a from scale-rotated-ellipse

Percentage Accurate: 2.7% → 10.8%

Time: 29.5s

Alternatives: 10

Speedup: 6.6×

• 30.2% 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: 2.7% 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, 2.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(a\_m \cdot b\_m\right)}{y-scale \cdot x-scale\_m} \cdot \frac{\left(-a\_m\right) \cdot b\_m}{y-scale \cdot x-scale\_m}\\ t_1 := \mathsf{fma}\left({a\_m}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\_m\right)}^{2}\right)\\ t_2 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_3 := \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;a\_m \leq 4.55 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_0\right) \cdot \left(\left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\right)\right) \cdot \frac{\sqrt{{t\_1}^{2}} + t\_1}{{y-scale}^{2}}}}{t\_0}\\ \mathbf{elif}\;a\_m \leq 6.6 \cdot 10^{+149}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({a\_m}^{4} \cdot \left(\left(0.5 + \sqrt{{\left(t\_2 - 0.5\right)}^{2}}\right) - t\_2\right)\right)}\right)}{{a\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\left(\left(x-scale\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left({a\_m}^{4} \cdot \frac{\left|0.5 \cdot \frac{t\_3 - 1}{y-scale \cdot y-scale}\right| + \frac{0.5 - 0.5 \cdot t\_3}{y-scale \cdot y-scale}}{y-scale \cdot y-scale}\right) \cdot 8}\right) \cdot b\_m}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (/ (* 4.0 (* a_m b_m)) (* y-scale x-scale_m))
          (/ (* (- a_m) b_m) (* y-scale x-scale_m))))
        (t_1
         (fma
          (pow a_m 2.0)
          (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0)
          (pow (* (sin (* (* PI angle) 0.005555555555555556)) b_m) 2.0)))
        (t_2 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
        (t_3 (cos (* 0.011111111111111112 (* PI angle)))))
   (if (<= a_m 4.55e-97)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_0) (* (* b_m a_m) (* b_m (- a_m))))
         (/ (+ (sqrt (pow t_1 2.0)) t_1) (pow y-scale 2.0)))))
      t_0)
     (if (<= a_m 6.6e+149)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (sqrt
            (*
             8.0
             (* (pow a_m 4.0) (- (+ 0.5 (sqrt (pow (- t_2 0.5) 2.0))) t_2))))))
         (pow a_m 2.0)))
       (*
        (/ 0.25 a_m)
        (/
         (*
          (*
           (* x-scale_m (* y-scale y-scale))
           (sqrt
            (*
             (*
              (pow a_m 4.0)
              (/
               (+
                (fabs (* 0.5 (/ (- t_3 1.0) (* y-scale y-scale))))
                (/ (- 0.5 (* 0.5 t_3)) (* y-scale y-scale)))
               (* y-scale y-scale)))
             8.0)))
          b_m)
         a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	double t_1 = fma(pow(a_m, 2.0), pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0), pow((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * b_m), 2.0));
	double t_2 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_3 = cos((0.011111111111111112 * (((double) M_PI) * angle)));
	double tmp;
	if (a_m <= 4.55e-97) {
		tmp = -sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * ((sqrt(pow(t_1, 2.0)) + t_1) / pow(y_45_scale, 2.0)))) / t_0;
	} else if (a_m <= 6.6e+149) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * sqrt((8.0 * (pow(a_m, 4.0) * ((0.5 + sqrt(pow((t_2 - 0.5), 2.0))) - t_2)))))) / pow(a_m, 2.0));
	} else {
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * sqrt(((pow(a_m, 4.0) * ((fabs((0.5 * ((t_3 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_3)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(Float64(4.0 * Float64(a_m * b_m)) / Float64(y_45_scale * x_45_scale_m)) * Float64(Float64(Float64(-a_m) * b_m) / Float64(y_45_scale * x_45_scale_m)))
	t_1 = fma((a_m ^ 2.0), (cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0), (Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * b_m) ^ 2.0))
	t_2 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_3 = cos(Float64(0.011111111111111112 * Float64(pi * angle)))
	tmp = 0.0
	if (a_m <= 4.55e-97)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_0) * Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))) * Float64(Float64(sqrt((t_1 ^ 2.0)) + t_1) / (y_45_scale ^ 2.0))))) / t_0);
	elseif (a_m <= 6.6e+149)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((a_m ^ 4.0) * Float64(Float64(0.5 + sqrt((Float64(t_2 - 0.5) ^ 2.0))) - t_2)))))) / (a_m ^ 2.0)));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(x_45_scale_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64((a_m ^ 4.0) * Float64(Float64(abs(Float64(0.5 * Float64(Float64(t_3 - 1.0) / Float64(y_45_scale * y_45_scale)))) + Float64(Float64(0.5 - Float64(0.5 * t_3)) / Float64(y_45_scale * y_45_scale))) / Float64(y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(4.0 * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[((-a$95$m) * b$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 4.55e-97], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[a$95$m, 6.6e+149], N[(0.25 * N[(N[(b$95$m * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(0.5 + N[Sqrt[N[Power[N[(t$95$2 - 0.5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(x$45$scale$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(N[Abs[N[(0.5 * N[(N[(t$95$3 - 1.0), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.54999999999999999e-97

   1. Initial program 2.7%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 4.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 4.5

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

   6. Applied rewrites 4.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 6.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 6.7

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

   8. Applied rewrites 6.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lower-pow.f64 N/A

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

   10. Applied rewrites 6.7%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. *-commutative N/A

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

       13. metadata-eval N/A

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

       14. mult-flip N/A

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

       15. lower-pow.f64 N/A

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

   12. Applied rewrites 7.1%

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

   if 4.54999999999999999e-97 < a < 6.6e149

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   10. Taylor expanded in y-scale around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

   12. Applied rewrites 6.8%

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

   if 6.6e149 < a

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   11. Applied rewrites 4.8%

