a from scale-rotated-ellipse

Percentage Accurate: 2.5% → 10.8%

Time: 40.6s

Alternatives: 13

Speedup: 6.6×

• 30.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 10.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := 2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\\ t_2 := \cos t\_0\\ t_3 := \frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\\ t_4 := \sin t\_0\\ t_5 := 0.5 \cdot \cos t\_1\\ t_6 := 0.5 - t\_5\\ t_7 := 0.5 + t\_5\\ \frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot t\_4\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_4\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin t\_1}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(t\_6, b \cdot b, t\_7 \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(t\_6, a \cdot a, t\_7 \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{t\_3} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (* 2.0 (* PI (* 0.005555555555555556 angle))))
        (t_2 (cos t_0))
        (t_3
         (*
          (/ (* 4.0 (* a b)) (* y-scale x-scale))
          (/ (* (- a) b) (* y-scale x-scale))))
        (t_4 (sin t_0))
        (t_5 (* 0.5 (cos t_1)))
        (t_6 (- 0.5 t_5))
        (t_7 (+ 0.5 t_5)))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_3) (* (* b a) (* b (- a))))
       (+
        (+
         (/ (/ (+ (pow (* a t_4) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale)
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_4) 2.0)) y-scale) y-scale))
        (hypot
         (/ (* (* (+ b a) (- b a)) (sin t_1)) (* y-scale x-scale))
         (-
          (/ (fma t_6 (* b b) (* t_7 (* a a))) (* y-scale y-scale))
          (/ (fma t_6 (* a a) (* t_7 (* b b))) (* x-scale x-scale))))))))
    t_3)))

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 = 2.0 * (((double) M_PI) * (0.005555555555555556 * angle));
	double t_2 = cos(t_0);
	double t_3 = ((4.0 * (a * b)) / (y_45_scale * x_45_scale)) * ((-a * b) / (y_45_scale * x_45_scale));
	double t_4 = sin(t_0);
	double t_5 = 0.5 * cos(t_1);
	double t_6 = 0.5 - t_5;
	double t_7 = 0.5 + t_5;
	return -sqrt((((2.0 * t_3) * ((b * a) * (b * -a))) * (((((pow((a * t_4), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale) + (((pow((a * t_2), 2.0) + pow((b * t_4), 2.0)) / y_45_scale) / y_45_scale)) + hypot(((((b + a) * (b - a)) * sin(t_1)) / (y_45_scale * x_45_scale)), ((fma(t_6, (b * b), (t_7 * (a * a))) / (y_45_scale * y_45_scale)) - (fma(t_6, (a * a), (t_7 * (b * b))) / (x_45_scale * x_45_scale))))))) / t_3;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * angle)))
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(4.0 * Float64(a * b)) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(Float64(-a) * b) / Float64(y_45_scale * x_45_scale)))
	t_4 = sin(t_0)
	t_5 = Float64(0.5 * cos(t_1))
	t_6 = Float64(0.5 - t_5)
	t_7 = Float64(0.5 + t_5)
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * Float64(Float64(Float64(Float64(Float64((Float64(a * t_4) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale) + Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_4) ^ 2.0)) / y_45_scale) / y_45_scale)) + hypot(Float64(Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(t_1)) / Float64(y_45_scale * x_45_scale)), Float64(Float64(fma(t_6, Float64(b * b), Float64(t_7 * Float64(a * a))) / Float64(y_45_scale * y_45_scale)) - Float64(fma(t_6, Float64(a * a), Float64(t_7 * Float64(b * b))) / Float64(x_45_scale * x_45_scale)))))))) / t_3)
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(0.5 - t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(0.5 + t$95$5), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$3), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Power[N[(a * t$95$4), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] + N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[(N[(t$95$6 * N[(b * b), $MachinePrecision] + N[(t$95$7 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$6 * N[(a * a), $MachinePrecision] + N[(t$95$7 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Initial program 2.5%

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

3. Applied rewrites 6.0%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \color{blue}{\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 7.3

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{x-scale \cdot y-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 7.3

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{x-scale \cdot y-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lift-*.f64 7.3

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{\color{blue}{y-scale \cdot x-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

5. Applied rewrites 7.3%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 10.8

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{x-scale \cdot y-scale}}} \]

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 10.8

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{x-scale \cdot y-scale}} \]

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lift-*.f64 10.8

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{\color{blue}{y-scale \cdot x-scale}}} \]

7. Applied rewrites 10.8%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\color{blue}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}}} \]
8. Add Preprocessing

Alternative 2: 10.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + 0.5\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5, a \cdot a, \left(0.5 + 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{t\_3} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (*
          (/ (* 4.0 (* a b)) (* y-scale x-scale))
          (/ (* (- a) b) (* y-scale x-scale)))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_3) (* (* b a) (* b (- a))))
       (+
        (+
         (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale)
         (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale))
        (hypot
         (/
          (*
           (* (+ b a) (- b a))
           (sin (* 2.0 (* PI (* 0.005555555555555556 angle)))))
          (* y-scale x-scale))
         (-
          (/
           (fma (- 0.5 0.5) (* b b) (* (+ 0.5 0.5) (* a a)))
           (* y-scale y-scale))
          (/
           (fma (- 0.5 0.5) (* a a) (* (+ 0.5 0.5) (* b b)))
           (* x-scale x-scale))))))))
    t_3)))

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

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(4.0 * Float64(a * b)) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(Float64(-a) * b) / Float64(y_45_scale * x_45_scale)))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * Float64(Float64(Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale) + Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)) + hypot(Float64(Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * angle))))) / Float64(y_45_scale * x_45_scale)), Float64(Float64(fma(Float64(0.5 - 0.5), Float64(b * b), Float64(Float64(0.5 + 0.5) * Float64(a * a))) / Float64(y_45_scale * y_45_scale)) - Float64(fma(Float64(0.5 - 0.5), Float64(a * a), Float64(Float64(0.5 + 0.5) * Float64(b * b))) / Float64(x_45_scale * x_45_scale)))))))) / t_3)
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$3), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] + N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(0.5 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.5 - 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]]]]]

Derivation

1. Initial program 2.5%

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

3. Applied rewrites 6.0%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \color{blue}{\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 7.3

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{x-scale \cdot y-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 7.3

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{x-scale \cdot y-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lift-*.f64 7.3

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{\color{blue}{y-scale \cdot x-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

5. Applied rewrites 7.3%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 10.8

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{x-scale \cdot y-scale}}} \]

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 10.8

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{x-scale \cdot y-scale}} \]

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lift-*.f64 10.8

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{\color{blue}{y-scale \cdot x-scale}}} \]

7. Applied rewrites 10.8%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\color{blue}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}}} \]

8. Taylor expanded in angle around 0

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

10. Applied rewrites 10.7%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - \color{blue}{0.5}, b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

11. Taylor expanded in angle around 0

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

13. Applied rewrites 10.7%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + \color{blue}{0.5}\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

