a from scale-rotated-ellipse

Percentage Accurate: 2.7% → 19.0%

Time: 31.1s

Alternatives: 12

Speedup: 11.4×

• 69.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 2.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 19.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := b \cdot \left|a\right|\\ t_1 := -\left|a\right|\\ t_2 := \left(t\_0 \cdot b\right) \cdot t\_1\\ t_3 := {\left(\left|y-scale\right|\right)}^{2}\\ t_4 := x-scale \cdot \left|y-scale\right|\\ t_5 := \left|y-scale\right| \cdot \left|y-scale\right|\\ t_6 := \frac{\left|a\right|}{t\_5}\\ t_7 := \frac{b}{x-scale \cdot x-scale}\\ t_8 := 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\\ t_9 := \mathsf{fma}\left(0.5 - t\_8, \left|a\right| \cdot \left|a\right|, \left(0.5 + t\_8\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 4.7 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(t\_2 \cdot \left(\left(4 \cdot \frac{t\_2}{\left(t\_5 \cdot x-scale\right) \cdot x-scale}\right) \cdot 2\right)\right) \cdot \mathsf{fma}\left(b, t\_7, \mathsf{fma}\left(\left|a\right|, t\_6, \left|\left|a\right| \cdot t\_6 - b \cdot t\_7\right|\right)\right)}}{\frac{t\_0 \cdot 4}{t\_4}}}{t\_1 \cdot \frac{b}{t\_4}}\\ \mathbf{elif}\;\left|y-scale\right| \leq 96:\\ \;\;\;\;0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(t\_3 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{{\left(\left|y-scale\right|\right)}^{4}}} + \frac{1}{t\_3}}{{x-scale}^{2} \cdot t\_3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{t\_0} \cdot \frac{\frac{\sqrt{8 \cdot \left({t\_0}^{4} \cdot \left(\left|t\_9\right| + t\_9\right)\right)}}{\left|\left|y-scale\right|\right|} \cdot t\_5}{t\_0}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* b (fabs a)))
        (t_1 (- (fabs a)))
        (t_2 (* (* t_0 b) t_1))
        (t_3 (pow (fabs y-scale) 2.0))
        (t_4 (* x-scale (fabs y-scale)))
        (t_5 (* (fabs y-scale) (fabs y-scale)))
        (t_6 (/ (fabs a) t_5))
        (t_7 (/ b (* x-scale x-scale)))
        (t_8 (* 0.5 (cos (* 2.0 (* (* PI angle) 0.005555555555555556)))))
        (t_9 (fma (- 0.5 t_8) (* (fabs a) (fabs a)) (* (+ 0.5 t_8) (* b b)))))
   (if (<= (fabs y-scale) 4.7e-132)
     (/
      (/
       (-
        (sqrt
         (*
          (* t_2 (* (* 4.0 (/ t_2 (* (* t_5 x-scale) x-scale))) 2.0))
          (fma
           b
           t_7
           (fma (fabs a) t_6 (fabs (- (* (fabs a) t_6) (* b t_7))))))))
       (/ (* t_0 4.0) t_4))
      (* t_1 (/ b t_4)))
     (if (<= (fabs y-scale) 96.0)
       (*
        0.25
        (*
         (fabs a)
         (*
          (pow x-scale 2.0)
          (*
           t_3
           (sqrt
            (*
             8.0
             (/
              (+ (sqrt (/ 1.0 (pow (fabs y-scale) 4.0))) (/ 1.0 t_3))
              (* (pow x-scale 2.0) t_3))))))))
       (*
        (/ 0.25 t_0)
        (/
         (*
          (/
           (sqrt (* 8.0 (* (pow t_0 4.0) (+ (fabs t_9) t_9))))
           (fabs (fabs y-scale)))
          t_5)
         t_0))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * fabs(a);
	double t_1 = -fabs(a);
	double t_2 = (t_0 * b) * t_1;
	double t_3 = pow(fabs(y_45_scale), 2.0);
	double t_4 = x_45_scale * fabs(y_45_scale);
	double t_5 = fabs(y_45_scale) * fabs(y_45_scale);
	double t_6 = fabs(a) / t_5;
	double t_7 = b / (x_45_scale * x_45_scale);