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

Alternative 2: 10.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(a\_m \cdot b\_m\right)}{y-scale \cdot x-scale\_m} \cdot \frac{\left(-a\_m\right) \cdot b\_m}{y-scale \cdot x-scale\_m}\\ t_1 := \mathsf{fma}\left({a\_m}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\left(0.005555555555555556 \cdot \left(angle \cdot \left(b\_m \cdot \pi\right)\right)\right)}^{2}\right)\\ t_2 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_3 := \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;a\_m \leq 4.55 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_0\right) \cdot \left(\left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\right)\right) \cdot \frac{\sqrt{{t\_1}^{2}} + t\_1}{{y-scale}^{2}}}}{t\_0}\\ \mathbf{elif}\;a\_m \leq 6.6 \cdot 10^{+149}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({a\_m}^{4} \cdot \left(\left(0.5 + \sqrt{{\left(t\_2 - 0.5\right)}^{2}}\right) - t\_2\right)\right)}\right)}{{a\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\left(\left(x-scale\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left({a\_m}^{4} \cdot \frac{\left|0.5 \cdot \frac{t\_3 - 1}{y-scale \cdot y-scale}\right| + \frac{0.5 - 0.5 \cdot t\_3}{y-scale \cdot y-scale}}{y-scale \cdot y-scale}\right) \cdot 8}\right) \cdot b\_m}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (/ (* 4.0 (* a_m b_m)) (* y-scale x-scale_m))
          (/ (* (- a_m) b_m) (* y-scale x-scale_m))))
        (t_1
         (fma
          (pow a_m 2.0)
          (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0)
          (pow (* 0.005555555555555556 (* angle (* b_m PI))) 2.0)))
        (t_2 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
        (t_3 (cos (* 0.011111111111111112 (* PI angle)))))
   (if (<= a_m 4.55e-97)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_0) (* (* b_m a_m) (* b_m (- a_m))))
         (/ (+ (sqrt (pow t_1 2.0)) t_1) (pow y-scale 2.0)))))
      t_0)
     (if (<= a_m 6.6e+149)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (sqrt
            (*
             8.0
             (* (pow a_m 4.0) (- (+ 0.5 (sqrt (pow (- t_2 0.5) 2.0))) t_2))))))
         (pow a_m 2.0)))
       (*
        (/ 0.25 a_m)
        (/
         (*
          (*
           (* x-scale_m (* y-scale y-scale))
           (sqrt
            (*
             (*
              (pow a_m 4.0)
              (/
               (+
                (fabs (* 0.5 (/ (- t_3 1.0) (* y-scale y-scale))))
                (/ (- 0.5 (* 0.5 t_3)) (* y-scale y-scale)))
               (* y-scale y-scale)))
             8.0)))
          b_m)
         a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	double t_1 = fma(pow(a_m, 2.0), pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0), pow((0.005555555555555556 * (angle * (b_m * ((double) M_PI)))), 2.0));
	double t_2 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_3 = cos((0.011111111111111112 * (((double) M_PI) * angle)));
	double tmp;
	if (a_m <= 4.55e-97) {
		tmp = -sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * ((sqrt(pow(t_1, 2.0)) + t_1) / pow(y_45_scale, 2.0)))) / t_0;
	} else if (a_m <= 6.6e+149) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * sqrt((8.0 * (pow(a_m, 4.0) * ((0.5 + sqrt(pow((t_2 - 0.5), 2.0))) - t_2)))))) / pow(a_m, 2.0));
	} else {
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * sqrt(((pow(a_m, 4.0) * ((fabs((0.5 * ((t_3 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_3)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(Float64(4.0 * Float64(a_m * b_m)) / Float64(y_45_scale * x_45_scale_m)) * Float64(Float64(Float64(-a_m) * b_m) / Float64(y_45_scale * x_45_scale_m)))
	t_1 = fma((a_m ^ 2.0), (cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0), (Float64(0.005555555555555556 * Float64(angle * Float64(b_m * pi))) ^ 2.0))
	t_2 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_3 = cos(Float64(0.011111111111111112 * Float64(pi * angle)))
	tmp = 0.0
	if (a_m <= 4.55e-97)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_0) * Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))) * Float64(Float64(sqrt((t_1 ^ 2.0)) + t_1) / (y_45_scale ^ 2.0))))) / t_0);
	elseif (a_m <= 6.6e+149)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((a_m ^ 4.0) * Float64(Float64(0.5 + sqrt((Float64(t_2 - 0.5) ^ 2.0))) - t_2)))))) / (a_m ^ 2.0)));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(x_45_scale_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64((a_m ^ 4.0) * Float64(Float64(abs(Float64(0.5 * Float64(Float64(t_3 - 1.0) / Float64(y_45_scale * y_45_scale)))) + Float64(Float64(0.5 - Float64(0.5 * t_3)) / Float64(y_45_scale * y_45_scale))) / Float64(y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(4.0 * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[((-a$95$m) * b$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(angle * N[(b$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 4.55e-97], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[a$95$m, 6.6e+149], N[(0.25 * N[(N[(b$95$m * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(0.5 + N[Sqrt[N[Power[N[(t$95$2 - 0.5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(x$45$scale$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(N[Abs[N[(0.5 * N[(N[(t$95$3 - 1.0), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.54999999999999999e-97

   1. Initial program 2.7%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 4.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 4.5

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

   6. Applied rewrites 4.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 6.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 6.7

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

   8. Applied rewrites 6.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lower-pow.f64 N/A

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

   10. Applied rewrites 6.7%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. *-commutative N/A

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

       13. metadata-eval N/A

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

       14. mult-flip N/A

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

       15. lower-pow.f64 N/A

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

   12. Applied rewrites 7.1%

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

   13. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 7.1

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

   15. Applied rewrites 7.1%

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

   16. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-PI.f64 7.1

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

   18. Applied rewrites 7.1%

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

   if 4.54999999999999999e-97 < a < 6.6e149

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   10. Taylor expanded in y-scale around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

   12. Applied rewrites 6.8%

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

   if 6.6e149 < a

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   11. Applied rewrites 4.8%

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

Alternative 3: 10.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := {b\_m}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ t_1 := \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\\ t_2 := \frac{4 \cdot \left(a\_m \cdot b\_m\right)}{y-scale \cdot x-scale\_m} \cdot \frac{\left(-a\_m\right) \cdot b\_m}{y-scale \cdot x-scale\_m}\\ t_3 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 4.55 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot \left(\left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\right)\right) \cdot \frac{\sqrt{{t\_0}^{2}} + t\_0}{{y-scale}^{2}}}}{t\_2}\\ \mathbf{elif}\;a\_m \leq 6.6 \cdot 10^{+149}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({a\_m}^{4} \cdot \left(\left(0.5 + \sqrt{{\left(t\_3 - 0.5\right)}^{2}}\right) - t\_3\right)\right)}\right)}{{a\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\left(\left(x-scale\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left({a\_m}^{4} \cdot \frac{\left|0.5 \cdot \frac{t\_1 - 1}{y-scale \cdot y-scale}\right| + \frac{0.5 - 0.5 \cdot t\_1}{y-scale \cdot y-scale}}{y-scale \cdot y-scale}\right) \cdot 8}\right) \cdot b\_m}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (pow b_m 2.0)
          (pow (sin (* 0.005555555555555556 (* angle PI))) 2.0)))
        (t_1 (cos (* 0.011111111111111112 (* PI angle))))
        (t_2
         (*
          (/ (* 4.0 (* a_m b_m)) (* y-scale x-scale_m))
          (/ (* (- a_m) b_m) (* y-scale x-scale_m))))
        (t_3 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
   (if (<= a_m 4.55e-97)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_2) (* (* b_m a_m) (* b_m (- a_m))))
         (/ (+ (sqrt (pow t_0 2.0)) t_0) (pow y-scale 2.0)))))
      t_2)
     (if (<= a_m 6.6e+149)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (sqrt
            (*
             8.0
             (* (pow a_m 4.0) (- (+ 0.5 (sqrt (pow (- t_3 0.5) 2.0))) t_3))))))
         (pow a_m 2.0)))
       (*
        (/ 0.25 a_m)
        (/
         (*
          (*
           (* x-scale_m (* y-scale y-scale))
           (sqrt
            (*
             (*
              (pow a_m 4.0)
              (/
               (+
                (fabs (* 0.5 (/ (- t_1 1.0) (* y-scale y-scale))))
                (/ (- 0.5 (* 0.5 t_1)) (* y-scale y-scale)))
               (* y-scale y-scale)))
             8.0)))
          b_m)
         a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = pow(b_m, 2.0) * pow(sin((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0);
	double t_1 = cos((0.011111111111111112 * (((double) M_PI) * angle)));
	double t_2 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	double t_3 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 4.55e-97) {
		tmp = -sqrt((((2.0 * t_2) * ((b_m * a_m) * (b_m * -a_m))) * ((sqrt(pow(t_0, 2.0)) + t_0) / pow(y_45_scale, 2.0)))) / t_2;
	} else if (a_m <= 6.6e+149) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * sqrt((8.0 * (pow(a_m, 4.0) * ((0.5 + sqrt(pow((t_3 - 0.5), 2.0))) - t_3)))))) / pow(a_m, 2.0));
	} else {
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * sqrt(((pow(a_m, 4.0) * ((fabs((0.5 * ((t_1 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_1)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = Math.pow(b_m, 2.0) * Math.pow(Math.sin((0.005555555555555556 * (angle * Math.PI))), 2.0);
	double t_1 = Math.cos((0.011111111111111112 * (Math.PI * angle)));
	double t_2 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	double t_3 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double tmp;
	if (a_m <= 4.55e-97) {
		tmp = -Math.sqrt((((2.0 * t_2) * ((b_m * a_m) * (b_m * -a_m))) * ((Math.sqrt(Math.pow(t_0, 2.0)) + t_0) / Math.pow(y_45_scale, 2.0)))) / t_2;
	} else if (a_m <= 6.6e+149) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * Math.sqrt((8.0 * (Math.pow(a_m, 4.0) * ((0.5 + Math.sqrt(Math.pow((t_3 - 0.5), 2.0))) - t_3)))))) / Math.pow(a_m, 2.0));
	} else {
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * Math.sqrt(((Math.pow(a_m, 4.0) * ((Math.abs((0.5 * ((t_1 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_1)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale):
	t_0 = math.pow(b_m, 2.0) * math.pow(math.sin((0.005555555555555556 * (angle * math.pi))), 2.0)
	t_1 = math.cos((0.011111111111111112 * (math.pi * angle)))
	t_2 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m))
	t_3 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	tmp = 0
	if a_m <= 4.55e-97:
		tmp = -math.sqrt((((2.0 * t_2) * ((b_m * a_m) * (b_m * -a_m))) * ((math.sqrt(math.pow(t_0, 2.0)) + t_0) / math.pow(y_45_scale, 2.0)))) / t_2
	elif a_m <= 6.6e+149:
		tmp = 0.25 * ((b_m * (x_45_scale_m * math.sqrt((8.0 * (math.pow(a_m, 4.0) * ((0.5 + math.sqrt(math.pow((t_3 - 0.5), 2.0))) - t_3)))))) / math.pow(a_m, 2.0))
	else:
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * math.sqrt(((math.pow(a_m, 4.0) * ((math.fabs((0.5 * ((t_1 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_1)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m)
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64((b_m ^ 2.0) * (sin(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0))
	t_1 = cos(Float64(0.011111111111111112 * Float64(pi * angle)))
	t_2 = Float64(Float64(Float64(4.0 * Float64(a_m * b_m)) / Float64(y_45_scale * x_45_scale_m)) * Float64(Float64(Float64(-a_m) * b_m) / Float64(y_45_scale * x_45_scale_m)))
	t_3 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (a_m <= 4.55e-97)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))) * Float64(Float64(sqrt((t_0 ^ 2.0)) + t_0) / (y_45_scale ^ 2.0))))) / t_2);
	elseif (a_m <= 6.6e+149)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((a_m ^ 4.0) * Float64(Float64(0.5 + sqrt((Float64(t_3 - 0.5) ^ 2.0))) - t_3)))))) / (a_m ^ 2.0)));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(x_45_scale_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64((a_m ^ 4.0) * Float64(Float64(abs(Float64(0.5 * Float64(Float64(t_1 - 1.0) / Float64(y_45_scale * y_45_scale)))) + Float64(Float64(0.5 - Float64(0.5 * t_1)) / Float64(y_45_scale * y_45_scale))) / Float64(y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = (b_m ^ 2.0) * (sin((0.005555555555555556 * (angle * pi))) ^ 2.0);
	t_1 = cos((0.011111111111111112 * (pi * angle)));
	t_2 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	t_3 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	tmp = 0.0;
	if (a_m <= 4.55e-97)
		tmp = -sqrt((((2.0 * t_2) * ((b_m * a_m) * (b_m * -a_m))) * ((sqrt((t_0 ^ 2.0)) + t_0) / (y_45_scale ^ 2.0)))) / t_2;
	elseif (a_m <= 6.6e+149)
		tmp = 0.25 * ((b_m * (x_45_scale_m * sqrt((8.0 * ((a_m ^ 4.0) * ((0.5 + sqrt(((t_3 - 0.5) ^ 2.0))) - t_3)))))) / (a_m ^ 2.0));
	else
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * sqrt((((a_m ^ 4.0) * ((abs((0.5 * ((t_1 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_1)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[Power[b$95$m, 2.0], $MachinePrecision] * N[Power[N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(4.0 * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[((-a$95$m) * b$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 4.55e-97], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[a$95$m, 6.6e+149], N[(0.25 * N[(N[(b$95$m * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(0.5 + N[Sqrt[N[Power[N[(t$95$3 - 0.5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(x$45$scale$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(N[Abs[N[(0.5 * N[(N[(t$95$1 - 1.0), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.54999999999999999e-97

