ^ 2.0) + ((((((2.0 * ((abs(b) ^ 2.0) - (a ^ 2.0))) * t_2) * t_7) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_9) <= Inf)
		tmp = -sqrt((((2.0 * t_6) * t_0) * (((abs(t_5) + t_5) / y_45_scale) / y_45_scale))) / t_6;
	else
		tmp = 0.25 * (((abs(b) * (x_45_scale * x_45_scale)) / a) * ((sqrt((8.0 * (((0.5 - (cos(((pi * angle) * 0.011111111111111112)) * 0.5)) + sqrt((t_4 ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale)) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(Pi * angle), $MachinePrecision] * 1/180), $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$4 * N[Abs[b], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[(N[(N[(1/2 - N[(-1/2 * N[Cos[N[(t$95$3 * 2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision] * 4), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[((-a) * N[Abs[b], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$8 = N[(N[(N[(N[Power[N[(a * t$95$7), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(N[Abs[b], $MachinePrecision] * t$95$2), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$9 = N[(N[(4 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(N[Abs[b], $MachinePrecision] * t$95$7), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[N[((-N[Sqrt[N[(N[(N[(2 * t$95$9), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(t$95$10 + t$95$8), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$10 - t$95$8), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[a, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$7), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$9), $MachinePrecision], Infinity], N[((-N[Sqrt[N[(N[(N[(2 * t$95$6), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[Abs[t$95$5], $MachinePrecision] + t$95$5), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision], N[(1/4 * N[(N[(N[(N[Abs[b], $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(N[(N[Sqrt[N[(8 * N[(N[(N[(1/2 - N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[t$95$4, 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[a, 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64)))) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (+.f64 (+.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64)))))))) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64)))) < +inf.0

   1. Initial program 2.6%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 4.2%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 4.2%

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 4.2%

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

   6. Applied rewrites 4.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 6.4%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 6.4%

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 6.4%

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

   8. Applied rewrites 6.4%

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

   10. Applied rewrites 8.4%

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

   12. Applied rewrites 8.4%

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

   14. Applied rewrites 9.4%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64)))) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (+.f64 (+.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64)))))))) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64))))