14. Taylor expanded in angle around 0

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

16. Applied rewrites 10.7%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + 0.5\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - \color{blue}{0.5}, a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

17. Taylor expanded in angle around 0

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

19. Applied rewrites 10.7%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + 0.5\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5, a \cdot a, \left(0.5 + \color{blue}{0.5}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
20. Add Preprocessing

Alternative 3: 10.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + 0.5\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5, a \cdot a, \left(0.5 + 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1
         (*
          (/ (* 4.0 (* a b)) (* y-scale x-scale))
          (/ (* (- a) b) (* y-scale x-scale)))))
   (/
    (-
     (sqrt
      (*
       (* (* 2.0 t_1) (* (* b a) (* b (- a))))
       (+
        (+
         (/
          (/ (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0)) x-scale)
          x-scale)
         (/ (/ (pow a 2.0) y-scale) y-scale))
        (hypot
         (/
          (*
           (* (+ b a) (- b a))
           (sin (* 2.0 (* PI (* 0.005555555555555556 angle)))))
          (* y-scale x-scale))
         (-
          (/
           (fma (- 0.5 0.5) (* b b) (* (+ 0.5 0.5) (* a a)))
           (* y-scale y-scale))
          (/
           (fma (- 0.5 0.5) (* a a) (* (+ 0.5 0.5) (* b b)))
           (* x-scale x-scale))))))))
    t_1)))

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 = ((4.0 * (a * b)) / (y_45_scale * x_45_scale)) * ((-a * b) / (y_45_scale * x_45_scale));
	return -sqrt((((2.0 * t_1) * ((b * a) * (b * -a))) * (((((pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0)) / x_45_scale) / x_45_scale) + ((pow(a, 2.0) / y_45_scale) / y_45_scale)) + hypot(((((b + a) * (b - a)) * sin((2.0 * (((double) M_PI) * (0.005555555555555556 * angle))))) / (y_45_scale * x_45_scale)), ((fma((0.5 - 0.5), (b * b), ((0.5 + 0.5) * (a * a))) / (y_45_scale * y_45_scale)) - (fma((0.5 - 0.5), (a * a), ((0.5 + 0.5) * (b * b))) / (x_45_scale * x_45_scale))))))) / t_1;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = Float64(Float64(Float64(4.0 * Float64(a * b)) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(Float64(-a) * b) / Float64(y_45_scale * x_45_scale)))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * Float64(Float64(Float64(Float64(Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0)) / x_45_scale) / x_45_scale) + Float64(Float64((a ^ 2.0) / y_45_scale) / y_45_scale)) + hypot(Float64(Float64(Float64(Float64(b + a) * Float64(b - a)) * sin(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * angle))))) / Float64(y_45_scale * x_45_scale)), Float64(Float64(fma(Float64(0.5 - 0.5), Float64(b * b), Float64(Float64(0.5 + 0.5) * Float64(a * a))) / Float64(y_45_scale * y_45_scale)) - Float64(fma(Float64(0.5 - 0.5), Float64(a * a), Float64(Float64(0.5 + 0.5) * Float64(b * b))) / Float64(x_45_scale * x_45_scale)))))))) / t_1)
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] + N[(N[(N[Power[a, 2.0], $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[(N[(N[(b + a), $MachinePrecision] * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(2.0 * N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(N[(N[(N[(0.5 - 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(N[(0.5 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.5 - 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(0.5 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]

Derivation

1. Initial program 2.5%

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

3. Applied rewrites 6.0%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \color{blue}{\mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 7.3

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{x-scale \cdot y-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 7.3

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{x-scale \cdot y-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lift-*.f64 7.3

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{\color{blue}{y-scale \cdot x-scale}}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

5. Applied rewrites 7.3%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lower-/.f64 10.8

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \color{blue}{\frac{b \cdot \left(-a\right)}{x-scale \cdot y-scale}}} \]

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 10.8

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\color{blue}{\left(-a\right) \cdot b}}{x-scale \cdot y-scale}} \]

   21. lift-*.f64 N/A

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

   22. *-commutative N/A

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

   23. lift-*.f64 10.8

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{\color{blue}{y-scale \cdot x-scale}}} \]

7. Applied rewrites 10.8%

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\color{blue}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}}} \]

8. Taylor expanded in angle around 0

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

10. Applied rewrites 10.7%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - \color{blue}{0.5}, b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

11. Taylor expanded in angle around 0

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

13. Applied rewrites 10.7%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + \color{blue}{0.5}\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

14. Taylor expanded in angle around 0

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

16. Applied rewrites 10.7%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + 0.5\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - \color{blue}{0.5}, a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

17. Taylor expanded in angle around 0

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

19. Applied rewrites 10.7%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + 0.5\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5, a \cdot a, \left(0.5 + \color{blue}{0.5}\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

20. Taylor expanded in angle around 0

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

    1. lower-/.f64 N/A

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

    2. lower-pow.f64 10.7

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + 0.5\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5, a \cdot a, \left(0.5 + 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]

22. Applied rewrites 10.7%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale}\right) + \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5, b \cdot b, \left(0.5 + 0.5\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5, a \cdot a, \left(0.5 + 0.5\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale}\right)\right)}}{\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}} \]
23. Add Preprocessing