   1. Initial program 2.7%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 4.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 4.5

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

   6. Applied rewrites 4.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 6.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 6.7

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

   8. Applied rewrites 6.7%

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

   9. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 6.5

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

   11. Applied rewrites 6.5%

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

   12. Taylor expanded in a around 0

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-PI.f64 4.0

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

   14. Applied rewrites 4.0%

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

   if 4.54999999999999999e-97 < a < 6.6e149

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   10. Taylor expanded in y-scale around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

   12. Applied rewrites 6.8%

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

   if 6.6e149 < a

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   11. Applied rewrites 4.8%

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

Alternative 4: 10.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(a\_m \cdot b\_m\right)}{y-scale \cdot x-scale\_m} \cdot \frac{\left(-a\_m\right) \cdot b\_m}{y-scale \cdot x-scale\_m}\\ t_1 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;a\_m \leq 4.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_0\right) \cdot \left(\left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\right)\right) \cdot \frac{\sqrt{{a\_m}^{4}} + \mathsf{fma}\left({a\_m}^{2}, {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\_m\right)}^{2}\right)}{{y-scale}^{2}}}}{t\_0}\\ \mathbf{elif}\;a\_m \leq 6.6 \cdot 10^{+149}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({a\_m}^{4} \cdot \left(\left(0.5 + \sqrt{{\left(t\_1 - 0.5\right)}^{2}}\right) - t\_1\right)\right)}\right)}{{a\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\left(\left(x-scale\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left({a\_m}^{4} \cdot \frac{\left|0.5 \cdot \frac{t\_2 - 1}{y-scale \cdot y-scale}\right| + \frac{0.5 - 0.5 \cdot t\_2}{y-scale \cdot y-scale}}{y-scale \cdot y-scale}\right) \cdot 8}\right) \cdot b\_m}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (/ (* 4.0 (* a_m b_m)) (* y-scale x-scale_m))
          (/ (* (- a_m) b_m) (* y-scale x-scale_m))))
        (t_1 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
        (t_2 (cos (* 0.011111111111111112 (* PI angle)))))
   (if (<= a_m 4.1e-97)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_0) (* (* b_m a_m) (* b_m (- a_m))))
         (/
          (+
           (sqrt (pow a_m 4.0))
           (fma
            (pow a_m 2.0)
            (pow (cos (* 0.005555555555555556 (* angle PI))) 2.0)
            (pow (* (sin (* (* PI angle) 0.005555555555555556)) b_m) 2.0)))
          (pow y-scale 2.0)))))
      t_0)
     (if (<= a_m 6.6e+149)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (sqrt
            (*
             8.0
             (* (pow a_m 4.0) (- (+ 0.5 (sqrt (pow (- t_1 0.5) 2.0))) t_1))))))
         (pow a_m 2.0)))
       (*
        (/ 0.25 a_m)
        (/
         (*
          (*
           (* x-scale_m (* y-scale y-scale))
           (sqrt
            (*
             (*
              (pow a_m 4.0)
              (/
               (+
                (fabs (* 0.5 (/ (- t_2 1.0) (* y-scale y-scale))))
                (/ (- 0.5 (* 0.5 t_2)) (* y-scale y-scale)))
               (* y-scale y-scale)))
             8.0)))
          b_m)
         a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	double t_1 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_2 = cos((0.011111111111111112 * (((double) M_PI) * angle)));
	double tmp;
	if (a_m <= 4.1e-97) {
		tmp = -sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * ((sqrt(pow(a_m, 4.0)) + fma(pow(a_m, 2.0), pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0), pow((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * b_m), 2.0))) / pow(y_45_scale, 2.0)))) / t_0;
	} else if (a_m <= 6.6e+149) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * sqrt((8.0 * (pow(a_m, 4.0) * ((0.5 + sqrt(pow((t_1 - 0.5), 2.0))) - t_1)))))) / pow(a_m, 2.0));
	} else {
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * sqrt(((pow(a_m, 4.0) * ((fabs((0.5 * ((t_2 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_2)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(Float64(4.0 * Float64(a_m * b_m)) / Float64(y_45_scale * x_45_scale_m)) * Float64(Float64(Float64(-a_m) * b_m) / Float64(y_45_scale * x_45_scale_m)))
	t_1 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_2 = cos(Float64(0.011111111111111112 * Float64(pi * angle)))
	tmp = 0.0
	if (a_m <= 4.1e-97)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_0) * Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))) * Float64(Float64(sqrt((a_m ^ 4.0)) + fma((a_m ^ 2.0), (cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0), (Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * b_m) ^ 2.0))) / (y_45_scale ^ 2.0))))) / t_0);
	elseif (a_m <= 6.6e+149)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((a_m ^ 4.0) * Float64(Float64(0.5 + sqrt((Float64(t_1 - 0.5) ^ 2.0))) - t_1)))))) / (a_m ^ 2.0)));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(x_45_scale_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64((a_m ^ 4.0) * Float64(Float64(abs(Float64(0.5 * Float64(Float64(t_2 - 1.0) / Float64(y_45_scale * y_45_scale)))) + Float64(Float64(0.5 - Float64(0.5 * t_2)) / Float64(y_45_scale * y_45_scale))) / Float64(y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(4.0 * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[((-a$95$m) * b$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 4.1e-97], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[Power[a$95$m, 4.0], $MachinePrecision]], $MachinePrecision] + N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[a$95$m, 6.6e+149], N[(0.25 * N[(N[(b$95$m * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(0.5 + N[Sqrt[N[Power[N[(t$95$1 - 0.5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(x$45$scale$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(N[Abs[N[(0.5 * N[(N[(t$95$2 - 1.0), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.09999999999999993e-97