   1. Initial program 2.6%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 8.3%

      \[\leadsto \frac{1}{4} \cdot \left(\frac{b \cdot \left(x-scale \cdot x-scale\right)}{a} \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(\frac{1}{2} - \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{1}{2}\right) + \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{\color{blue}{a}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 15.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|b\right| \cdot a\right) \cdot \left(\left|b\right| \cdot \left(-a\right)\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ t_2 := \sin t\_1\\ t_3 := \frac{a \cdot a}{y-scale \cdot y-scale}\\ t_4 := \frac{\left|b\right| \cdot \left|b\right|}{x-scale \cdot x-scale}\\ t_5 := \cos t\_1\\ t_6 := \frac{\frac{{\left(a \cdot t\_5\right)}^{2} + {\left(\left|b\right| \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_7 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ t_8 := \left(2 \cdot t\_7\right) \cdot t\_0\\ t_9 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(\left|b\right| \cdot t\_5\right)}^{2}}{x-scale}}{x-scale}\\ \mathbf{if}\;\frac{-\sqrt{t\_8 \cdot \left(\left(t\_9 + t\_6\right) + \sqrt{{\left(t\_9 - t\_6\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({\left(\left|b\right|\right)}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_5}{x-scale}}{y-scale}\right)}^{2}}\right)}}{t\_7} \leq \infty:\\ \;\;\;\;\frac{-\sqrt{t\_8 \cdot \left(\left(t\_3 + t\_4\right) + \left|t\_4 - t\_3\right|\right)}}{t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{4} \cdot \left(\frac{\left|b\right| \cdot \left(x-scale \cdot x-scale\right)}{a} \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(\frac{1}{2} - \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{1}{2}\right) + \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{a}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* (fabs b) a) (* (fabs b) (- a))))
       (t_1 (* (/ angle 180) PI))
       (t_2 (sin t_1))
       (t_3 (/ (* a a) (* y-scale y-scale)))
       (t_4 (/ (* (fabs b) (fabs b)) (* x-scale x-scale)))
       (t_5 (cos t_1))
       (t_6
        (/
         (/ (+ (pow (* a t_5) 2) (pow (* (fabs b) t_2) 2)) y-scale)
         y-scale))
       (t_7 (/ (* 4 t_0) (pow (* x-scale y-scale) 2)))
       (t_8 (* (* 2 t_7) t_0))
       (t_9
        (/
         (/ (+ (pow (* a t_2) 2) (pow (* (fabs b) t_5) 2)) x-scale)
         x-scale)))
  (if (<=
       (/
        (-
         (sqrt
          (*
           t_8
           (+
            (+ t_9 t_6)
            (sqrt
             (+
              (pow (- t_9 t_6) 2)
              (pow
               (/
                (/
                 (* (* (* 2 (- (pow (fabs b) 2) (pow a 2))) t_2) t_5)
                 x-scale)
                y-scale)
               2)))))))
        t_7)
       INFINITY)
    (/ (- (sqrt (* t_8 (+ (+ t_3 t_4) (fabs (- t_4 t_3)))))) t_7)
    (*
     1/4
     (*
      (/ (* (fabs b) (* x-scale x-scale)) a)
      (/
       (/
        (sqrt
         (*
          8
          (*
           (+
            (- 1/2 (* (cos (* (* PI angle) 1/90)) 1/2))
            (sqrt (pow (sin (* (* PI angle) 1/180)) 4)))
           (pow a 4))))
        (fabs x-scale))
       a))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (fabs(b) * a) * (fabs(b) * -a);
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double t_2 = sin(t_1);
	double t_3 = (a * a) / (y_45_scale * y_45_scale);
	double t_4 = (fabs(b) * fabs(b)) / (x_45_scale * x_45_scale);
	double t_5 = cos(t_1);
	double t_6 = ((pow((a * t_5), 2.0) + pow((fabs(b) * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_7 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double t_8 = (2.0 * t_7) * t_0;
	double t_9 = ((pow((a * t_2), 2.0) + pow((fabs(b) * t_5), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((-sqrt((t_8 * ((t_9 + t_6) + sqrt((pow((t_9 - t_6), 2.0) + pow((((((2.0 * (pow(fabs(b), 2.0) - pow(a, 2.0))) * t_2) * t_5) / x_45_scale) / y_45_scale), 2.0)))))) / t_7) <= ((double) INFINITY)) {
		tmp = -sqrt((t_8 * ((t_3 + t_4) + fabs((t_4 - t_3))))) / t_7;
	} else {
		tmp = 0.25 * (((fabs(b) * (x_45_scale * x_45_scale)) / a) * ((sqrt((8.0 * (((0.5 - (cos(((((double) M_PI) * angle) * 0.011111111111111112)) * 0.5)) + sqrt(pow(sin(((((double) M_PI) * angle) * 0.005555555555555556)), 4.0))) * pow(a, 4.0)))) / fabs(x_45_scale)) / a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.abs(b) * a) * (Math.abs(b) * -a);
	double t_1 = (angle / 180.0) * Math.PI;
	double t_2 = Math.sin(t_1);
	double t_3 = (a * a) / (y_45_scale * y_45_scale);
	double t_4 = (Math.abs(b) * Math.abs(b)) / (x_45_scale * x_45_scale);
	double t_5 = Math.cos(t_1);
	double t_6 = ((Math.pow((a * t_5), 2.0) + Math.pow((Math.abs(b) * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_7 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 2.0);
	double t_8 = (2.0 * t_7) * t_0;
	double t_9 = ((Math.pow((a * t_2), 2.0) + Math.pow((Math.abs(b) * t_5), 2.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if ((-Math.sqrt((t_8 * ((t_9 + t_6) + Math.sqrt((Math.pow((t_9 - t_6), 2.0) + Math.pow((((((2.0 * (Math.pow(Math.abs(b), 2.0) - Math.pow(a, 2.0))) * t_2) * t_5) / x_45_scale) / y_45_scale), 2.0)))))) / t_7) <= Double.POSITIVE_INFINITY) {
		tmp = -Math.sqrt((t_8 * ((t_3 + t_4) + Math.abs((t_4 - t_3))))) / t_7;