Alternative 4: 8.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := \left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)\\ t_2 := \frac{a}{y-scale \cdot y-scale}\\ t_3 := \left(\pi \cdot angle\right) \cdot 0.011111111111111112\\ t_4 := \cos t\_3\\ t_5 := \frac{t\_4}{x-scale \cdot x-scale}\\ t_6 := \left(\left(-a\right) \cdot b\right) \cdot b\\ t_7 := t\_6 \cdot a\\ t_8 := \frac{0.5}{x-scale \cdot x-scale}\\ \mathbf{if}\;a \leq 1.15 \cdot 10^{-104}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(b, t\_0, \mathsf{fma}\left(a, t\_2, \left|a \cdot t\_2 - b \cdot t\_0\right|\right)\right) \cdot \left(\frac{t\_7}{t\_1} \cdot 8\right)\right) \cdot t\_7}}{\left(\left(\left(b \cdot a\right) \cdot 4\right) \cdot a\right) \cdot b} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\sqrt{\left(t\_6 \cdot \left(a \cdot \left(\left(\left(\left(\mathsf{hypot}\left(\frac{\sin t\_3}{x-scale \cdot y-scale}, t\_8 - \mathsf{fma}\left(0.5, t\_5 + \frac{t\_4}{y-scale \cdot y-scale}, \frac{0.5}{y-scale \cdot y-scale}\right)\right) + t\_8\right) + \frac{\mathsf{fma}\left(t\_4, 0.5, 0.5\right)}{y-scale \cdot y-scale}\right) - t\_5 \cdot 0.5\right) \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(8 \cdot t\_7\right)}}{\left|x-scale \cdot y-scale\right|}}{\left(4 \cdot a\right) \cdot b}}{b \cdot a} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale)))
        (t_1 (* (* x-scale y-scale) (* x-scale y-scale)))
        (t_2 (/ a (* y-scale y-scale)))
        (t_3 (* (* PI angle) 0.011111111111111112))
        (t_4 (cos t_3))
        (t_5 (/ t_4 (* x-scale x-scale)))
        (t_6 (* (* (- a) b) b))
        (t_7 (* t_6 a))
        (t_8 (/ 0.5 (* x-scale x-scale))))
   (if (<= a 1.15e-104)
     (*
      (/
       (sqrt
        (*
         (*
          (fma b t_0 (fma a t_2 (fabs (- (* a t_2) (* b t_0)))))
          (* (/ t_7 t_1) 8.0))
         t_7))
       (* (* (* (* b a) 4.0) a) b))
      t_1)
     (*
      (/
       (/
        (/
         (sqrt
          (*
           (*
            t_6
            (*
             a
             (*
              (-
               (+
                (+
                 (hypot
                  (/ (sin t_3) (* x-scale y-scale))
                  (-
                   t_8
                   (fma
                    0.5
                    (+ t_5 (/ t_4 (* y-scale y-scale)))
                    (/ 0.5 (* y-scale y-scale)))))
                 t_8)
                (/ (fma t_4 0.5 0.5) (* y-scale y-scale)))
               (* t_5 0.5))
              (* a a))))
           (* 8.0 t_7)))
         (fabs (* x-scale y-scale)))
        (* (* 4.0 a) b))
       (* b a))
      (* (* (* 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 = b / (x_45_scale * x_45_scale);
	double t_1 = (x_45_scale * y_45_scale) * (x_45_scale * y_45_scale);
	double t_2 = a / (y_45_scale * y_45_scale);
	double t_3 = (((double) M_PI) * angle) * 0.011111111111111112;
	double t_4 = cos(t_3);
	double t_5 = t_4 / (x_45_scale * x_45_scale);
	double t_6 = (-a * b) * b;
	double t_7 = t_6 * a;
	double t_8 = 0.5 / (x_45_scale * x_45_scale);
	double tmp;
	if (a <= 1.15e-104) {
		tmp = (sqrt(((fma(b, t_0, fma(a, t_2, fabs(((a * t_2) - (b * t_0))))) * ((t_7 / t_1) * 8.0)) * t_7)) / ((((b * a) * 4.0) * a) * b)) * t_1;
	} else {
		tmp = (((sqrt(((t_6 * (a * ((((hypot((sin(t_3) / (x_45_scale * y_45_scale)), (t_8 - fma(0.5, (t_5 + (t_4 / (y_45_scale * y_45_scale))), (0.5 / (y_45_scale * y_45_scale))))) + t_8) + (fma(t_4, 0.5, 0.5) / (y_45_scale * y_45_scale))) - (t_5 * 0.5)) * (a * a)))) * (8.0 * t_7))) / fabs((x_45_scale * y_45_scale))) / ((4.0 * a) * b)) / (b * a)) * (((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(b / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(Float64(x_45_scale * y_45_scale) * Float64(x_45_scale * y_45_scale))
	t_2 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(Float64(pi * angle) * 0.011111111111111112)
	t_4 = cos(t_3)
	t_5 = Float64(t_4 / Float64(x_45_scale * x_45_scale))
	t_6 = Float64(Float64(Float64(-a) * b) * b)
	t_7 = Float64(t_6 * a)
	t_8 = Float64(0.5 / Float64(x_45_scale * x_45_scale))
	tmp = 0.0
	if (a <= 1.15e-104)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(b, t_0, fma(a, t_2, abs(Float64(Float64(a * t_2) - Float64(b * t_0))))) * Float64(Float64(t_7 / t_1) * 8.0)) * t_7)) / Float64(Float64(Float64(Float64(b * a) * 4.0) * a) * b)) * t_1);
	else
		tmp = Float64(Float64(Float64(Float64(sqrt(Float64(Float64(t_6 * Float64(a * Float64(Float64(Float64(Float64(hypot(Float64(sin(t_3) / Float64(x_45_scale * y_45_scale)), Float64(t_8 - fma(0.5, Float64(t_5 + Float64(t_4 / Float64(y_45_scale * y_45_scale))), Float64(0.5 / Float64(y_45_scale * y_45_scale))))) + t_8) + Float64(fma(t_4, 0.5, 0.5) / Float64(y_45_scale * y_45_scale))) - Float64(t_5 * 0.5)) * Float64(a * a)))) * Float64(8.0 * t_7))) / abs(Float64(x_45_scale * y_45_scale))) / Float64(Float64(4.0 * a) * b)) / Float64(b * a)) * Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$45$scale * y$45$scale), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 * a), $MachinePrecision]}, Block[{t$95$8 = N[(0.5 / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 1.15e-104], N[(N[(N[Sqrt[N[(N[(N[(b * t$95$0 + N[(a * t$95$2 + N[Abs[N[(N[(a * t$95$2), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$7 / t$95$1), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] * t$95$7), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[(N[(N[Sqrt[N[(N[(t$95$6 * N[(a * N[(N[(N[(N[(N[Sqrt[N[(N[Sin[t$95$3], $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(t$95$8 - N[(0.5 * N[(t$95$5 + N[(t$95$4 / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] + t$95$8), $MachinePrecision] + N[(N[(t$95$4 * 0.5 + 0.5), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$5 * 0.5), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(8.0 * t$95$7), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(x$45$scale * y$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $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 a < 1.15e-104

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 1.3%

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

   8. Applied rewrites 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-pow.f64 N/A

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

      6. associate-*l* N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. pow-prod-down N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lower-*.f64 4.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 4.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 4.2