   1. Initial program 2.7%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 4.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 4.5

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

   6. Applied rewrites 4.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 6.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 6.7

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

   8. Applied rewrites 6.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-prod-down N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-PI.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lower-pow.f64 N/A

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

   10. Applied rewrites 6.7%

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

       1. lift-*.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. pow-prod-down N/A

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

       5. lift-sin.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lift-*.f64 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. *-commutative N/A

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

       12. *-commutative N/A

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

       13. metadata-eval N/A

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

       14. mult-flip N/A

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

       15. lower-pow.f64 N/A

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

   12. Applied rewrites 7.1%

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

   13. Taylor expanded in angle around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-pow.f64 7.0

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

   15. Applied rewrites 7.0%

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

   if 4.09999999999999993e-97 < a < 6.6e149

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   10. Taylor expanded in y-scale around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

   12. Applied rewrites 6.8%

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

   if 6.6e149 < a

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   11. Applied rewrites 4.8%

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

Alternative 5: 9.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(a\_m \cdot b\_m\right)}{y-scale \cdot x-scale\_m} \cdot \frac{\left(-a\_m\right) \cdot b\_m}{y-scale \cdot x-scale\_m}\\ t_1 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_2 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ t_3 := \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;a\_m \leq 4.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_0\right) \cdot \left(\left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\right)\right) \cdot \frac{{b\_m}^{2} \cdot \left(\sqrt{{t\_2}^{4}} + {t\_2}^{2}\right)}{{y-scale}^{2}}}}{t\_0}\\ \mathbf{elif}\;a\_m \leq 6.6 \cdot 10^{+149}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({a\_m}^{4} \cdot \left(\left(0.5 + \sqrt{{\left(t\_1 - 0.5\right)}^{2}}\right) - t\_1\right)\right)}\right)}{{a\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\left(\left(x-scale\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left({a\_m}^{4} \cdot \frac{\left|0.5 \cdot \frac{t\_3 - 1}{y-scale \cdot y-scale}\right| + \frac{0.5 - 0.5 \cdot t\_3}{y-scale \cdot y-scale}}{y-scale \cdot y-scale}\right) \cdot 8}\right) \cdot b\_m}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (/ (* 4.0 (* a_m b_m)) (* y-scale x-scale_m))
          (/ (* (- a_m) b_m) (* y-scale x-scale_m))))
        (t_1 (* 0.5 (cos (* 0.011111111111111112 (* angle PI)))))
        (t_2 (sin (* 0.005555555555555556 (* angle PI))))
        (t_3 (cos (* 0.011111111111111112 (* PI angle)))))
   (if (<= a_m 4.1e-97)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_0) (* (* b_m a_m) (* b_m (- a_m))))
         (/
          (* (pow b_m 2.0) (+ (sqrt (pow t_2 4.0)) (pow t_2 2.0)))
          (pow y-scale 2.0)))))
      t_0)
     (if (<= a_m 6.6e+149)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (sqrt
            (*
             8.0
             (* (pow a_m 4.0) (- (+ 0.5 (sqrt (pow (- t_1 0.5) 2.0))) t_1))))))
         (pow a_m 2.0)))
       (*
        (/ 0.25 a_m)
        (/
         (*
          (*
           (* x-scale_m (* y-scale y-scale))
           (sqrt
            (*
             (*
              (pow a_m 4.0)
              (/
               (+
                (fabs (* 0.5 (/ (- t_3 1.0) (* y-scale y-scale))))
                (/ (- 0.5 (* 0.5 t_3)) (* y-scale y-scale)))
               (* y-scale y-scale)))
             8.0)))
          b_m)
         a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	double t_1 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_2 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double t_3 = cos((0.011111111111111112 * (((double) M_PI) * angle)));
	double tmp;
	if (a_m <= 4.1e-97) {
		tmp = -sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * ((pow(b_m, 2.0) * (sqrt(pow(t_2, 4.0)) + pow(t_2, 2.0))) / pow(y_45_scale, 2.0)))) / t_0;
	} else if (a_m <= 6.6e+149) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * sqrt((8.0 * (pow(a_m, 4.0) * ((0.5 + sqrt(pow((t_1 - 0.5), 2.0))) - t_1)))))) / pow(a_m, 2.0));
	} else {
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * sqrt(((pow(a_m, 4.0) * ((fabs((0.5 * ((t_3 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_3)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	}
	return tmp;
}

Java

a_m = Math.abs(a);
b_m = Math.abs(b);
x-scale_m = Math.abs(x_45_scale);
public static double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	double t_1 = 0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI)));
	double t_2 = Math.sin((0.005555555555555556 * (angle * Math.PI)));
	double t_3 = Math.cos((0.011111111111111112 * (Math.PI * angle)));
	double tmp;
	if (a_m <= 4.1e-97) {
		tmp = -Math.sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * ((Math.pow(b_m, 2.0) * (Math.sqrt(Math.pow(t_2, 4.0)) + Math.pow(t_2, 2.0))) / Math.pow(y_45_scale, 2.0)))) / t_0;
	} else if (a_m <= 6.6e+149) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * Math.sqrt((8.0 * (Math.pow(a_m, 4.0) * ((0.5 + Math.sqrt(Math.pow((t_1 - 0.5), 2.0))) - t_1)))))) / Math.pow(a_m, 2.0));
	} else {
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * Math.sqrt(((Math.pow(a_m, 4.0) * ((Math.abs((0.5 * ((t_3 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_3)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	}
	return tmp;
}

Python

a_m = math.fabs(a)
b_m = math.fabs(b)
x-scale_m = math.fabs(x_45_scale)
def code(a_m, b_m, angle, x_45_scale_m, y_45_scale):
	t_0 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m))
	t_1 = 0.5 * math.cos((0.011111111111111112 * (angle * math.pi)))
	t_2 = math.sin((0.005555555555555556 * (angle * math.pi)))
	t_3 = math.cos((0.011111111111111112 * (math.pi * angle)))
	tmp = 0
	if a_m <= 4.1e-97:
		tmp = -math.sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * ((math.pow(b_m, 2.0) * (math.sqrt(math.pow(t_2, 4.0)) + math.pow(t_2, 2.0))) / math.pow(y_45_scale, 2.0)))) / t_0
	elif a_m <= 6.6e+149:
		tmp = 0.25 * ((b_m * (x_45_scale_m * math.sqrt((8.0 * (math.pow(a_m, 4.0) * ((0.5 + math.sqrt(math.pow((t_1 - 0.5), 2.0))) - t_1)))))) / math.pow(a_m, 2.0))
	else:
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * math.sqrt(((math.pow(a_m, 4.0) * ((math.fabs((0.5 * ((t_3 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_3)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m)
	return tmp