	} else {
		tmp = 0.25 * (((Math.abs(b) * (x_45_scale * x_45_scale)) / a) * ((Math.sqrt((8.0 * (((0.5 - (Math.cos(((Math.PI * angle) * 0.011111111111111112)) * 0.5)) + Math.sqrt(Math.pow(Math.sin(((Math.PI * angle) * 0.005555555555555556)), 4.0))) * Math.pow(a, 4.0)))) / Math.abs(x_45_scale)) / a));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (math.fabs(b) * a) * (math.fabs(b) * -a)
	t_1 = (angle / 180.0) * math.pi
	t_2 = math.sin(t_1)
	t_3 = (a * a) / (y_45_scale * y_45_scale)
	t_4 = (math.fabs(b) * math.fabs(b)) / (x_45_scale * x_45_scale)
	t_5 = math.cos(t_1)
	t_6 = ((math.pow((a * t_5), 2.0) + math.pow((math.fabs(b) * t_2), 2.0)) / y_45_scale) / y_45_scale
	t_7 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 2.0)
	t_8 = (2.0 * t_7) * t_0
	t_9 = ((math.pow((a * t_2), 2.0) + math.pow((math.fabs(b) * t_5), 2.0)) / x_45_scale) / x_45_scale
	tmp = 0
	if (-math.sqrt((t_8 * ((t_9 + t_6) + math.sqrt((math.pow((t_9 - t_6), 2.0) + math.pow((((((2.0 * (math.pow(math.fabs(b), 2.0) - math.pow(a, 2.0))) * t_2) * t_5) / x_45_scale) / y_45_scale), 2.0)))))) / t_7) <= math.inf:
		tmp = -math.sqrt((t_8 * ((t_3 + t_4) + math.fabs((t_4 - t_3))))) / t_7
	else:
		tmp = 0.25 * (((math.fabs(b) * (x_45_scale * x_45_scale)) / a) * ((math.sqrt((8.0 * (((0.5 - (math.cos(((math.pi * angle) * 0.011111111111111112)) * 0.5)) + math.sqrt(math.pow(math.sin(((math.pi * angle) * 0.005555555555555556)), 4.0))) * math.pow(a, 4.0)))) / math.fabs(x_45_scale)) / a))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(abs(b) * a) * Float64(abs(b) * Float64(-a)))
	t_1 = Float64(Float64(angle / 180.0) * pi)
	t_2 = sin(t_1)
	t_3 = Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale))
	t_4 = Float64(Float64(abs(b) * abs(b)) / Float64(x_45_scale * x_45_scale))
	t_5 = cos(t_1)
	t_6 = Float64(Float64(Float64((Float64(a * t_5) ^ 2.0) + (Float64(abs(b) * t_2) ^ 2.0)) / y_45_scale) / y_45_scale)
	t_7 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	t_8 = Float64(Float64(2.0 * t_7) * t_0)
	t_9 = Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(abs(b) * t_5) ^ 2.0)) / x_45_scale) / x_45_scale)
	tmp = 0.0
	if (Float64(Float64(-sqrt(Float64(t_8 * Float64(Float64(t_9 + t_6) + sqrt(Float64((Float64(t_9 - t_6) ^ 2.0) + (Float64(Float64(Float64(Float64(Float64(2.0 * Float64((abs(b) ^ 2.0) - (a ^ 2.0))) * t_2) * t_5) / x_45_scale) / y_45_scale) ^ 2.0))))))) / t_7) <= Inf)
		tmp = Float64(Float64(-sqrt(Float64(t_8 * Float64(Float64(t_3 + t_4) + abs(Float64(t_4 - t_3)))))) / t_7);
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(abs(b) * Float64(x_45_scale * x_45_scale)) / a) * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) * 0.5)) + sqrt((sin(Float64(Float64(pi * angle) * 0.005555555555555556)) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale)) / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (abs(b) * a) * (abs(b) * -a);
	t_1 = (angle / 180.0) * pi;
	t_2 = sin(t_1);
	t_3 = (a * a) / (y_45_scale * y_45_scale);
	t_4 = (abs(b) * abs(b)) / (x_45_scale * x_45_scale);
	t_5 = cos(t_1);
	t_6 = ((((a * t_5) ^ 2.0) + ((abs(b) * t_2) ^ 2.0)) / y_45_scale) / y_45_scale;
	t_7 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 2.0);
	t_8 = (2.0 * t_7) * t_0;
	t_9 = ((((a * t_2) ^ 2.0) + ((abs(b) * t_5) ^ 2.0)) / x_45_scale) / x_45_scale;
	tmp = 0.0;
	if ((-sqrt((t_8 * ((t_9 + t_6) + sqrt((((t_9 - t_6) ^ 2.0) + ((((((2.0 * ((abs(b) ^ 2.0) - (a ^ 2.0))) * t_2) * t_5) / x_45_scale) / y_45_scale) ^ 2.0)))))) / t_7) <= Inf)
		tmp = -sqrt((t_8 * ((t_3 + t_4) + abs((t_4 - t_3))))) / t_7;
	else
		tmp = 0.25 * (((abs(b) * (x_45_scale * x_45_scale)) / a) * ((sqrt((8.0 * (((0.5 - (cos(((pi * angle) * 0.011111111111111112)) * 0.5)) + sqrt((sin(((pi * angle) * 0.005555555555555556)) ^ 4.0))) * (a ^ 4.0)))) / abs(x_45_scale)) / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(N[Power[N[(a * t$95$5), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(N[Abs[b], $MachinePrecision] * t$95$2), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$7 = N[(N[(4 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[(2 * t$95$7), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(N[Abs[b], $MachinePrecision] * t$95$5), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[N[((-N[Sqrt[N[(t$95$8 * N[(N[(t$95$9 + t$95$6), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(t$95$9 - t$95$6), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(2 * N[(N[Power[N[Abs[b], $MachinePrecision], 2], $MachinePrecision] - N[Power[a, 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$5), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$7), $MachinePrecision], Infinity], N[((-N[Sqrt[N[(t$95$8 * N[(N[(t$95$3 + t$95$4), $MachinePrecision] + N[Abs[N[(t$95$4 - t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$7), $MachinePrecision], N[(1/4 * N[(N[(N[(N[Abs[b], $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] * N[(N[(N[Sqrt[N[(8 * N[(N[(N[(1/2 - N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 1/180), $MachinePrecision]], $MachinePrecision], 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[a, 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64)))) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (+.f64 (+.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64)))))))) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64)))) < +inf.0