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

   10. Applied rewrites 4.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-pow.f64 N/A

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

       6. associate-*l* N/A

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. pow-prod-down N/A

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

       10. lift-*.f64 N/A

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

       11. pow2 N/A

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

       12. lower-*.f64 5.8

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

       13. lift-*.f64 N/A

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

       14. *-commutative N/A

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

       15. lift-*.f64 5.8

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

       16. lift-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lift-*.f64 5.8

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

   12. Applied rewrites 5.8%

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

   if 1.15e-104 < a

   1. Initial program 2.5%

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

   3. Applied rewrites 4.7%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 3.9%

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

   8. Applied rewrites 8.9%

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

Alternative 5: 7.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := \frac{1}{{x-scale}^{2}}\\ t_2 := \frac{1}{{y-scale}^{2}}\\ t_3 := \left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ t_4 := \frac{b}{x-scale \cdot y-scale}\\ t_5 := \left(\left(-a\right) \cdot t\_4\right) \cdot \left(a \cdot t\_4\right)\\ t_6 := \frac{a}{y-scale \cdot y-scale}\\ t_7 := \frac{a \cdot a}{y-scale \cdot y-scale}\\ \mathbf{if}\;x-scale \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(b, t\_0, \mathsf{fma}\left(a, t\_6, \left|a \cdot t\_6 - b \cdot t\_0\right|\right)\right) \cdot \left(\frac{t\_3}{x-scale \cdot y-scale} \cdot \frac{8}{x-scale \cdot y-scale}\right)\right) \cdot t\_3}}{\left(\left(\left(b \cdot a\right) \cdot 4\right) \cdot a\right) \cdot b} \cdot \left(\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale\right)\\ \mathbf{elif}\;x-scale \leq 5 \cdot 10^{+219}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, t\_7\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - t\_7\right|\right) \cdot \left(\left(t\_5 \cdot 8\right) \cdot \left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{-4 \cdot t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(8 \cdot t\_3\right) \cdot \left(t\_3 \cdot \left({a}^{2} \cdot \left(\sqrt{{\left(0.5 \cdot t\_1 - \mathsf{fma}\left(0.5, t\_1, t\_2\right)\right)}^{2}} + t\_2\right)\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale)))
        (t_1 (/ 1.0 (pow x-scale 2.0)))
        (t_2 (/ 1.0 (pow y-scale 2.0)))
        (t_3 (* (* (* (- a) b) b) a))
        (t_4 (/ b (* x-scale y-scale)))
        (t_5 (* (* (- a) t_4) (* a t_4)))
        (t_6 (/ a (* y-scale y-scale)))
        (t_7 (/ (* a a) (* y-scale y-scale))))
   (if (<= x-scale 5e-105)
     (*
      (/
       (sqrt
        (*
         (*
          (fma b t_0 (fma a t_6 (fabs (- (* a t_6) (* b t_0)))))
          (* (/ t_3 (* x-scale y-scale)) (/ 8.0 (* x-scale y-scale))))
         t_3))
       (* (* (* (* b a) 4.0) a) b))
      (* (* (* y-scale y-scale) x-scale) x-scale))
     (if (<= x-scale 5e+219)
       (/
        (sqrt
         (*
          (+
           (fma (/ b x-scale) (/ b x-scale) t_7)
           (fabs (- (/ (* b b) (* x-scale x-scale)) t_7)))
          (* (* t_5 8.0) (* (* (* a b) b) (- a)))))
        (* -4.0 t_5))
       (*
        (/
         (/
          (sqrt
           (*
            (* 8.0 t_3)
            (*
             t_3
             (*
              (pow a 2.0)
              (+ (sqrt (pow (- (* 0.5 t_1) (fma 0.5 t_1 t_2)) 2.0)) t_2)))))
          (fabs (* y-scale x-scale)))
         (* (* (* a b) 4.0) (* a b)))
        (* (* (* 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 = b / (x_45_scale * x_45_scale);
	double t_1 = 1.0 / pow(x_45_scale, 2.0);
	double t_2 = 1.0 / pow(y_45_scale, 2.0);
	double t_3 = ((-a * b) * b) * a;
	double t_4 = b / (x_45_scale * y_45_scale);
	double t_5 = (-a * t_4) * (a * t_4);
	double t_6 = a / (y_45_scale * y_45_scale);
	double t_7 = (a * a) / (y_45_scale * y_45_scale);
	double tmp;
	if (x_45_scale <= 5e-105) {
		tmp = (sqrt(((fma(b, t_0, fma(a, t_6, fabs(((a * t_6) - (b * t_0))))) * ((t_3 / (x_45_scale * y_45_scale)) * (8.0 / (x_45_scale * y_45_scale)))) * t_3)) / ((((b * a) * 4.0) * a) * b)) * (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale);
	} else if (x_45_scale <= 5e+219) {
		tmp = sqrt(((fma((b / x_45_scale), (b / x_45_scale), t_7) + fabs((((b * b) / (x_45_scale * x_45_scale)) - t_7))) * ((t_5 * 8.0) * (((a * b) * b) * -a)))) / (-4.0 * t_5);
	} else {
		tmp = ((sqrt(((8.0 * t_3) * (t_3 * (pow(a, 2.0) * (sqrt(pow(((0.5 * t_1) - fma(0.5, t_1, t_2)), 2.0)) + t_2))))) / fabs((y_45_scale * x_45_scale))) / (((a * b) * 4.0) * (a * b))) * (((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(b / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(1.0 / (x_45_scale ^ 2.0))
	t_2 = Float64(1.0 / (y_45_scale ^ 2.0))
	t_3 = Float64(Float64(Float64(Float64(-a) * b) * b) * a)
	t_4 = Float64(b / Float64(x_45_scale * y_45_scale))
	t_5 = Float64(Float64(Float64(-a) * t_4) * Float64(a * t_4))
	t_6 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_7 = Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale))
	tmp = 0.0
	if (x_45_scale <= 5e-105)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(b, t_0, fma(a, t_6, abs(Float64(Float64(a * t_6) - Float64(b * t_0))))) * Float64(Float64(t_3 / Float64(x_45_scale * y_45_scale)) * Float64(8.0 / Float64(x_45_scale * y_45_scale)))) * t_3)) / Float64(Float64(Float64(Float64(b * a) * 4.0) * a) * b)) * Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale));
	elseif (x_45_scale <= 5e+219)
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(b / x_45_scale), Float64(b / x_45_scale), t_7) + abs(Float64(Float64(Float64(b * b) / Float64(x_45_scale * x_45_scale)) - t_7))) * Float64(Float64(t_5 * 8.0) * Float64(Float64(Float64(a * b) * b) * Float64(-a))))) / Float64(-4.0 * t_5));
	else
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * t_3) * Float64(t_3 * Float64((a ^ 2.0) * Float64(sqrt((Float64(Float64(0.5 * t_1) - fma(0.5, t_1, t_2)) ^ 2.0)) + t_2))))) / abs(Float64(y_45_scale * x_45_scale))) / Float64(Float64(Float64(a * b) * 4.0) * Float64(a * b))) * Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$4 = N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[((-a) * t$95$4), $MachinePrecision] * N[(a * t$95$4), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale, 5e-105], N[(N[(N[Sqrt[N[(N[(N[(b * t$95$0 + N[(a * t$95$6 + N[Abs[N[(N[(a * t$95$6), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(8.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$45$scale, 5e+219], N[(N[Sqrt[N[(N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision] + t$95$7), $MachinePrecision] + N[Abs[N[(N[(N[(b * b), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - t$95$7), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$5 * 8.0), $MachinePrecision] * N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(-4.0 * t$95$5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sqrt[N[(N[(8.0 * t$95$3), $MachinePrecision] * N[(t$95$3 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[Sqrt[N[Power[N[(N[(0.5 * t$95$1), $MachinePrecision] - N[(0.5 * t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if x-scale < 4.99999999999999963e-105

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 1.3%

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

   8. Applied rewrites 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 5.1%

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

   if 4.99999999999999963e-105 < x-scale < 5e219

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 1.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 2.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 7.6%