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(Float64(4.0 * Float64(a_m * b_m)) / Float64(y_45_scale * x_45_scale_m)) * Float64(Float64(Float64(-a_m) * b_m) / Float64(y_45_scale * x_45_scale_m)))
	t_1 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	t_2 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	t_3 = cos(Float64(0.011111111111111112 * Float64(pi * angle)))
	tmp = 0.0
	if (a_m <= 4.1e-97)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_0) * Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))) * Float64(Float64((b_m ^ 2.0) * Float64(sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0))) / (y_45_scale ^ 2.0))))) / t_0);
	elseif (a_m <= 6.6e+149)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((a_m ^ 4.0) * Float64(Float64(0.5 + sqrt((Float64(t_1 - 0.5) ^ 2.0))) - t_1)))))) / (a_m ^ 2.0)));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(x_45_scale_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64((a_m ^ 4.0) * Float64(Float64(abs(Float64(0.5 * Float64(Float64(t_3 - 1.0) / Float64(y_45_scale * y_45_scale)))) + Float64(Float64(0.5 - Float64(0.5 * t_3)) / Float64(y_45_scale * y_45_scale))) / Float64(y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m));
	end
	return tmp
end

MATLAB

a_m = abs(a);
b_m = abs(b);
x-scale_m = abs(x_45_scale);
function tmp_2 = code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * ((-a_m * b_m) / (y_45_scale * x_45_scale_m));
	t_1 = 0.5 * cos((0.011111111111111112 * (angle * pi)));
	t_2 = sin((0.005555555555555556 * (angle * pi)));
	t_3 = cos((0.011111111111111112 * (pi * angle)));
	tmp = 0.0;
	if (a_m <= 4.1e-97)
		tmp = -sqrt((((2.0 * t_0) * ((b_m * a_m) * (b_m * -a_m))) * (((b_m ^ 2.0) * (sqrt((t_2 ^ 4.0)) + (t_2 ^ 2.0))) / (y_45_scale ^ 2.0)))) / t_0;
	elseif (a_m <= 6.6e+149)
		tmp = 0.25 * ((b_m * (x_45_scale_m * sqrt((8.0 * ((a_m ^ 4.0) * ((0.5 + sqrt(((t_1 - 0.5) ^ 2.0))) - t_1)))))) / (a_m ^ 2.0));
	else
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * sqrt((((a_m ^ 4.0) * ((abs((0.5 * ((t_3 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_3)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	end
	tmp_2 = tmp;
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(4.0 * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[((-a$95$m) * b$95$m), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a$95$m, 4.1e-97], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[b$95$m, 2.0], $MachinePrecision] * N[(N[Sqrt[N[Power[t$95$2, 4.0], $MachinePrecision]], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[a$95$m, 6.6e+149], N[(0.25 * N[(N[(b$95$m * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(0.5 + N[Sqrt[N[Power[N[(t$95$1 - 0.5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(x$45$scale$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(N[Abs[N[(0.5 * N[(N[(t$95$3 - 1.0), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.09999999999999993e-97

   1. Initial program 2.7%

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

   2. Taylor expanded in y-scale around 0

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

   4. Applied rewrites 3.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 4.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 4.5

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

   6. Applied rewrites 4.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 6.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 6.7

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

   8. Applied rewrites 6.7%

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

   9. Taylor expanded in b around inf

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-+.f64 N/A

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

   11. Applied rewrites 4.0%

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

   if 4.09999999999999993e-97 < a < 6.6e149

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   10. Taylor expanded in y-scale around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

   12. Applied rewrites 6.8%

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

   if 6.6e149 < a

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   11. Applied rewrites 4.8%

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

Alternative 6: 9.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \frac{a\_m}{y-scale \cdot y-scale}\\ t_1 := \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\\ t_2 := \left(-a\_m\right) \cdot b\_m\\ t_3 := \frac{b\_m}{x-scale\_m \cdot x-scale\_m}\\ t_4 := 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;a\_m \leq 4.55 \cdot 10^{-97}:\\ \;\;\;\;\left(\frac{-\sqrt{\mathsf{fma}\left(b\_m, t\_3, \mathsf{fma}\left(a\_m, t\_0, \left|a\_m \cdot t\_0 - b\_m \cdot t\_3\right|\right)\right) \cdot \left(\left(\left(\left(\frac{b\_m \cdot a\_m}{x-scale\_m \cdot y-scale} \cdot \left(\left(-a\_m\right) \cdot \frac{b\_m}{x-scale\_m \cdot y-scale}\right)\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(t\_2 \cdot b\_m\right) \cdot a\_m\right)\right)}}{t\_2 \cdot \left(\left(b\_m \cdot a\_m\right) \cdot 4\right)} \cdot \left(\left(x-scale\_m \cdot y-scale\right) \cdot x-scale\_m\right)\right) \cdot y-scale\\ \mathbf{elif}\;a\_m \leq 6.6 \cdot 10^{+149}:\\ \;\;\;\;0.25 \cdot \frac{b\_m \cdot \left(x-scale\_m \cdot \sqrt{8 \cdot \left({a\_m}^{4} \cdot \left(\left(0.5 + \sqrt{{\left(t\_4 - 0.5\right)}^{2}}\right) - t\_4\right)\right)}\right)}{{a\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\left(\left(x-scale\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left({a\_m}^{4} \cdot \frac{\left|0.5 \cdot \frac{t\_1 - 1}{y-scale \cdot y-scale}\right| + \frac{0.5 - 0.5 \cdot t\_1}{y-scale \cdot y-scale}}{y-scale \cdot y-scale}\right) \cdot 8}\right) \cdot b\_m}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (/ a_m (* y-scale y-scale)))
        (t_1 (cos (* 0.011111111111111112 (* PI angle))))
        (t_2 (* (- a_m) b_m))
        (t_3 (/ b_m (* x-scale_m x-scale_m)))
        (t_4 (* 0.5 (cos (* 0.011111111111111112 (* angle PI))))))
   (if (<= a_m 4.55e-97)
     (*
      (*
       (/
        (-
         (sqrt
          (*
           (fma b_m t_3 (fma a_m t_0 (fabs (- (* a_m t_0) (* b_m t_3)))))
           (*
            (*
             (*
              (*
               (/ (* b_m a_m) (* x-scale_m y-scale))
               (* (- a_m) (/ b_m (* x-scale_m y-scale))))
              4.0)
             2.0)
            (* (* t_2 b_m) a_m)))))
        (* t_2 (* (* b_m a_m) 4.0)))
       (* (* x-scale_m y-scale) x-scale_m))
      y-scale)
     (if (<= a_m 6.6e+149)
       (*
        0.25
        (/
         (*
          b_m
          (*
           x-scale_m
           (sqrt
            (*
             8.0
             (* (pow a_m 4.0) (- (+ 0.5 (sqrt (pow (- t_4 0.5) 2.0))) t_4))))))
         (pow a_m 2.0)))
       (*
        (/ 0.25 a_m)
        (/
         (*
          (*
           (* x-scale_m (* y-scale y-scale))
           (sqrt
            (*
             (*
              (pow a_m 4.0)
              (/
               (+
                (fabs (* 0.5 (/ (- t_1 1.0) (* y-scale y-scale))))
                (/ (- 0.5 (* 0.5 t_1)) (* y-scale y-scale)))
               (* y-scale y-scale)))
             8.0)))
          b_m)
         a_m))))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = a_m / (y_45_scale * y_45_scale);
	double t_1 = cos((0.011111111111111112 * (((double) M_PI) * angle)));
	double t_2 = -a_m * b_m;
	double t_3 = b_m / (x_45_scale_m * x_45_scale_m);
	double t_4 = 0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double tmp;
	if (a_m <= 4.55e-97) {
		tmp = ((-sqrt((fma(b_m, t_3, fma(a_m, t_0, fabs(((a_m * t_0) - (b_m * t_3))))) * ((((((b_m * a_m) / (x_45_scale_m * y_45_scale)) * (-a_m * (b_m / (x_45_scale_m * y_45_scale)))) * 4.0) * 2.0) * ((t_2 * b_m) * a_m)))) / (t_2 * ((b_m * a_m) * 4.0))) * ((x_45_scale_m * y_45_scale) * x_45_scale_m)) * y_45_scale;
	} else if (a_m <= 6.6e+149) {
		tmp = 0.25 * ((b_m * (x_45_scale_m * sqrt((8.0 * (pow(a_m, 4.0) * ((0.5 + sqrt(pow((t_4 - 0.5), 2.0))) - t_4)))))) / pow(a_m, 2.0));
	} else {
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * sqrt(((pow(a_m, 4.0) * ((fabs((0.5 * ((t_1 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_1)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(a_m / Float64(y_45_scale * y_45_scale))
	t_1 = cos(Float64(0.011111111111111112 * Float64(pi * angle)))
	t_2 = Float64(Float64(-a_m) * b_m)
	t_3 = Float64(b_m / Float64(x_45_scale_m * x_45_scale_m))
	t_4 = Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))
	tmp = 0.0
	if (a_m <= 4.55e-97)
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(fma(b_m, t_3, fma(a_m, t_0, abs(Float64(Float64(a_m * t_0) - Float64(b_m * t_3))))) * Float64(Float64(Float64(Float64(Float64(Float64(b_m * a_m) / Float64(x_45_scale_m * y_45_scale)) * Float64(Float64(-a_m) * Float64(b_m / Float64(x_45_scale_m * y_45_scale)))) * 4.0) * 2.0) * Float64(Float64(t_2 * b_m) * a_m))))) / Float64(t_2 * Float64(Float64(b_m * a_m) * 4.0))) * Float64(Float64(x_45_scale_m * y_45_scale) * x_45_scale_m)) * y_45_scale);
	elseif (a_m <= 6.6e+149)
		tmp = Float64(0.25 * Float64(Float64(b_m * Float64(x_45_scale_m * sqrt(Float64(8.0 * Float64((a_m ^ 4.0) * Float64(Float64(0.5 + sqrt((Float64(t_4 - 0.5) ^ 2.0))) - t_4)))))) / (a_m ^ 2.0)));
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(x_45_scale_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64((a_m ^ 4.0) * Float64(Float64(abs(Float64(0.5 * Float64(Float64(t_1 - 1.0) / Float64(y_45_scale * y_45_scale)))) + Float64(Float64(0.5 - Float64(0.5 * t_1)) / Float64(y_45_scale * y_45_scale))) / Float64(y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[(a$95$m / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[((-a$95$m) * b$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(b$95$m / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a$95$m, 4.55e-97], N[(N[(N[((-N[Sqrt[N[(N[(b$95$m * t$95$3 + N[(a$95$m * t$95$0 + N[Abs[N[(N[(a$95$m * t$95$0), $MachinePrecision] - N[(b$95$m * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(b$95$m * a$95$m), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[((-a$95$m) * N[(b$95$m / N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$2 * b$95$m), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(t$95$2 * N[(N[(b$95$m * a$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x$45$scale$95$m * y$45$scale), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision], If[LessEqual[a$95$m, 6.6e+149], N[(0.25 * N[(N[(b$95$m * N[(x$45$scale$95$m * N[Sqrt[N[(8.0 * N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(0.5 + N[Sqrt[N[Power[N[(t$95$4 - 0.5), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[a$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(x$45$scale$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(N[Abs[N[(0.5 * N[(N[(t$95$1 - 1.0), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 4.54999999999999999e-97