   1. Initial program 2.6%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 4.1%

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 4.1%

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 4.1%

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 4.1%

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

      13. lift-pow.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 4.1%

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

      16. lift-sqrt.f64 N/A

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

      17. lift-pow.f64 N/A

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

   6. Applied rewrites 4.3%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64)))) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (+.f64 (+.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) (sqrt.f64 (+.f64 (pow.f64 (-.f64 (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) x-scale) x-scale) (/.f64 (/.f64 (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) #s(literal 2 binary64))) y-scale) y-scale)) #s(literal 2 binary64)) (pow.f64 (/.f64 (/.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) (-.f64 (pow.f64 b #s(literal 2 binary64)) (pow.f64 a #s(literal 2 binary64)))) (sin.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) (cos.f64 (*.f64 (/.f64 angle #s(literal 180 binary64)) (PI.f64)))) x-scale) y-scale) #s(literal 2 binary64)))))))) (/.f64 (*.f64 #s(literal 4 binary64) (*.f64 (*.f64 b a) (*.f64 b (neg.f64 a)))) (pow.f64 (*.f64 x-scale y-scale) #s(literal 2 binary64))))

   1. Initial program 2.6%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 8.3%

      \[\leadsto \frac{1}{4} \cdot \left(\frac{b \cdot \left(x-scale \cdot x-scale\right)}{a} \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(\frac{1}{2} - \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{1}{2}\right) + \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|}}{\color{blue}{a}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 11.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \left|a\right|\\ t_1 := \left(\frac{1}{2} - 1 \cdot \frac{1}{2}\right) \cdot \left(\left|b\right| \cdot \left|b\right|\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot 1\right) \cdot \left|a\right|\right) \cdot \left|a\right|\\ t_2 := \left|b\right| \cdot \frac{\left|b\right|}{x-scale \cdot x-scale}\\ t_3 := -\left|a\right|\\ t_4 := t\_3 \cdot \left|b\right|\\ t_5 := 4 \cdot t\_0\\ t_6 := \left|a\right| \cdot \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_7 := \frac{\left(\left|a\right| \cdot \left|b\right|\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{t\_4}{y-scale \cdot x-scale}\\ \mathbf{if}\;\left|a\right| \leq \frac{4417117661945961}{27606985387162255149739023449108101809804435888681546220650096895197184}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left(\left(\left(\frac{\left|b\right|}{\left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot t\_3\right) \cdot \left(t\_5 \cdot 2\right)\right) \cdot \left(t\_0 \cdot \left|b\right|\right)\right) \cdot t\_3\right) \cdot \left(\left|t\_6 - t\_2\right| + \left(t\_2 + t\_6\right)\right)}}{t\_5}}{t\_4} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right)\\ \mathbf{elif}\;\left|a\right| \leq 48000000000000001987463513459421542544804992225473439482941353636669046511737368705419469347170308598973901747830710504197303976962715123653608443215872:\\ \;\;\;\;\frac{1}{4} \cdot \left(\left|b\right| \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(\frac{1}{2} - \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{1}{2}\right) + \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right) \cdot {\left(\left|a\right|\right)}^{4}\right)}}{\left|x-scale\right|} \cdot \left(x-scale \cdot x-scale\right)}{\left|a\right| \cdot \left|a\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_7\right) \cdot \left(t\_0 \cdot \left(\left|b\right| \cdot t\_3\right)\right)\right) \cdot \frac{\frac{\left|t\_1\right| + t\_1}{y-scale}}{y-scale}}}{t\_7}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs b) (fabs a)))
       (t_1
        (+
         (* (- 1/2 (* 1 1/2)) (* (fabs b) (fabs b)))
         (* (* (- 1/2 (* -1/2 1)) (fabs a)) (fabs a))))
       (t_2 (* (fabs b) (/ (fabs b) (* x-scale x-scale))))
       (t_3 (- (fabs a)))
       (t_4 (* t_3 (fabs b)))
       (t_5 (* 4 t_0))
       (t_6 (* (fabs a) (/ (fabs a) (* y-scale y-scale))))
       (t_7
        (*
         (/ (* (* (fabs a) (fabs b)) 4) (* y-scale x-scale))
         (/ t_4 (* y-scale x-scale)))))
  (if (<=
       (fabs a)
       4417117661945961/27606985387162255149739023449108101809804435888681546220650096895197184)
    (*
     (/
      (/
       (-
        (sqrt
         (*
          (*
           (*
            (*
             (*
              (/ (fabs b) (* (* (* x-scale y-scale) x-scale) y-scale))
              t_3)
             (* t_5 2))
            (* t_0 (fabs b)))
           t_3)
          (+ (fabs (- t_6 t_2)) (+ t_2 t_6)))))
       t_5)
      t_4)
     (* (* (* y-scale x-scale) x-scale) y-scale))
    (if (<=
         (fabs a)
         48000000000000001987463513459421542544804992225473439482941353636669046511737368705419469347170308598973901747830710504197303976962715123653608443215872)
      (*
       1/4
       (*
        (fabs b)
        (/
         (*
          (/
           (sqrt
            (*
             8
             (*
              (+
               (- 1/2 (* (cos (* (* PI angle) 1/90)) 1/2))
               (sqrt (pow (sin (* (* PI angle) 1/180)) 4)))
              (pow (fabs a) 4))))
           (fabs x-scale))
          (* x-scale x-scale))
         (* (fabs a) (fabs a)))))
      (/
       (-
        (sqrt
         (*
          (* (* 2 t_7) (* t_0 (* (fabs b) t_3)))
          (/ (/ (+ (fabs t_1) t_1) y-scale) y-scale))))
       t_7)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) * fabs(a);
	double t_1 = ((0.5 - (1.0 * 0.5)) * (fabs(b) * fabs(b))) + (((0.5 - (-0.5 * 1.0)) * fabs(a)) * fabs(a));
	double t_2 = fabs(b) * (fabs(b) / (x_45_scale * x_45_scale));
	double t_3 = -fabs(a);
	double t_4 = t_3 * fabs(b);
	double t_5 = 4.0 * t_0;
	double t_6 = fabs(a) * (fabs(a) / (y_45_scale * y_45_scale));
	double t_7 = (((fabs(a) * fabs(b)) * 4.0) / (y_45_scale * x_45_scale)) * (t_4 / (y_45_scale * x_45_scale));
	double tmp;
	if (fabs(a) <= 1.6e-55) {
		tmp = ((-sqrt(((((((fabs(b) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * t_3) * (t_5 * 2.0)) * (t_0 * fabs(b))) * t_3) * (fabs((t_6 - t_2)) + (t_2 + t_6)))) / t_5) / t_4) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
	} else if (fabs(a) <= 4.8e+151) {
		tmp = 0.25 * (fabs(b) * (((sqrt((8.0 * (((0.5 - (cos(((((double) M_PI) * angle) * 0.011111111111111112)) * 0.5)) + sqrt(pow(sin(((((double) M_PI) * angle) * 0.005555555555555556)), 4.0))) * pow(fabs(a), 4.0)))) / fabs(x_45_scale)) * (x_45_scale * x_45_scale)) / (fabs(a) * fabs(a))));
	} else {
		tmp = -sqrt((((2.0 * t_7) * (t_0 * (fabs(b) * t_3))) * (((fabs(t_1) + t_1) / y_45_scale) / y_45_scale))) / t_7;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs(b) * Math.abs(a);