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

   if 5e219 < x-scale

   1. Initial program 2.5%

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

   3. Applied rewrites 4.7%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 3.9%

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

   7. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

   9. Applied rewrites 4.1%

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

Alternative 6: 6.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{x-scale \cdot y-scale}\\ t_1 := \left(\left(-a\right) \cdot t\_0\right) \cdot \left(a \cdot t\_0\right)\\ t_2 := \frac{a \cdot a}{y-scale \cdot y-scale}\\ t_3 := \left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ t_4 := \frac{1}{{x-scale}^{2}}\\ t_5 := \mathsf{fma}\left(0.5, t\_4, \frac{1}{{y-scale}^{2}}\right)\\ \mathbf{if}\;b \leq 2.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{\sqrt{\left(8 \cdot t\_3\right) \cdot \left(t\_3 \cdot \left({a}^{2} \cdot \left(\left(\sqrt{{\left(0.5 \cdot t\_4 - t\_5\right)}^{2}} + t\_5\right) - 0.5 \cdot \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{x-scale}^{2}}\right)\right)\right)}}{\left|y-scale \cdot x-scale\right|}}{\left(\left(a \cdot b\right) \cdot 4\right) \cdot \left(a \cdot b\right)} \cdot \left(\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, t\_2\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - t\_2\right|\right) \cdot \left(\left(t\_1 \cdot 8\right) \cdot \left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{-4 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale y-scale)))
        (t_1 (* (* (- a) t_0) (* a t_0)))
        (t_2 (/ (* a a) (* y-scale y-scale)))
        (t_3 (* (* (* (- a) b) b) a))
        (t_4 (/ 1.0 (pow x-scale 2.0)))
        (t_5 (fma 0.5 t_4 (/ 1.0 (pow y-scale 2.0)))))
   (if (<= b 2.8e-32)
     (*
      (/
       (/
        (sqrt
         (*
          (* 8.0 t_3)
          (*
           t_3
           (*
            (pow a 2.0)
            (-
             (+ (sqrt (pow (- (* 0.5 t_4) t_5) 2.0)) t_5)
             (*
              0.5
              (/
               (cos (* 0.011111111111111112 (* angle PI)))
               (pow x-scale 2.0))))))))
        (fabs (* y-scale x-scale)))
       (* (* (* a b) 4.0) (* a b)))
      (* (* (* y-scale x-scale) x-scale) y-scale))
     (/
      (sqrt
       (*
        (+
         (fma (/ b x-scale) (/ b x-scale) t_2)
         (fabs (- (/ (* b b) (* x-scale x-scale)) t_2)))
        (* (* t_1 8.0) (* (* (* a b) b) (- a)))))
      (* -4.0 t_1)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * y_45_scale);
	double t_1 = (-a * t_0) * (a * t_0);
	double t_2 = (a * a) / (y_45_scale * y_45_scale);
	double t_3 = ((-a * b) * b) * a;
	double t_4 = 1.0 / pow(x_45_scale, 2.0);
	double t_5 = fma(0.5, t_4, (1.0 / pow(y_45_scale, 2.0)));
	double tmp;
	if (b <= 2.8e-32) {
		tmp = ((sqrt(((8.0 * t_3) * (t_3 * (pow(a, 2.0) * ((sqrt(pow(((0.5 * t_4) - t_5), 2.0)) + t_5) - (0.5 * (cos((0.011111111111111112 * (angle * ((double) M_PI)))) / pow(x_45_scale, 2.0)))))))) / fabs((y_45_scale * x_45_scale))) / (((a * b) * 4.0) * (a * b))) * (((y_45_scale * x_45_scale) * x_45_scale) * y_45_scale);
	} else {
		tmp = sqrt(((fma((b / x_45_scale), (b / x_45_scale), t_2) + fabs((((b * b) / (x_45_scale * x_45_scale)) - t_2))) * ((t_1 * 8.0) * (((a * b) * b) * -a)))) / (-4.0 * t_1);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * y_45_scale))
	t_1 = Float64(Float64(Float64(-a) * t_0) * Float64(a * t_0))
	t_2 = Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(Float64(Float64(Float64(-a) * b) * b) * a)
	t_4 = Float64(1.0 / (x_45_scale ^ 2.0))
	t_5 = fma(0.5, t_4, Float64(1.0 / (y_45_scale ^ 2.0)))
	tmp = 0.0
	if (b <= 2.8e-32)
		tmp = Float64(Float64(Float64(sqrt(Float64(Float64(8.0 * t_3) * Float64(t_3 * Float64((a ^ 2.0) * Float64(Float64(sqrt((Float64(Float64(0.5 * t_4) - t_5) ^ 2.0)) + t_5) - Float64(0.5 * Float64(cos(Float64(0.011111111111111112 * Float64(angle * pi))) / (x_45_scale ^ 2.0)))))))) / abs(Float64(y_45_scale * x_45_scale))) / Float64(Float64(Float64(a * b) * 4.0) * Float64(a * b))) * Float64(Float64(Float64(y_45_scale * x_45_scale) * x_45_scale) * y_45_scale));
	else
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(b / x_45_scale), Float64(b / x_45_scale), t_2) + abs(Float64(Float64(Float64(b * b) / Float64(x_45_scale * x_45_scale)) - t_2))) * Float64(Float64(t_1 * 8.0) * Float64(Float64(Float64(a * b) * b) * Float64(-a))))) / Float64(-4.0 * t_1));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-a) * t$95$0), $MachinePrecision] * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 * t$95$4 + N[(1.0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.8e-32], N[(N[(N[(N[Sqrt[N[(N[(8.0 * t$95$3), $MachinePrecision] * N[(t$95$3 * N[(N[Power[a, 2.0], $MachinePrecision] * N[(N[(N[Sqrt[N[Power[N[(N[(0.5 * t$95$4), $MachinePrecision] - t$95$5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + t$95$5), $MachinePrecision] - N[(0.5 * N[(N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[x$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(y$45$scale * x$45$scale), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(a * b), $MachinePrecision] * 4.0), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision] + t$95$2), $MachinePrecision] + N[Abs[N[(N[(N[(b * b), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * 8.0), $MachinePrecision] * N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(-4.0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.7999999999999999e-32

   1. Initial program 2.5%

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

   3. Applied rewrites 4.7%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 3.9%

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

   7. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

   9. Applied rewrites 5.0%

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

   if 2.7999999999999999e-32 < b

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 1.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 2.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 7.6%