   1. Initial program 2.7%

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

   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 4.2%

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

   8. Applied rewrites 4.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. times-frac N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-/.f64 N/A

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

   10. Applied rewrites 6.7%

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

   if 4.54999999999999999e-97 < a < 6.6e149

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   10. Taylor expanded in y-scale around 0

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

       1. lower-sqrt.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-*.f64 N/A

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

   12. Applied rewrites 6.8%

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

   if 6.6e149 < a

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   11. Applied rewrites 4.8%

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

Alternative 7: 9.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\\ t_1 := \frac{b\_m}{x-scale\_m \cdot x-scale\_m}\\ t_2 := \frac{a\_m}{y-scale \cdot y-scale}\\ t_3 := \left(-a\_m\right) \cdot b\_m\\ \mathbf{if}\;a\_m \leq 8 \cdot 10^{+140}:\\ \;\;\;\;\left(\frac{-\sqrt{\mathsf{fma}\left(b\_m, t\_1, \mathsf{fma}\left(a\_m, t\_2, \left|a\_m \cdot t\_2 - b\_m \cdot t\_1\right|\right)\right) \cdot \left(\left(\left(\left(\frac{b\_m \cdot a\_m}{x-scale\_m \cdot y-scale} \cdot \left(\left(-a\_m\right) \cdot \frac{b\_m}{x-scale\_m \cdot y-scale}\right)\right) \cdot 4\right) \cdot 2\right) \cdot \left(\left(t\_3 \cdot b\_m\right) \cdot a\_m\right)\right)}}{t\_3 \cdot \left(\left(b\_m \cdot a\_m\right) \cdot 4\right)} \cdot \left(\left(x-scale\_m \cdot y-scale\right) \cdot x-scale\_m\right)\right) \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{a\_m} \cdot \frac{\left(\left(x-scale\_m \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \sqrt{\left({a\_m}^{4} \cdot \frac{\left|0.5 \cdot \frac{t\_0 - 1}{y-scale \cdot y-scale}\right| + \frac{0.5 - 0.5 \cdot t\_0}{y-scale \cdot y-scale}}{y-scale \cdot y-scale}\right) \cdot 8}\right) \cdot b\_m}{a\_m}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (cos (* 0.011111111111111112 (* PI angle))))
        (t_1 (/ b_m (* x-scale_m x-scale_m)))
        (t_2 (/ a_m (* y-scale y-scale)))
        (t_3 (* (- a_m) b_m)))
   (if (<= a_m 8e+140)
     (*
      (*
       (/
        (-
         (sqrt
          (*
           (fma b_m t_1 (fma a_m t_2 (fabs (- (* a_m t_2) (* b_m t_1)))))
           (*
            (*
             (*
              (*
               (/ (* b_m a_m) (* x-scale_m y-scale))
               (* (- a_m) (/ b_m (* x-scale_m y-scale))))
              4.0)
             2.0)
            (* (* t_3 b_m) a_m)))))
        (* t_3 (* (* b_m a_m) 4.0)))
       (* (* x-scale_m y-scale) x-scale_m))
      y-scale)
     (*
      (/ 0.25 a_m)
      (/
       (*
        (*
         (* x-scale_m (* y-scale y-scale))
         (sqrt
          (*
           (*
            (pow a_m 4.0)
            (/
             (+
              (fabs (* 0.5 (/ (- t_0 1.0) (* y-scale y-scale))))
              (/ (- 0.5 (* 0.5 t_0)) (* y-scale y-scale)))
             (* y-scale y-scale)))
           8.0)))
        b_m)
       a_m)))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = cos((0.011111111111111112 * (((double) M_PI) * angle)));
	double t_1 = b_m / (x_45_scale_m * x_45_scale_m);
	double t_2 = a_m / (y_45_scale * y_45_scale);
	double t_3 = -a_m * b_m;
	double tmp;
	if (a_m <= 8e+140) {
		tmp = ((-sqrt((fma(b_m, t_1, fma(a_m, t_2, fabs(((a_m * t_2) - (b_m * t_1))))) * ((((((b_m * a_m) / (x_45_scale_m * y_45_scale)) * (-a_m * (b_m / (x_45_scale_m * y_45_scale)))) * 4.0) * 2.0) * ((t_3 * b_m) * a_m)))) / (t_3 * ((b_m * a_m) * 4.0))) * ((x_45_scale_m * y_45_scale) * x_45_scale_m)) * y_45_scale;
	} else {
		tmp = (0.25 / a_m) * ((((x_45_scale_m * (y_45_scale * y_45_scale)) * sqrt(((pow(a_m, 4.0) * ((fabs((0.5 * ((t_0 - 1.0) / (y_45_scale * y_45_scale)))) + ((0.5 - (0.5 * t_0)) / (y_45_scale * y_45_scale))) / (y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m);
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = cos(Float64(0.011111111111111112 * Float64(pi * angle)))
	t_1 = Float64(b_m / Float64(x_45_scale_m * x_45_scale_m))
	t_2 = Float64(a_m / Float64(y_45_scale * y_45_scale))
	t_3 = Float64(Float64(-a_m) * b_m)
	tmp = 0.0
	if (a_m <= 8e+140)
		tmp = Float64(Float64(Float64(Float64(-sqrt(Float64(fma(b_m, t_1, fma(a_m, t_2, abs(Float64(Float64(a_m * t_2) - Float64(b_m * t_1))))) * Float64(Float64(Float64(Float64(Float64(Float64(b_m * a_m) / Float64(x_45_scale_m * y_45_scale)) * Float64(Float64(-a_m) * Float64(b_m / Float64(x_45_scale_m * y_45_scale)))) * 4.0) * 2.0) * Float64(Float64(t_3 * b_m) * a_m))))) / Float64(t_3 * Float64(Float64(b_m * a_m) * 4.0))) * Float64(Float64(x_45_scale_m * y_45_scale) * x_45_scale_m)) * y_45_scale);
	else
		tmp = Float64(Float64(0.25 / a_m) * Float64(Float64(Float64(Float64(x_45_scale_m * Float64(y_45_scale * y_45_scale)) * sqrt(Float64(Float64((a_m ^ 4.0) * Float64(Float64(abs(Float64(0.5 * Float64(Float64(t_0 - 1.0) / Float64(y_45_scale * y_45_scale)))) + Float64(Float64(0.5 - Float64(0.5 * t_0)) / Float64(y_45_scale * y_45_scale))) / Float64(y_45_scale * y_45_scale))) * 8.0))) * b_m) / a_m));
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(b$95$m / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a$95$m / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-a$95$m) * b$95$m), $MachinePrecision]}, If[LessEqual[a$95$m, 8e+140], N[(N[(N[((-N[Sqrt[N[(N[(b$95$m * t$95$1 + N[(a$95$m * t$95$2 + N[Abs[N[(N[(a$95$m * t$95$2), $MachinePrecision] - N[(b$95$m * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(b$95$m * a$95$m), $MachinePrecision] / N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[((-a$95$m) * N[(b$95$m / N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] * 2.0), $MachinePrecision] * N[(N[(t$95$3 * b$95$m), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(t$95$3 * N[(N[(b$95$m * a$95$m), $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x$45$scale$95$m * y$45$scale), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision], N[(N[(0.25 / a$95$m), $MachinePrecision] * N[(N[(N[(N[(x$45$scale$95$m * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(N[Power[a$95$m, 4.0], $MachinePrecision] * N[(N[(N[Abs[N[(0.5 * N[(N[(t$95$0 - 1.0), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * b$95$m), $MachinePrecision] / a$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 8.00000000000000047e140