	double t_8 = 0.5 * cos((2.0 * ((((double) M_PI) * angle) * 0.005555555555555556)));
	double t_9 = fma((0.5 - t_8), (fabs(a) * fabs(a)), ((0.5 + t_8) * (b * b)));
	double tmp;
	if (fabs(y_45_scale) <= 4.7e-132) {
		tmp = (-sqrt(((t_2 * ((4.0 * (t_2 / ((t_5 * x_45_scale) * x_45_scale))) * 2.0)) * fma(b, t_7, fma(fabs(a), t_6, fabs(((fabs(a) * t_6) - (b * t_7))))))) / ((t_0 * 4.0) / t_4)) / (t_1 * (b / t_4));
	} else if (fabs(y_45_scale) <= 96.0) {
		tmp = 0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (t_3 * sqrt((8.0 * ((sqrt((1.0 / pow(fabs(y_45_scale), 4.0))) + (1.0 / t_3)) / (pow(x_45_scale, 2.0) * t_3)))))));
	} else {
		tmp = (0.25 / t_0) * (((sqrt((8.0 * (pow(t_0, 4.0) * (fabs(t_9) + t_9)))) / fabs(fabs(y_45_scale))) * t_5) / t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * abs(a))
	t_1 = Float64(-abs(a))
	t_2 = Float64(Float64(t_0 * b) * t_1)
	t_3 = abs(y_45_scale) ^ 2.0
	t_4 = Float64(x_45_scale * abs(y_45_scale))
	t_5 = Float64(abs(y_45_scale) * abs(y_45_scale))
	t_6 = Float64(abs(a) / t_5)
	t_7 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_8 = Float64(0.5 * cos(Float64(2.0 * Float64(Float64(pi * angle) * 0.005555555555555556))))
	t_9 = fma(Float64(0.5 - t_8), Float64(abs(a) * abs(a)), Float64(Float64(0.5 + t_8) * Float64(b * b)))
	tmp = 0.0
	if (abs(y_45_scale) <= 4.7e-132)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(t_2 * Float64(Float64(4.0 * Float64(t_2 / Float64(Float64(t_5 * x_45_scale) * x_45_scale))) * 2.0)) * fma(b, t_7, fma(abs(a), t_6, abs(Float64(Float64(abs(a) * t_6) - Float64(b * t_7)))))))) / Float64(Float64(t_0 * 4.0) / t_4)) / Float64(t_1 * Float64(b / t_4)));
	elseif (abs(y_45_scale) <= 96.0)
		tmp = Float64(0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(t_3 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(1.0 / (abs(y_45_scale) ^ 4.0))) + Float64(1.0 / t_3)) / Float64((x_45_scale ^ 2.0) * t_3))))))));
	else
		tmp = Float64(Float64(0.25 / t_0) * Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((t_0 ^ 4.0) * Float64(abs(t_9) + t_9)))) / abs(abs(y_45_scale))) * t_5) / t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$2 = N[(N[(t$95$0 * b), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Abs[a], $MachinePrecision] / t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(0.5 * N[Cos[N[(2.0 * N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(0.5 - t$95$8), $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + t$95$8), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 4.7e-132], N[(N[((-N[Sqrt[N[(N[(t$95$2 * N[(N[(4.0 * N[(t$95$2 / N[(N[(t$95$5 * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(b * t$95$7 + N[(N[Abs[a], $MachinePrecision] * t$95$6 + N[Abs[N[(N[(N[Abs[a], $MachinePrecision] * t$95$6), $MachinePrecision] - N[(b * t$95$7), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(t$95$0 * 4.0), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(b / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 96.0], N[(0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(t$95$3 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(1.0 / N[Power[N[Abs[y$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / t$95$0), $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[t$95$0, 4.0], $MachinePrecision] * N[(N[Abs[t$95$9], $MachinePrecision] + t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 4.7000000000000002e-132