	double t_1 = ((0.5 - (1.0 * 0.5)) * (Math.abs(b) * Math.abs(b))) + (((0.5 - (-0.5 * 1.0)) * Math.abs(a)) * Math.abs(a));
	double t_2 = Math.abs(b) * (Math.abs(b) / (x_45_scale * x_45_scale));
	double t_3 = -Math.abs(a);
	double t_4 = t_3 * Math.abs(b);
	double t_5 = 4.0 * t_0;
	double t_6 = Math.abs(a) * (Math.abs(a) / (y_45_scale * y_45_scale));
	double t_7 = (((Math.abs(a) * Math.abs(b)) * 4.0) / (y_45_scale * x_45_scale)) * (t_4 / (y_45_scale * x_45_scale));
	double tmp;
	if (Math.abs(a) <= 1.6e-55) {
		tmp = ((-Math.sqrt(((((((Math.abs(b) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * t_3) * (t_5 * 2.0)) * (t_0 * Math.abs(b))) * t_3) * (Math.abs((t_6 - t_2)) + (t_2 + t_6)))) / t_5) / t_4) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
	} else if (Math.abs(a) <= 4.8e+151) {
		tmp = 0.25 * (Math.abs(b) * (((Math.sqrt((8.0 * (((0.5 - (Math.cos(((Math.PI * angle) * 0.011111111111111112)) * 0.5)) + Math.sqrt(Math.pow(Math.sin(((Math.PI * angle) * 0.005555555555555556)), 4.0))) * Math.pow(Math.abs(a), 4.0)))) / Math.abs(x_45_scale)) * (x_45_scale * x_45_scale)) / (Math.abs(a) * Math.abs(a))));
	} else {
		tmp = -Math.sqrt((((2.0 * t_7) * (t_0 * (Math.abs(b) * t_3))) * (((Math.abs(t_1) + t_1) / y_45_scale) / y_45_scale))) / t_7;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(b) * math.fabs(a)
	t_1 = ((0.5 - (1.0 * 0.5)) * (math.fabs(b) * math.fabs(b))) + (((0.5 - (-0.5 * 1.0)) * math.fabs(a)) * math.fabs(a))
	t_2 = math.fabs(b) * (math.fabs(b) / (x_45_scale * x_45_scale))
	t_3 = -math.fabs(a)
	t_4 = t_3 * math.fabs(b)
	t_5 = 4.0 * t_0
	t_6 = math.fabs(a) * (math.fabs(a) / (y_45_scale * y_45_scale))
	t_7 = (((math.fabs(a) * math.fabs(b)) * 4.0) / (y_45_scale * x_45_scale)) * (t_4 / (y_45_scale * x_45_scale))
	tmp = 0
	if math.fabs(a) <= 1.6e-55:
		tmp = ((-math.sqrt(((((((math.fabs(b) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * t_3) * (t_5 * 2.0)) * (t_0 * math.fabs(b))) * t_3) * (math.fabs((t_6 - t_2)) + (t_2 + t_6)))) / t_5) / t_4) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)
	elif math.fabs(a) <= 4.8e+151:
		tmp = 0.25 * (math.fabs(b) * (((math.sqrt((8.0 * (((0.5 - (math.cos(((math.pi * angle) * 0.011111111111111112)) * 0.5)) + math.sqrt(math.pow(math.sin(((math.pi * angle) * 0.005555555555555556)), 4.0))) * math.pow(math.fabs(a), 4.0)))) / math.fabs(x_45_scale)) * (x_45_scale * x_45_scale)) / (math.fabs(a) * math.fabs(a))))
	else:
		tmp = -math.sqrt((((2.0 * t_7) * (t_0 * (math.fabs(b) * t_3))) * (((math.fabs(t_1) + t_1) / y_45_scale) / y_45_scale))) / t_7
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) * abs(a))
	t_1 = Float64(Float64(Float64(0.5 - Float64(1.0 * 0.5)) * Float64(abs(b) * abs(b))) + Float64(Float64(Float64(0.5 - Float64(-0.5 * 1.0)) * abs(a)) * abs(a)))
	t_2 = Float64(abs(b) * Float64(abs(b) / Float64(x_45_scale * x_45_scale)))
	t_3 = Float64(-abs(a))
	t_4 = Float64(t_3 * abs(b))
	t_5 = Float64(4.0 * t_0)
	t_6 = Float64(abs(a) * Float64(abs(a) / Float64(y_45_scale * y_45_scale)))
	t_7 = Float64(Float64(Float64(Float64(abs(a) * abs(b)) * 4.0) / Float64(y_45_scale * x_45_scale)) * Float64(t_4 / Float64(y_45_scale * x_45_scale)))
	tmp = 0.0
	if (abs(a) <= 1.6e-55)
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(abs(b) / Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * t_3) * Float64(t_5 * 2.0)) * Float64(t_0 * abs(b))) * t_3) * Float64(abs(Float64(t_6 - t_2)) + Float64(t_2 + t_6))))) / t_5) / t_4) * Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale));
	elseif (abs(a) <= 4.8e+151)
		tmp = Float64(0.25 * Float64(abs(b) * Float64(Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(pi * angle) * 0.011111111111111112)) * 0.5)) + sqrt((sin(Float64(Float64(pi * angle) * 0.005555555555555556)) ^ 4.0))) * (abs(a) ^ 4.0)))) / abs(x_45_scale)) * Float64(x_45_scale * x_45_scale)) / Float64(abs(a) * abs(a)))));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_7) * Float64(t_0 * Float64(abs(b) * t_3))) * Float64(Float64(Float64(abs(t_1) + t_1) / y_45_scale) / y_45_scale)))) / t_7);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) * abs(a);
	t_1 = ((0.5 - (1.0 * 0.5)) * (abs(b) * abs(b))) + (((0.5 - (-0.5 * 1.0)) * abs(a)) * abs(a));
	t_2 = abs(b) * (abs(b) / (x_45_scale * x_45_scale));
	t_3 = -abs(a);
	t_4 = t_3 * abs(b);
	t_5 = 4.0 * t_0;
	t_6 = abs(a) * (abs(a) / (y_45_scale * y_45_scale));
	t_7 = (((abs(a) * abs(b)) * 4.0) / (y_45_scale * x_45_scale)) * (t_4 / (y_45_scale * x_45_scale));
	tmp = 0.0;
	if (abs(a) <= 1.6e-55)
		tmp = ((-sqrt(((((((abs(b) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * t_3) * (t_5 * 2.0)) * (t_0 * abs(b))) * t_3) * (abs((t_6 - t_2)) + (t_2 + t_6)))) / t_5) / t_4) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
	elseif (abs(a) <= 4.8e+151)
		tmp = 0.25 * (abs(b) * (((sqrt((8.0 * (((0.5 - (cos(((pi * angle) * 0.011111111111111112)) * 0.5)) + sqrt((sin(((pi * angle) * 0.005555555555555556)) ^ 4.0))) * (abs(a) ^ 4.0)))) / abs(x_45_scale)) * (x_45_scale * x_45_scale)) / (abs(a) * abs(a))));
	else
		tmp = -sqrt((((2.0 * t_7) * (t_0 * (abs(b) * t_3))) * (((abs(t_1) + t_1) / y_45_scale) / y_45_scale))) / t_7;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1/2 - N[(1 * 1/2), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1/2 - N[(-1/2 * 1), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$4 = N[(t$95$3 * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(4 * t$95$0), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[(N[(N[Abs[a], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * 4), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 4417117661945961/27606985387162255149739023449108101809804435888681546220650096895197184], N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[(N[Abs[b], $MachinePrecision] / N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$5 * 2), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(N[Abs[N[(t$95$6 - t$95$2), $MachinePrecision]], $MachinePrecision] + N[(t$95$2 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$5), $MachinePrecision] / t$95$4), $MachinePrecision] * N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 48000000000000001987463513459421542544804992225473439482941353636669046511737368705419469347170308598973901747830710504197303976962715123653608443215872], N[(1/4 * N[(N[Abs[b], $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(8 * N[(N[(N[(1/2 - N[(N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 1/90), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[Power[N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 1/180), $MachinePrecision]], $MachinePrecision], 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2 * t$95$7), $MachinePrecision] * N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[t$95$1], $MachinePrecision] + t$95$1), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$7), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.6000000000000001e-55