   1. Initial program 2.7%

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

   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 4.2%

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

   8. Applied rewrites 4.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lift-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. pow2 N/A

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

      16. *-commutative N/A

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

      17. times-frac N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-/.f64 N/A

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

   10. Applied rewrites 6.7%

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

   if 8.00000000000000047e140 < a

   1. Initial program 2.7%

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

   3. Applied rewrites 3.1%

      \[\leadsto \frac{-\sqrt{\color{blue}{\left(\frac{\left(\left(\left(-a\right) \cdot b\right) \cdot b\right) \cdot a}{\left(\left(y-scale \cdot x-scale\right) \cdot x-scale\right) \cdot y-scale} \cdot 8\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)}}}{\frac{4 \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. Taylor expanded in b around inf

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

   6. Applied rewrites 1.2%

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

   7. Taylor expanded in x-scale around inf

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

   9. Applied rewrites 2.6%

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

   11. Applied rewrites 4.8%

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

Alternative 8: 8.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} a_m = \left|a\right| \\ b_m = \left|b\right| \\ x-scale_m = \left|x-scale\right| \\ \begin{array}{l} t_0 := \left(-a\_m\right) \cdot b\_m\\ t_1 := \frac{a\_m}{y-scale \cdot y-scale}\\ t_2 := \frac{4 \cdot \left(a\_m \cdot b\_m\right)}{y-scale \cdot x-scale\_m} \cdot \frac{t\_0}{y-scale \cdot x-scale\_m}\\ \mathbf{if}\;x-scale\_m \leq 2 \cdot 10^{+191}:\\ \;\;\;\;\left(\left(\frac{\frac{\sqrt{\left(\mathsf{fma}\left(\frac{b\_m}{x-scale\_m \cdot x-scale\_m}, b\_m, \mathsf{fma}\left(t\_1, a\_m, \left|\frac{b\_m \cdot b\_m}{x-scale\_m \cdot x-scale\_m} - t\_1 \cdot a\_m\right|\right)\right) \cdot \left(\left(\left(\left(b\_m \cdot a\_m\right) \cdot b\_m\right) \cdot \frac{-a\_m}{\left(\left(x-scale\_m \cdot y-scale\right) \cdot x-scale\_m\right) \cdot y-scale}\right) \cdot 8\right)\right) \cdot \left(\left(t\_0 \cdot b\_m\right) \cdot a\_m\right)}}{b\_m \cdot a\_m}}{\left(4 \cdot a\_m\right) \cdot b\_m} \cdot \left(x-scale\_m \cdot y-scale\right)\right) \cdot x-scale\_m\right) \cdot y-scale\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot \left(\left(b\_m \cdot a\_m\right) \cdot \left(b\_m \cdot \left(-a\_m\right)\right)\right)\right) \cdot \left({a\_m}^{2} \cdot \left(\sqrt{\frac{1}{{y-scale}^{4}}} + \frac{1}{{y-scale}^{2}}\right)\right)}}{t\_2}\\ \end{array} \end{array} \]

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (* (- a_m) b_m))
        (t_1 (/ a_m (* y-scale y-scale)))
        (t_2
         (*
          (/ (* 4.0 (* a_m b_m)) (* y-scale x-scale_m))
          (/ t_0 (* y-scale x-scale_m)))))
   (if (<= x-scale_m 2e+191)
     (*
      (*
       (*
        (/
         (/
          (sqrt
           (*
            (*
             (fma
              (/ b_m (* x-scale_m x-scale_m))
              b_m
              (fma
               t_1
               a_m
               (fabs (- (/ (* b_m b_m) (* x-scale_m x-scale_m)) (* t_1 a_m)))))
             (*
              (*
               (* (* b_m a_m) b_m)
               (/ (- a_m) (* (* (* x-scale_m y-scale) x-scale_m) y-scale)))
              8.0))
            (* (* t_0 b_m) a_m)))
          (* b_m a_m))
         (* (* 4.0 a_m) b_m))
        (* x-scale_m y-scale))
       x-scale_m)
      y-scale)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_2) (* (* b_m a_m) (* b_m (- a_m))))
         (*
          (pow a_m 2.0)
          (+ (sqrt (/ 1.0 (pow y-scale 4.0))) (/ 1.0 (pow y-scale 2.0)))))))
      t_2))))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = -a_m * b_m;
	double t_1 = a_m / (y_45_scale * y_45_scale);
	double t_2 = ((4.0 * (a_m * b_m)) / (y_45_scale * x_45_scale_m)) * (t_0 / (y_45_scale * x_45_scale_m));
	double tmp;
	if (x_45_scale_m <= 2e+191) {
		tmp = ((((sqrt(((fma((b_m / (x_45_scale_m * x_45_scale_m)), b_m, fma(t_1, a_m, fabs((((b_m * b_m) / (x_45_scale_m * x_45_scale_m)) - (t_1 * a_m))))) * ((((b_m * a_m) * b_m) * (-a_m / (((x_45_scale_m * y_45_scale) * x_45_scale_m) * y_45_scale))) * 8.0)) * ((t_0 * b_m) * a_m))) / (b_m * a_m)) / ((4.0 * a_m) * b_m)) * (x_45_scale_m * y_45_scale)) * x_45_scale_m) * y_45_scale;
	} else {
		tmp = -sqrt((((2.0 * t_2) * ((b_m * a_m) * (b_m * -a_m))) * (pow(a_m, 2.0) * (sqrt((1.0 / pow(y_45_scale, 4.0))) + (1.0 / pow(y_45_scale, 2.0)))))) / t_2;
	}
	return tmp;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(Float64(-a_m) * b_m)
	t_1 = Float64(a_m / Float64(y_45_scale * y_45_scale))
	t_2 = Float64(Float64(Float64(4.0 * Float64(a_m * b_m)) / Float64(y_45_scale * x_45_scale_m)) * Float64(t_0 / Float64(y_45_scale * x_45_scale_m)))
	tmp = 0.0
	if (x_45_scale_m <= 2e+191)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(fma(Float64(b_m / Float64(x_45_scale_m * x_45_scale_m)), b_m, fma(t_1, a_m, abs(Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale_m * x_45_scale_m)) - Float64(t_1 * a_m))))) * Float64(Float64(Float64(Float64(b_m * a_m) * b_m) * Float64(Float64(-a_m) / Float64(Float64(Float64(x_45_scale_m * y_45_scale) * x_45_scale_m) * y_45_scale))) * 8.0)) * Float64(Float64(t_0 * b_m) * a_m))) / Float64(b_m * a_m)) / Float64(Float64(4.0 * a_m) * b_m)) * Float64(x_45_scale_m * y_45_scale)) * x_45_scale_m) * y_45_scale);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * Float64(Float64(b_m * a_m) * Float64(b_m * Float64(-a_m)))) * Float64((a_m ^ 2.0) * Float64(sqrt(Float64(1.0 / (y_45_scale ^ 4.0))) + Float64(1.0 / (y_45_scale ^ 2.0))))))) / t_2);
	end
	return tmp
end