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

Alternative 7: 6.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{a \cdot a}{y-scale \cdot y-scale}\\ t_1 := \frac{a}{y-scale \cdot y-scale}\\ t_2 := \frac{b}{x-scale \cdot y-scale}\\ t_3 := \left(\left(-a\right) \cdot t\_2\right) \cdot \left(a \cdot t\_2\right)\\ t_4 := \left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ t_5 := \frac{b}{x-scale \cdot x-scale}\\ \mathbf{if}\;b \leq 7.8 \cdot 10^{-123}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(b, t\_5, \mathsf{fma}\left(a, t\_1, \left|a \cdot t\_1 - b \cdot t\_5\right|\right)\right) \cdot \left(\frac{t\_4}{x-scale \cdot y-scale} \cdot \frac{8}{x-scale \cdot y-scale}\right)\right) \cdot t\_4}}{\left(\left(\left(b \cdot a\right) \cdot 4\right) \cdot a\right) \cdot b} \cdot \left(\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, t\_0\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - t\_0\right|\right) \cdot \left(\left(t\_3 \cdot 8\right) \cdot \left(\left(\left(a \cdot b\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{-4 \cdot t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* a a) (* y-scale y-scale)))
        (t_1 (/ a (* y-scale y-scale)))
        (t_2 (/ b (* x-scale y-scale)))
        (t_3 (* (* (- a) t_2) (* a t_2)))
        (t_4 (* (* (* (- a) b) b) a))
        (t_5 (/ b (* x-scale x-scale))))
   (if (<= b 7.8e-123)
     (*
      (/
       (sqrt
        (*
         (*
          (fma b t_5 (fma a t_1 (fabs (- (* a t_1) (* b t_5)))))
          (* (/ t_4 (* x-scale y-scale)) (/ 8.0 (* x-scale y-scale))))
         t_4))
       (* (* (* (* b a) 4.0) a) b))
      (* (* (* y-scale y-scale) x-scale) x-scale))
     (/
      (sqrt
       (*
        (+
         (fma (/ b x-scale) (/ b x-scale) t_0)
         (fabs (- (/ (* b b) (* x-scale x-scale)) t_0)))
        (* (* t_3 8.0) (* (* (* a b) b) (- a)))))
      (* -4.0 t_3)))))

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 = a / (y_45_scale * y_45_scale);
	double t_2 = b / (x_45_scale * y_45_scale);
	double t_3 = (-a * t_2) * (a * t_2);
	double t_4 = ((-a * b) * b) * a;
	double t_5 = b / (x_45_scale * x_45_scale);
	double tmp;
	if (b <= 7.8e-123) {
		tmp = (sqrt(((fma(b, t_5, fma(a, t_1, fabs(((a * t_1) - (b * t_5))))) * ((t_4 / (x_45_scale * y_45_scale)) * (8.0 / (x_45_scale * y_45_scale)))) * t_4)) / ((((b * a) * 4.0) * a) * b)) * (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale);
	} else {
		tmp = sqrt(((fma((b / x_45_scale), (b / x_45_scale), t_0) + fabs((((b * b) / (x_45_scale * x_45_scale)) - t_0))) * ((t_3 * 8.0) * (((a * b) * b) * -a)))) / (-4.0 * t_3);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(a * a) / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_2 = Float64(b / Float64(x_45_scale * y_45_scale))
	t_3 = Float64(Float64(Float64(-a) * t_2) * Float64(a * t_2))
	t_4 = Float64(Float64(Float64(Float64(-a) * b) * b) * a)
	t_5 = Float64(b / Float64(x_45_scale * x_45_scale))
	tmp = 0.0
	if (b <= 7.8e-123)
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(b, t_5, fma(a, t_1, abs(Float64(Float64(a * t_1) - Float64(b * t_5))))) * Float64(Float64(t_4 / Float64(x_45_scale * y_45_scale)) * Float64(8.0 / Float64(x_45_scale * y_45_scale)))) * t_4)) / Float64(Float64(Float64(Float64(b * a) * 4.0) * a) * b)) * Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale));
	else
		tmp = Float64(sqrt(Float64(Float64(fma(Float64(b / x_45_scale), Float64(b / x_45_scale), t_0) + abs(Float64(Float64(Float64(b * b) / Float64(x_45_scale * x_45_scale)) - t_0))) * Float64(Float64(t_3 * 8.0) * Float64(Float64(Float64(a * b) * b) * Float64(-a))))) / Float64(-4.0 * t_3));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[((-a) * t$95$2), $MachinePrecision] * N[(a * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$5 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 7.8e-123], N[(N[(N[Sqrt[N[(N[(N[(b * t$95$5 + N[(a * t$95$1 + N[Abs[N[(N[(a * t$95$1), $MachinePrecision] - N[(b * t$95$5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$4 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(8.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(b / x$45$scale), $MachinePrecision] * N[(b / x$45$scale), $MachinePrecision] + t$95$0), $MachinePrecision] + N[Abs[N[(N[(N[(b * b), $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$3 * 8.0), $MachinePrecision] * N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(-4.0 * t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 7.79999999999999952e-123

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 1.3%

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

   8. Applied rewrites 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 5.1%

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

   if 7.79999999999999952e-123 < b

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 1.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 N/A

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

   8. Applied rewrites 2.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. unswap-sqr N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. times-frac N/A

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

      15. lift-/.f64 N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 7.6%

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

Alternative 8: 6.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := \left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale\\ t_2 := \frac{a}{y-scale \cdot y-scale}\\ t_3 := \frac{b}{x-scale \cdot y-scale}\\ t_4 := \left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ t_5 := \mathsf{fma}\left(b, t\_0, \mathsf{fma}\left(a, t\_2, \left|a \cdot t\_2 - b \cdot t\_0\right|\right)\right)\\ \mathbf{if}\;y-scale \leq 8.2 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(t\_5 \cdot \left(\frac{t\_4}{t\_1} \cdot 8\right)\right) \cdot t\_4}}{\left(4 \cdot a\right) \cdot t\_3}}{\left(-a\right) \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(t\_5 \cdot \left(\frac{t\_4}{x-scale \cdot y-scale} \cdot \frac{8}{x-scale \cdot y-scale}\right)\right) \cdot t\_4}}{\left(\left(\left(b \cdot a\right) \cdot 4\right) \cdot a\right) \cdot b} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale)))
        (t_1 (* (* (* y-scale y-scale) x-scale) x-scale))
        (t_2 (/ a (* y-scale y-scale)))
        (t_3 (/ b (* x-scale y-scale)))
        (t_4 (* (* (* (- a) b) b) a))
        (t_5 (fma b t_0 (fma a t_2 (fabs (- (* a t_2) (* b t_0)))))))
   (if (<= y-scale 8.2e-100)
     (/
      (/ (- (sqrt (* (* t_5 (* (/ t_4 t_1) 8.0)) t_4))) (* (* 4.0 a) t_3))
      (* (- a) t_3))
     (*
      (/
       (sqrt
        (*
         (* t_5 (* (/ t_4 (* x-scale y-scale)) (/ 8.0 (* x-scale y-scale))))
         t_4))
       (* (* (* (* b a) 4.0) a) b))
      t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * x_45_scale);
	double t_1 = ((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale;
	double t_2 = a / (y_45_scale * y_45_scale);
	double t_3 = b / (x_45_scale * y_45_scale);
	double t_4 = ((-a * b) * b) * a;
	double t_5 = fma(b, t_0, fma(a, t_2, fabs(((a * t_2) - (b * t_0)))));
	double tmp;
	if (y_45_scale <= 8.2e-100) {
		tmp = (-sqrt(((t_5 * ((t_4 / t_1) * 8.0)) * t_4)) / ((4.0 * a) * t_3)) / (-a * t_3);
	} else {
		tmp = (sqrt(((t_5 * ((t_4 / (x_45_scale * y_45_scale)) * (8.0 / (x_45_scale * y_45_scale)))) * t_4)) / ((((b * a) * 4.0) * a) * b)) * t_1;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)
	t_2 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(b / Float64(x_45_scale * y_45_scale))
	t_4 = Float64(Float64(Float64(Float64(-a) * b) * b) * a)
	t_5 = fma(b, t_0, fma(a, t_2, abs(Float64(Float64(a * t_2) - Float64(b * t_0)))))
	tmp = 0.0
	if (y_45_scale <= 8.2e-100)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(t_5 * Float64(Float64(t_4 / t_1) * 8.0)) * t_4))) / Float64(Float64(4.0 * a) * t_3)) / Float64(Float64(-a) * t_3));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(t_5 * Float64(Float64(t_4 / Float64(x_45_scale * y_45_scale)) * Float64(8.0 / Float64(x_45_scale * y_45_scale)))) * t_4)) / Float64(Float64(Float64(Float64(b * a) * 4.0) * a) * b)) * t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$5 = N[(b * t$95$0 + N[(a * t$95$2 + N[Abs[N[(N[(a * t$95$2), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 8.2e-100], N[(N[((-N[Sqrt[N[(N[(t$95$5 * N[(N[(t$95$4 / t$95$1), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * a), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[((-a) * t$95$3), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(t$95$5 * N[(N[(t$95$4 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(8.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 8.1999999999999998e-100