   1. Initial program 2.6%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.1%

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

   6. Applied rewrites 3.7%

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

   8. Applied rewrites 7.7%

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

   if 1.6000000000000001e-55 < a < 4.8000000000000002e151

   1. Initial program 2.6%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 1.6%

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

   9. Applied rewrites 3.8%

      \[\leadsto \frac{1}{4} \cdot \left(b \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\left(\frac{1}{2} - \cos \left(\left(\pi \cdot angle\right) \cdot \frac{1}{90}\right) \cdot \frac{1}{2}\right) + \sqrt{{\sin \left(\left(\pi \cdot angle\right) \cdot \frac{1}{180}\right)}^{4}}\right) \cdot {a}^{4}\right)}}{\left|x-scale\right|} \cdot \left(x-scale \cdot x-scale\right)}{\color{blue}{a \cdot a}}\right) \]

   if 4.8000000000000002e151 < a

   1. Initial program 2.6%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 4.2%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 4.2%

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 4.2%

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

   6. Applied rewrites 4.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 6.4%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 6.4%

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 6.4%

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

   8. Applied rewrites 6.4%

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

   10. Applied rewrites 8.4%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 6.1%

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

   14. Taylor expanded in angle around 0

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

   16. Applied rewrites 6.1%

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

   17. Taylor expanded in angle around 0

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

   19. Applied rewrites 5.9%

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

   20. Taylor expanded in angle around 0

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

   22. Applied rewrites 5.9%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|\left(\frac{1}{2} - 1 \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot 1\right) \cdot a\right) \cdot a\right| + \left(\left(\frac{1}{2} - 1 \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot 1\right) \cdot a\right) \cdot a\right)}{y-scale}}{y-scale}}}{\frac{\left(a \cdot b\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 9.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot \frac{\left|b\right|}{x-scale \cdot x-scale}\\ t_1 := \left(-a\right) \cdot \left|b\right|\\ t_2 := \frac{\left(a \cdot \left|b\right|\right) \cdot 4}{y-scale \cdot x-scale} \cdot \frac{t\_1}{y-scale \cdot x-scale}\\ t_3 := \left|b\right| \cdot a\\ t_4 := 4 \cdot t\_3\\ t_5 := \left(\frac{1}{2} - 1 \cdot \frac{1}{2}\right) \cdot \left(\left|b\right| \cdot \left|b\right|\right) + \left(\left(\frac{1}{2} - \frac{-1}{2} \cdot 1\right) \cdot a\right) \cdot a\\ t_6 := a \cdot \frac{a}{y-scale \cdot y-scale}\\ \mathbf{if}\;\left|b\right| \leq \frac{3022314549036573}{4722366482869645213696}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot \left(t\_3 \cdot \left(\left|b\right| \cdot \left(-a\right)\right)\right)\right) \cdot \frac{\frac{\left|t\_5\right| + t\_5}{y-scale}}{y-scale}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(\left(\left(\left(\frac{\left|b\right|}{\left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot \left(-a\right)\right) \cdot \left(t\_4 \cdot 2\right)\right) \cdot \left(t\_3 \cdot \left|b\right|\right)\right) \cdot \left(-a\right)\right) \cdot \left(\left|t\_6 - t\_0\right| + \left(t\_0 + t\_6\right)\right)}}{t\_4}}{t\_1} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (fabs b) (/ (fabs b) (* x-scale x-scale))))
       (t_1 (* (- a) (fabs b)))
       (t_2
        (*
         (/ (* (* a (fabs b)) 4) (* y-scale x-scale))
         (/ t_1 (* y-scale x-scale))))
       (t_3 (* (fabs b) a))
       (t_4 (* 4 t_3))
       (t_5
        (+
         (* (- 1/2 (* 1 1/2)) (* (fabs b) (fabs b)))
         (* (* (- 1/2 (* -1/2 1)) a) a)))
       (t_6 (* a (/ a (* y-scale y-scale)))))
  (if (<= (fabs b) 3022314549036573/4722366482869645213696)
    (/
     (-
      (sqrt
       (*
        (* (* 2 t_2) (* t_3 (* (fabs b) (- a))))
        (/ (/ (+ (fabs t_5) t_5) y-scale) y-scale))))
     t_2)
    (*
     (/
      (/
       (-
        (sqrt
         (*
          (*
           (*
            (*
             (*
              (/ (fabs b) (* (* (* x-scale y-scale) x-scale) y-scale))
              (- a))
             (* t_4 2))
            (* t_3 (fabs b)))
           (- a))
          (+ (fabs (- t_6 t_0)) (+ t_0 t_6)))))
       t_4)
      t_1)
     (* (* (* y-scale x-scale) x-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) * (fabs(b) / (x_45_scale * x_45_scale));
	double t_1 = -a * fabs(b);
	double t_2 = (((a * fabs(b)) * 4.0) / (y_45_scale * x_45_scale)) * (t_1 / (y_45_scale * x_45_scale));
	double t_3 = fabs(b) * a;
	double t_4 = 4.0 * t_3;
	double t_5 = ((0.5 - (1.0 * 0.5)) * (fabs(b) * fabs(b))) + (((0.5 - (-0.5 * 1.0)) * a) * a);
	double t_6 = a * (a / (y_45_scale * y_45_scale));
	double tmp;
	if (fabs(b) <= 6.4e-7) {
		tmp = -sqrt((((2.0 * t_2) * (t_3 * (fabs(b) * -a))) * (((fabs(t_5) + t_5) / y_45_scale) / y_45_scale))) / t_2;
	} else {