Wolfram

a_m = N[Abs[a], $MachinePrecision]
b_m = N[Abs[b], $MachinePrecision]
x-scale_m = N[Abs[x$45$scale], $MachinePrecision]
code[a$95$m_, b$95$m_, angle_, x$45$scale$95$m_, y$45$scale_] := Block[{t$95$0 = N[((-a$95$m) * b$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(a$95$m / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(4.0 * N[(a$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(y$45$scale * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$45$scale$95$m, 2e+191], N[(N[(N[(N[(N[(N[Sqrt[N[(N[(N[(N[(b$95$m / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] * b$95$m + N[(t$95$1 * a$95$m + N[Abs[N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] / N[(x$45$scale$95$m * x$45$scale$95$m), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * a$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(b$95$m * a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision] * N[((-a$95$m) / N[(N[(N[(x$45$scale$95$m * y$45$scale), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8.0), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$0 * b$95$m), $MachinePrecision] * a$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(b$95$m * a$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * a$95$m), $MachinePrecision] * b$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale$95$m * y$45$scale), $MachinePrecision]), $MachinePrecision] * x$45$scale$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(N[(b$95$m * a$95$m), $MachinePrecision] * N[(b$95$m * (-a$95$m)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[a$95$m, 2.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / N[Power[y$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 2.00000000000000015e191

   1. Initial program 2.7%

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

   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 4.2%

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

   8. Applied rewrites 4.6%

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

   10. Applied rewrites 8.6%

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

   if 2.00000000000000015e191 < x-scale

   1. Initial program 2.7%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 5.5

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 5.5

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

   6. Applied rewrites 5.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 8.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 8.0

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

   8. Applied rewrites 8.0%

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

   9. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-sqrt.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lower-pow.f64 5.6

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

   11. Applied rewrites 5.6%

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

Alternative 9: 7.2% accurate, 6.3× speedup

Math

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

FPCore

a_m = (fabs.f64 a)
b_m = (fabs.f64 b)
x-scale_m = (fabs.f64 x-scale)
(FPCore (a_m b_m angle x-scale_m y-scale)
 :precision binary64
 (let* ((t_0 (/ a_m (* y-scale y-scale)))
        (t_1 (* (- a_m) b_m))
        (t_2 (/ b_m (* x-scale_m x-scale_m))))
   (*
    (*
     (/
      (-
       (sqrt
        (*
         (fma b_m t_2 (fma a_m t_0 (fabs (- (* a_m t_0) (* b_m t_2)))))
         (*
          (*
           (*
            (*
             (/ (* b_m a_m) (* x-scale_m y-scale))
             (* (- a_m) (/ b_m (* x-scale_m y-scale))))
            4.0)
           2.0)
          (* (* t_1 b_m) a_m)))))
      (* t_1 (* (* b_m a_m) 4.0)))
     (* (* x-scale_m y-scale) x-scale_m))
    y-scale)))

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = a_m / (y_45_scale * y_45_scale);
	double t_1 = -a_m * b_m;
	double t_2 = b_m / (x_45_scale_m * x_45_scale_m);
	return ((-sqrt((fma(b_m, t_2, fma(a_m, t_0, fabs(((a_m * t_0) - (b_m * t_2))))) * ((((((b_m * a_m) / (x_45_scale_m * y_45_scale)) * (-a_m * (b_m / (x_45_scale_m * y_45_scale)))) * 4.0) * 2.0) * ((t_1 * b_m) * a_m)))) / (t_1 * ((b_m * a_m) * 4.0))) * ((x_45_scale_m * y_45_scale) * x_45_scale_m)) * y_45_scale;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(a_m / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(Float64(-a_m) * b_m)
	t_2 = Float64(b_m / Float64(x_45_scale_m * x_45_scale_m))
	return Float64(Float64(Float64(Float64(-sqrt(Float64(fma(b_m, t_2, fma(a_m, t_0, abs(Float64(Float64(a_m * t_0) - Float64(b_m * t_2))))) * Float64(Float64(Float64(Float64(Float64(Float64(b_m * a_m) / Float64(x_45_scale_m * y_45_scale)) * Float64(Float64(-a_m) * Float64(b_m / Float64(x_45_scale_m * y_45_scale)))) * 4.0) * 2.0) * Float64(Float64(t_1 * b_m) * a_m))))) / Float64(t_1 * Float64(Float64(b_m * a_m) * 4.0))) * Float64(Float64(x_45_scale_m * y_45_scale) * x_45_scale_m)) * y_45_scale)
end

Wolfram

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

Derivation

1. Initial program 2.7%

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

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 4.2%

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

8. Applied rewrites 4.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. lift-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. pow2 N/A

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

   16. *-commutative N/A

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

   17. times-frac N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lift-/.f64 N/A

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

10. Applied rewrites 6.7%

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

Alternative 10: 6.7% accurate, 6.6× speedup

Math

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

FPCore

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

C

a_m = fabs(a);
b_m = fabs(b);
x-scale_m = fabs(x_45_scale);
double code(double a_m, double b_m, double angle, double x_45_scale_m, double y_45_scale) {
	double t_0 = a_m / (y_45_scale * y_45_scale);
	return ((((sqrt(((fma((b_m / (x_45_scale_m * x_45_scale_m)), b_m, fma(t_0, a_m, fabs((((b_m * b_m) / (x_45_scale_m * x_45_scale_m)) - (t_0 * a_m))))) * ((((b_m * a_m) * b_m) * (-a_m / (((x_45_scale_m * y_45_scale) * x_45_scale_m) * y_45_scale))) * 8.0)) * (((-a_m * b_m) * b_m) * a_m))) / (b_m * a_m)) / ((4.0 * a_m) * b_m)) * (x_45_scale_m * y_45_scale)) * x_45_scale_m) * y_45_scale;
}

Julia

a_m = abs(a)
b_m = abs(b)
x-scale_m = abs(x_45_scale)
function code(a_m, b_m, angle, x_45_scale_m, y_45_scale)
	t_0 = Float64(a_m / Float64(y_45_scale * y_45_scale))
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(Float64(fma(Float64(b_m / Float64(x_45_scale_m * x_45_scale_m)), b_m, fma(t_0, a_m, abs(Float64(Float64(Float64(b_m * b_m) / Float64(x_45_scale_m * x_45_scale_m)) - Float64(t_0 * a_m))))) * Float64(Float64(Float64(Float64(b_m * a_m) * b_m) * Float64(Float64(-a_m) / Float64(Float64(Float64(x_45_scale_m * y_45_scale) * x_45_scale_m) * y_45_scale))) * 8.0)) * Float64(Float64(Float64(Float64(-a_m) * b_m) * b_m) * a_m))) / Float64(b_m * a_m)) / Float64(Float64(4.0 * a_m) * b_m)) * Float64(x_45_scale_m * y_45_scale)) * x_45_scale_m) * y_45_scale)
end

Wolfram

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

Derivation

1. Initial program 2.7%

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

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 4.2%

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

8. Applied rewrites 4.6%

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

10. Applied rewrites 8.6%

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

Reproduce

herbie shell --seed 2025156 
(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))))