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 1.3%

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

   8. Applied rewrites 7.2%

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

   if 8.1999999999999998e-100 < y-scale

   1. Initial program 2.5%

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

   2. Taylor expanded in 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 4.4%

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

   6. Applied rewrites 1.3%

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

   8. Applied rewrites 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 5.1%

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

Alternative 9: 5.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := \left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale\\ t_2 := \left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ t_3 := \frac{a}{y-scale \cdot y-scale}\\ t_4 := \mathsf{fma}\left(b, t\_0, \mathsf{fma}\left(a, t\_3, \left|a \cdot t\_3 - b \cdot t\_0\right|\right)\right)\\ t_5 := \left(b \cdot a\right) \cdot 4\\ \mathbf{if}\;y-scale \leq 7 \cdot 10^{-149}:\\ \;\;\;\;\frac{-\sqrt{\left(t\_4 \cdot \left(\frac{t\_2}{t\_1} \cdot 8\right)\right) \cdot t\_2}}{t\_5 \cdot \left(\left(-a\right) \cdot \frac{b}{x-scale \cdot y-scale}\right)} \cdot \left(x-scale \cdot y-scale\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(t\_4 \cdot \left(\frac{t\_2}{x-scale \cdot y-scale} \cdot \frac{8}{x-scale \cdot y-scale}\right)\right) \cdot t\_2}}{\left(t\_5 \cdot a\right) \cdot b} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale)))
        (t_1 (* (* (* y-scale y-scale) x-scale) x-scale))
        (t_2 (* (* (* (- a) b) b) a))
        (t_3 (/ a (* y-scale y-scale)))
        (t_4 (fma b t_0 (fma a t_3 (fabs (- (* a t_3) (* b t_0))))))
        (t_5 (* (* b a) 4.0)))
   (if (<= y-scale 7e-149)
     (*
      (/
       (- (sqrt (* (* t_4 (* (/ t_2 t_1) 8.0)) t_2)))
       (* t_5 (* (- a) (/ b (* x-scale y-scale)))))
      (* x-scale y-scale))
     (*
      (/
       (sqrt
        (*
         (* t_4 (* (/ t_2 (* x-scale y-scale)) (/ 8.0 (* x-scale y-scale))))
         t_2))
       (* (* t_5 a) b))
      t_1))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * x_45_scale);
	double t_1 = ((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale;
	double t_2 = ((-a * b) * b) * a;
	double t_3 = a / (y_45_scale * y_45_scale);
	double t_4 = fma(b, t_0, fma(a, t_3, fabs(((a * t_3) - (b * t_0)))));
	double t_5 = (b * a) * 4.0;
	double tmp;
	if (y_45_scale <= 7e-149) {
		tmp = (-sqrt(((t_4 * ((t_2 / t_1) * 8.0)) * t_2)) / (t_5 * (-a * (b / (x_45_scale * y_45_scale))))) * (x_45_scale * y_45_scale);
	} else {
		tmp = (sqrt(((t_4 * ((t_2 / (x_45_scale * y_45_scale)) * (8.0 / (x_45_scale * y_45_scale)))) * t_2)) / ((t_5 * a) * b)) * t_1;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale)
	t_2 = Float64(Float64(Float64(Float64(-a) * b) * b) * a)
	t_3 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_4 = fma(b, t_0, fma(a, t_3, abs(Float64(Float64(a * t_3) - Float64(b * t_0)))))
	t_5 = Float64(Float64(b * a) * 4.0)
	tmp = 0.0
	if (y_45_scale <= 7e-149)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(t_4 * Float64(Float64(t_2 / t_1) * 8.0)) * t_2))) / Float64(t_5 * Float64(Float64(-a) * Float64(b / Float64(x_45_scale * y_45_scale))))) * Float64(x_45_scale * y_45_scale));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(t_4 * Float64(Float64(t_2 / Float64(x_45_scale * y_45_scale)) * Float64(8.0 / Float64(x_45_scale * y_45_scale)))) * t_2)) / Float64(Float64(t_5 * a) * b)) * t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$3 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(b * t$95$0 + N[(a * t$95$3 + N[Abs[N[(N[(a * t$95$3), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[y$45$scale, 7e-149], N[(N[((-N[Sqrt[N[(N[(t$95$4 * N[(N[(t$95$2 / t$95$1), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]) / N[(t$95$5 * N[((-a) * N[(b / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(t$95$4 * N[(N[(t$95$2 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(8.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / N[(N[(t$95$5 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 7e-149