		tmp = ((-sqrt(((((((fabs(b) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * -a) * (t_4 * 2.0)) * (t_3 * fabs(b))) * -a) * (fabs((t_6 - t_0)) + (t_0 + t_6)))) / t_4) / t_1) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: t_6
    real(8) :: tmp
    t_0 = abs(b) * (abs(b) / (x_45scale * x_45scale))
    t_1 = -a * abs(b)
    t_2 = (((a * abs(b)) * 4.0d0) / (y_45scale * x_45scale)) * (t_1 / (y_45scale * x_45scale))
    t_3 = abs(b) * a
    t_4 = 4.0d0 * t_3
    t_5 = ((0.5d0 - (1.0d0 * 0.5d0)) * (abs(b) * abs(b))) + (((0.5d0 - ((-0.5d0) * 1.0d0)) * a) * a)
    t_6 = a * (a / (y_45scale * y_45scale))
    if (abs(b) <= 6.4d-7) then
        tmp = -sqrt((((2.0d0 * t_2) * (t_3 * (abs(b) * -a))) * (((abs(t_5) + t_5) / y_45scale) / y_45scale))) / t_2
    else
        tmp = ((-sqrt(((((((abs(b) / (((x_45scale * y_45scale) * x_45scale) * y_45scale)) * -a) * (t_4 * 2.0d0)) * (t_3 * abs(b))) * -a) * (abs((t_6 - t_0)) + (t_0 + t_6)))) / t_4) / t_1) * (((y_45scale * x_45scale) * x_45scale) * y_45scale)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.abs(b) * (Math.abs(b) / (x_45_scale * x_45_scale));
	double t_1 = -a * Math.abs(b);
	double t_2 = (((a * Math.abs(b)) * 4.0) / (y_45_scale * x_45_scale)) * (t_1 / (y_45_scale * x_45_scale));
	double t_3 = Math.abs(b) * a;
	double t_4 = 4.0 * t_3;
	double t_5 = ((0.5 - (1.0 * 0.5)) * (Math.abs(b) * Math.abs(b))) + (((0.5 - (-0.5 * 1.0)) * a) * a);
	double t_6 = a * (a / (y_45_scale * y_45_scale));
	double tmp;
	if (Math.abs(b) <= 6.4e-7) {
		tmp = -Math.sqrt((((2.0 * t_2) * (t_3 * (Math.abs(b) * -a))) * (((Math.abs(t_5) + t_5) / y_45_scale) / y_45_scale))) / t_2;
	} else {
		tmp = ((-Math.sqrt(((((((Math.abs(b) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * -a) * (t_4 * 2.0)) * (t_3 * Math.abs(b))) * -a) * (Math.abs((t_6 - t_0)) + (t_0 + t_6)))) / t_4) / t_1) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.fabs(b) * (math.fabs(b) / (x_45_scale * x_45_scale))
	t_1 = -a * math.fabs(b)
	t_2 = (((a * math.fabs(b)) * 4.0) / (y_45_scale * x_45_scale)) * (t_1 / (y_45_scale * x_45_scale))
	t_3 = math.fabs(b) * a
	t_4 = 4.0 * t_3
	t_5 = ((0.5 - (1.0 * 0.5)) * (math.fabs(b) * math.fabs(b))) + (((0.5 - (-0.5 * 1.0)) * a) * a)
	t_6 = a * (a / (y_45_scale * y_45_scale))
	tmp = 0
	if math.fabs(b) <= 6.4e-7:
		tmp = -math.sqrt((((2.0 * t_2) * (t_3 * (math.fabs(b) * -a))) * (((math.fabs(t_5) + t_5) / y_45_scale) / y_45_scale))) / t_2
	else:
		tmp = ((-math.sqrt(((((((math.fabs(b) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * -a) * (t_4 * 2.0)) * (t_3 * math.fabs(b))) * -a) * (math.fabs((t_6 - t_0)) + (t_0 + t_6)))) / t_4) / t_1) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) * Float64(abs(b) / Float64(x_45_scale * x_45_scale)))
	t_1 = Float64(Float64(-a) * abs(b))
	t_2 = Float64(Float64(Float64(Float64(a * abs(b)) * 4.0) / Float64(y_45_scale * x_45_scale)) * Float64(t_1 / Float64(y_45_scale * x_45_scale)))
	t_3 = Float64(abs(b) * a)
	t_4 = Float64(4.0 * t_3)
	t_5 = Float64(Float64(Float64(0.5 - Float64(1.0 * 0.5)) * Float64(abs(b) * abs(b))) + Float64(Float64(Float64(0.5 - Float64(-0.5 * 1.0)) * a) * a))
	t_6 = Float64(a * Float64(a / Float64(y_45_scale * y_45_scale)))
	tmp = 0.0
	if (abs(b) <= 6.4e-7)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * Float64(t_3 * Float64(abs(b) * Float64(-a)))) * Float64(Float64(Float64(abs(t_5) + t_5) / y_45_scale) / y_45_scale)))) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(Float64(abs(b) / Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * Float64(-a)) * Float64(t_4 * 2.0)) * Float64(t_3 * abs(b))) * Float64(-a)) * Float64(abs(Float64(t_6 - t_0)) + Float64(t_0 + t_6))))) / t_4) / t_1) * Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = abs(b) * (abs(b) / (x_45_scale * x_45_scale));
	t_1 = -a * abs(b);
	t_2 = (((a * abs(b)) * 4.0) / (y_45_scale * x_45_scale)) * (t_1 / (y_45_scale * x_45_scale));
	t_3 = abs(b) * a;
	t_4 = 4.0 * t_3;
	t_5 = ((0.5 - (1.0 * 0.5)) * (abs(b) * abs(b))) + (((0.5 - (-0.5 * 1.0)) * a) * a);
	t_6 = a * (a / (y_45_scale * y_45_scale));
	tmp = 0.0;
	if (abs(b) <= 6.4e-7)
		tmp = -sqrt((((2.0 * t_2) * (t_3 * (abs(b) * -a))) * (((abs(t_5) + t_5) / y_45_scale) / y_45_scale))) / t_2;
	else
		tmp = ((-sqrt(((((((abs(b) / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale)) * -a) * (t_4 * 2.0)) * (t_3 * abs(b))) * -a) * (abs((t_6 - t_0)) + (t_0 + t_6)))) / t_4) / t_1) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-a) * N[Abs[b], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision] * 4), $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$3 = N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$4 = N[(4 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(1/2 - N[(1 * 1/2), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1/2 - N[(-1/2 * 1), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(a * N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 3022314549036573/4722366482869645213696], N[((-N[Sqrt[N[(N[(N[(2 * t$95$2), $MachinePrecision] * N[(t$95$3 * N[(N[Abs[b], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Abs[t$95$5], $MachinePrecision] + t$95$5), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(N[(N[Abs[b], $MachinePrecision] / N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision] * N[(t$95$4 * 2), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision] * N[(N[Abs[N[(t$95$6 - t$95$0), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 + t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 6.4000000000000001e-7