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.4%

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

   6. Applied rewrites 1.3%

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

   8. Applied rewrites 4.6%

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

   if 7e-149 < y-scale

   1. Initial program 2.5%

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

   2. Taylor expanded in 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 4.4%

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

   6. Applied rewrites 1.3%

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

   8. Applied rewrites 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 5.1%

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

Alternative 10: 5.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := \left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a\\ t_2 := \frac{a}{y-scale \cdot y-scale}\\ \mathbf{if}\;y-scale \leq 5 \cdot 10^{-150}:\\ \;\;\;\;\frac{-\sqrt{\frac{2 \cdot \left(t\_1 \cdot 4\right)}{\left(y-scale \cdot \left(x-scale \cdot y-scale\right)\right) \cdot x-scale} \cdot \left(t\_1 \cdot \frac{\mathsf{fma}\left(a, a, \sqrt{{a}^{4}}\right)}{y-scale \cdot y-scale}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(b, t\_0, \mathsf{fma}\left(a, t\_2, \left|a \cdot t\_2 - b \cdot t\_0\right|\right)\right) \cdot \left(\frac{t\_1}{x-scale \cdot y-scale} \cdot \frac{8}{x-scale \cdot y-scale}\right)\right) \cdot t\_1}}{\left(\left(\left(b \cdot a\right) \cdot 4\right) \cdot a\right) \cdot b} \cdot \left(\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale)))
        (t_1 (* (* (* (- a) b) b) a))
        (t_2 (/ a (* y-scale y-scale))))
   (if (<= y-scale 5e-150)
     (/
      (-
       (sqrt
        (*
         (/ (* 2.0 (* t_1 4.0)) (* (* y-scale (* x-scale y-scale)) x-scale))
         (* t_1 (/ (fma a a (sqrt (pow a 4.0))) (* y-scale y-scale))))))
      (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0)))
     (*
      (/
       (sqrt
        (*
         (*
          (fma b t_0 (fma a t_2 (fabs (- (* a t_2) (* b t_0)))))
          (* (/ t_1 (* x-scale y-scale)) (/ 8.0 (* x-scale y-scale))))
         t_1))
       (* (* (* (* b a) 4.0) a) b))
      (* (* (* y-scale y-scale) x-scale) x-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * x_45_scale);
	double t_1 = ((-a * b) * b) * a;
	double t_2 = a / (y_45_scale * y_45_scale);
	double tmp;
	if (y_45_scale <= 5e-150) {
		tmp = -sqrt((((2.0 * (t_1 * 4.0)) / ((y_45_scale * (x_45_scale * y_45_scale)) * x_45_scale)) * (t_1 * (fma(a, a, sqrt(pow(a, 4.0))) / (y_45_scale * y_45_scale))))) / ((4.0 * ((b * a) * (b * -a))) / pow((x_45_scale * y_45_scale), 2.0));
	} else {
		tmp = (sqrt(((fma(b, t_0, fma(a, t_2, fabs(((a * t_2) - (b * t_0))))) * ((t_1 / (x_45_scale * y_45_scale)) * (8.0 / (x_45_scale * y_45_scale)))) * t_1)) / ((((b * a) * 4.0) * a) * b)) * (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(Float64(Float64(Float64(-a) * b) * b) * a)
	t_2 = Float64(a / Float64(y_45_scale * y_45_scale))
	tmp = 0.0
	if (y_45_scale <= 5e-150)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * Float64(t_1 * 4.0)) / Float64(Float64(y_45_scale * Float64(x_45_scale * y_45_scale)) * x_45_scale)) * Float64(t_1 * Float64(fma(a, a, sqrt((a ^ 4.0))) / Float64(y_45_scale * y_45_scale)))))) / Float64(Float64(4.0 * Float64(Float64(b * a) * Float64(b * Float64(-a)))) / (Float64(x_45_scale * y_45_scale) ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(fma(b, t_0, fma(a, t_2, abs(Float64(Float64(a * t_2) - Float64(b * t_0))))) * Float64(Float64(t_1 / Float64(x_45_scale * y_45_scale)) * Float64(8.0 / Float64(x_45_scale * y_45_scale)))) * t_1)) / Float64(Float64(Float64(Float64(b * a) * 4.0) * a) * b)) * Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[((-a) * b), $MachinePrecision] * b), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$45$scale, 5e-150], N[((-N[Sqrt[N[(N[(N[(2.0 * N[(t$95$1 * 4.0), $MachinePrecision]), $MachinePrecision] / N[(N[(y$45$scale * N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(a * a + N[Sqrt[N[Power[a, 4.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(4.0 * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[(b * t$95$0 + N[(a * t$95$2 + N[Abs[N[(N[(a * t$95$2), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(8.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(N[(b * a), $MachinePrecision] * 4.0), $MachinePrecision] * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y-scale < 4.9999999999999999e-150

   1. Initial program 2.5%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 4.4%

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

   5. Taylor expanded in y-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-pow.f64 2.4

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

   7. Applied rewrites 2.4%

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

   9. Applied rewrites 2.5%

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

   if 4.9999999999999999e-150 < y-scale

   1. Initial program 2.5%

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

   2. Taylor expanded in 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 4.4%

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

   6. Applied rewrites 1.3%

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

   8. Applied rewrites 2.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. pow-prod-down N/A

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

      13. lift-*.f64 N/A

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

      14. pow2 N/A

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

      15. times-frac N/A

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

      16. lower-*.f64 N/A

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

   10. Applied rewrites 5.1%

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

Alternative 11: 4.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.5%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.4%

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

6. Applied rewrites 1.3%

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

8. Applied rewrites 2.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-pow.f64 N/A

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

   8. pow2 N/A

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

   9. pow-prod-down N/A

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

   10. lift-*.f64 N/A

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

   11. pow2 N/A

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

   12. lower-*.f64 4.2

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f64 4.2

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 4.2

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

10. Applied rewrites 4.2%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. pow2 N/A

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

    5. lift-pow.f64 N/A

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

    6. associate-*l* N/A

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

    7. lift-pow.f64 N/A

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

    8. pow2 N/A

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

    9. pow-prod-down N/A

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

    10. lift-*.f64 N/A

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

    11. pow2 N/A

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

    12. lower-*.f64 5.8

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

    13. lift-*.f64 N/A

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

    14. *-commutative N/A

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

    15. lift-*.f64 5.8

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

    16. lift-*.f64 N/A

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

    17. *-commutative N/A

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

    18. lift-*.f64 5.8

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

12. Applied rewrites 5.8%

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

Alternative 12: 4.4% accurate, 6.6× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* y-scale (* x-scale y-scale)))
        (t_1 (/ b (* x-scale x-scale)))
        (t_2 (* (* (- a) b) b)))
   (*
    (*
     (/
      (sqrt
       (*
        (*
         (* (* t_2 (/ a (* t_0 x-scale))) 8.0)
         (fma
          t_1
          b
          (fma
           (/ a (* y-scale y-scale))
           a
           (fabs (- (* t_1 b) (/ (* a a) (* y-scale y-scale)))))))
        (* t_2 a)))
      (* (* b a) (* (* 4.0 a) b)))
     t_0)
    x-scale)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.5%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.4%

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

6. Applied rewrites 1.3%

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

8. Applied rewrites 2.4%

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

10. Applied rewrites 4.7%

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

Alternative 13: 2.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 2.5%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 4.4%

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

5. Taylor expanded in y-scale around 0

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-pow.f64 2.4

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

7. Applied rewrites 2.4%

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

9. Applied rewrites 2.2%

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

Reproduce

herbie shell --seed 2025150 
(FPCore (a b angle x-scale y-scale)
  :name "a from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2.0 (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))) (* (* b a) (* b (- a)))) (+ (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0))))))) (/ (* 4.0 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2.0))))