   1. Initial program 2.6%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 4.2%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 4.2%

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 4.2%

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

   6. Applied rewrites 4.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lower-/.f64 6.4%

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f64 6.4%

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

      22. lift-*.f64 N/A

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

      23. *-commutative N/A

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

      24. lift-*.f64 6.4%

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

   8. Applied rewrites 6.4%

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

   10. Applied rewrites 8.4%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 6.1%

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

   14. Taylor expanded in angle around 0

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

   16. Applied rewrites 6.1%

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

   17. Taylor expanded in angle around 0

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

   19. Applied rewrites 5.9%

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

   20. Taylor expanded in angle around 0

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

   22. Applied rewrites 5.9%

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

   if 6.4000000000000001e-7 < b

   1. Initial program 2.6%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.1%

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

   6. Applied rewrites 3.7%

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

   8. Applied rewrites 7.7%

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

Alternative 6: 7.7% accurate, 12.0× speedup

Math

\[\begin{array}{l} t_0 := a \cdot \frac{a}{y-scale \cdot y-scale}\\ t_1 := b \cdot \frac{b}{x-scale \cdot x-scale}\\ t_2 := \left(-a\right) \cdot b\\ \frac{\frac{-\sqrt{\left(\left(\left(\left(8 \cdot \left(b \cdot a\right)\right) \cdot \left(-a\right)\right) \cdot \frac{b}{\left(\left(x-scale \cdot y-scale\right) \cdot x-scale\right) \cdot y-scale}\right) \cdot \left(\left(t\_2 \cdot b\right) \cdot a\right)\right) \cdot \left(\left|t\_0 - t\_1\right| + \left(t\_1 + t\_0\right)\right)}}{4 \cdot \left(b \cdot a\right)}}{t\_2} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right) \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* a (/ a (* y-scale y-scale))))
       (t_1 (* b (/ b (* x-scale x-scale))))
       (t_2 (* (- a) b)))
  (*
   (/
    (/
     (-
      (sqrt
       (*
        (*
         (*
          (* (* 8 (* b a)) (- a))
          (/ b (* (* (* x-scale y-scale) x-scale) y-scale)))
         (* (* t_2 b) a))
        (+ (fabs (- t_0 t_1)) (+ t_1 t_0)))))
     (* 4 (* b a)))
    t_2)
   (* (* (* y-scale x-scale) x-scale) y-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (a / (y_45_scale * y_45_scale));
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = -a * b;
	return ((-sqrt((((((8.0 * (b * a)) * -a) * (b / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale))) * ((t_2 * b) * a)) * (fabs((t_0 - t_1)) + (t_1 + t_0)))) / (4.0 * (b * a))) / t_2) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = a * (a / (y_45scale * y_45scale))
    t_1 = b * (b / (x_45scale * x_45scale))
    t_2 = -a * b
    code = ((-sqrt((((((8.0d0 * (b * a)) * -a) * (b / (((x_45scale * y_45scale) * x_45scale) * y_45scale))) * ((t_2 * b) * a)) * (abs((t_0 - t_1)) + (t_1 + t_0)))) / (4.0d0 * (b * a))) / t_2) * (((y_45scale * x_45scale) * x_45scale) * y_45scale)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (a / (y_45_scale * y_45_scale));
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = -a * b;
	return ((-Math.sqrt((((((8.0 * (b * a)) * -a) * (b / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale))) * ((t_2 * b) * a)) * (Math.abs((t_0 - t_1)) + (t_1 + t_0)))) / (4.0 * (b * a))) / t_2) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * (a / (y_45_scale * y_45_scale))
	t_1 = b * (b / (x_45_scale * x_45_scale))
	t_2 = -a * b
	return ((-math.sqrt((((((8.0 * (b * a)) * -a) * (b / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale))) * ((t_2 * b) * a)) * (math.fabs((t_0 - t_1)) + (t_1 + t_0)))) / (4.0 * (b * a))) / t_2) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a * Float64(a / Float64(y_45_scale * y_45_scale)))
	t_1 = Float64(b * Float64(b / Float64(x_45_scale * x_45_scale)))
	t_2 = Float64(Float64(-a) * b)
	return Float64(Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(Float64(8.0 * Float64(b * a)) * Float64(-a)) * Float64(b / Float64(Float64(Float64(x_45_scale * y_45_scale) * x_45_scale) * y_45_scale))) * Float64(Float64(t_2 * b) * a)) * Float64(abs(Float64(t_0 - t_1)) + Float64(t_1 + t_0))))) / Float64(4.0 * Float64(b * a))) / t_2) * Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * (a / (y_45_scale * y_45_scale));
	t_1 = b * (b / (x_45_scale * x_45_scale));
	t_2 = -a * b;
	tmp = ((-sqrt((((((8.0 * (b * a)) * -a) * (b / (((x_45_scale * y_45_scale) * x_45_scale) * y_45_scale))) * ((t_2 * b) * a)) * (abs((t_0 - t_1)) + (t_1 + t_0)))) / (4.0 * (b * a))) / t_2) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-a) * b), $MachinePrecision]}, N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(8 * N[(b * a), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision] * N[(b / N[(N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$2 * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(4 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 2.6%

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

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 4.1%

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

6. Applied rewrites 3.7%

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

7. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 3.7%

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

9. Applied rewrites 3.7%

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

11. Applied rewrites 7.7%

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

Alternative 7: 4.5% accurate, 12.3× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* a (/ a (* y-scale y-scale))))
       (t_1 (* b (/ b (* x-scale x-scale))))
       (t_2 (* (* x-scale y-scale) x-scale))
       (t_3 (* (- a) b)))
  (*
   (*
    (/
     (-
      (sqrt
       (*
        (*
         (* (* (* 8 (* b a)) (- a)) (/ b (* t_2 y-scale)))
         (* (* t_3 b) a))
        (+ (fabs (- t_0 t_1)) (+ t_1 t_0)))))
     (* (* t_3 4) (* b a)))
    t_2)
   y-scale)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    t_0 = a * (a / (y_45scale * y_45scale))
    t_1 = b * (b / (x_45scale * x_45scale))
    t_2 = (x_45scale * y_45scale) * x_45scale
    t_3 = -a * b
    code = ((-sqrt((((((8.0d0 * (b * a)) * -a) * (b / (t_2 * y_45scale))) * ((t_3 * b) * a)) * (abs((t_0 - t_1)) + (t_1 + t_0)))) / ((t_3 * 4.0d0) * (b * a))) * t_2) * y_45scale
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a * (a / (y_45_scale * y_45_scale));
	double t_1 = b * (b / (x_45_scale * x_45_scale));
	double t_2 = (x_45_scale * y_45_scale) * x_45_scale;
	double t_3 = -a * b;
	return ((-Math.sqrt((((((8.0 * (b * a)) * -a) * (b / (t_2 * y_45_scale))) * ((t_3 * b) * a)) * (Math.abs((t_0 - t_1)) + (t_1 + t_0)))) / ((t_3 * 4.0) * (b * a))) * t_2) * y_45_scale;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = a * (a / (y_45_scale * y_45_scale))
	t_1 = b * (b / (x_45_scale * x_45_scale))
	t_2 = (x_45_scale * y_45_scale) * x_45_scale
	t_3 = -a * b
	return ((-math.sqrt((((((8.0 * (b * a)) * -a) * (b / (t_2 * y_45_scale))) * ((t_3 * b) * a)) * (math.fabs((t_0 - t_1)) + (t_1 + t_0)))) / ((t_3 * 4.0) * (b * a))) * t_2) * y_45_scale

Julia

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

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = a * (a / (y_45_scale * y_45_scale));
	t_1 = b * (b / (x_45_scale * x_45_scale));
	t_2 = (x_45_scale * y_45_scale) * x_45_scale;
	t_3 = -a * b;
	tmp = ((-sqrt((((((8.0 * (b * a)) * -a) * (b / (t_2 * y_45_scale))) * ((t_3 * b) * a)) * (abs((t_0 - t_1)) + (t_1 + t_0)))) / ((t_3 * 4.0) * (b * a))) * t_2) * y_45_scale;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a * N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$3 = N[((-a) * b), $MachinePrecision]}, N[(N[(N[((-N[Sqrt[N[(N[(N[(N[(N[(8 * N[(b * a), $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision] * N[(b / N[(t$95$2 * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] + N[(t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(t$95$3 * 4), $MachinePrecision] * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * y$45$scale), $MachinePrecision]]]]]

Derivation

1. Initial program 2.6%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.1%

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

6. Applied rewrites 3.7%

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

7. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 3.7%

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

9. Applied rewrites 3.7%

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

11. Applied rewrites 4.5%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2 (/ (* 4 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180) PI))) 2) (pow (* b (cos (* (/ angle 180) PI))) 2)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180) PI))) 2) (pow (* b (sin (* (/ angle 180) PI))) 2)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180) PI))) 2) (pow (* b (cos (* (/ angle 180) PI))) 2)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180) PI))) 2) (pow (* b (sin (* (/ angle 180) PI))) 2)) y-scale) y-scale)) 2) (pow (/ (/ (* (* (* 2 (- (pow b 2) (pow a 2))) (sin (* (/ angle 180) PI))) (cos (* (/ angle 180) PI))) x-scale) y-scale) 2))))))) (/ (* 4 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2))))