   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 4.1%

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

      1. lift-/.f64 N/A

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

      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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 5.1%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 5.1%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 5.1%

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

   6. Applied rewrites 5.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 7.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 7.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 7.6%

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

   8. Applied rewrites 7.6%

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

   10. Applied rewrites 6.6%

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

   if 4.7000000000000002e-132 < y-scale < 96

   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

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

   4. Applied rewrites 0.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

   10. Applied rewrites 2.6%

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

   if 96 < y-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 x-scale around 0

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

   4. Applied rewrites 1.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   6. Applied rewrites 17.0%

      \[\leadsto \frac{0.25}{b \cdot a} \cdot \color{blue}{\frac{\frac{\sqrt{8 \cdot \left({\left(b \cdot a\right)}^{4} \cdot \left(\left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right| + \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right)\right)}}{\left|y-scale\right|} \cdot \left(y-scale \cdot y-scale\right)}{b \cdot a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 18.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{b}{x-scale \cdot x-scale}\\ t_1 := b \cdot \left|a\right|\\ t_2 := -\left|a\right|\\ t_3 := {\left(\left|y-scale\right|\right)}^{2}\\ t_4 := x-scale \cdot \left|y-scale\right|\\ t_5 := \left(t\_1 \cdot b\right) \cdot t\_2\\ t_6 := 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\\ t_7 := \left|y-scale\right| \cdot \left|y-scale\right|\\ t_8 := \frac{\left|a\right|}{t\_7}\\ t_9 := \mathsf{fma}\left(0.5 - t\_6, \left|a\right| \cdot \left|a\right|, \left(0.5 + t\_6\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{if}\;\left|y-scale\right| \leq 4.7 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(t\_5 \cdot \left(\left(4 \cdot \frac{t\_5}{\left(t\_7 \cdot x-scale\right) \cdot x-scale}\right) \cdot 2\right)\right) \cdot \mathsf{fma}\left(b, t\_0, \mathsf{fma}\left(\left|a\right|, t\_8, \left|\left|a\right| \cdot t\_8 - b \cdot t\_0\right|\right)\right)}}{\frac{t\_1 \cdot 4}{t\_4}}}{t\_2 \cdot \frac{b}{t\_4}}\\ \mathbf{elif}\;\left|y-scale\right| \leq 96:\\ \;\;\;\;0.25 \cdot \left(\left|a\right| \cdot \left({x-scale}^{2} \cdot \left(t\_3 \cdot \sqrt{8 \cdot \frac{\sqrt{\frac{1}{{\left(\left|y-scale\right|\right)}^{4}}} + \frac{1}{t\_3}}{{x-scale}^{2} \cdot t\_3}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\frac{\frac{\sqrt{8 \cdot \left({t\_1}^{4} \cdot \left(\left|t\_9\right| + t\_9\right)\right)}}{\left|\left|y-scale\right|\right|}}{t\_1} \cdot \frac{t\_7}{t\_1}\right)\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ b (* x-scale x-scale)))
        (t_1 (* b (fabs a)))
        (t_2 (- (fabs a)))
        (t_3 (pow (fabs y-scale) 2.0))
        (t_4 (* x-scale (fabs y-scale)))
        (t_5 (* (* t_1 b) t_2))
        (t_6 (* 0.5 (cos (* 2.0 (* (* PI angle) 0.005555555555555556)))))
        (t_7 (* (fabs y-scale) (fabs y-scale)))
        (t_8 (/ (fabs a) t_7))
        (t_9 (fma (- 0.5 t_6) (* (fabs a) (fabs a)) (* (+ 0.5 t_6) (* b b)))))
   (if (<= (fabs y-scale) 4.7e-132)
     (/
      (/
       (-
        (sqrt
         (*
          (* t_5 (* (* 4.0 (/ t_5 (* (* t_7 x-scale) x-scale))) 2.0))
          (fma
           b
           t_0
           (fma (fabs a) t_8 (fabs (- (* (fabs a) t_8) (* b t_0))))))))
       (/ (* t_1 4.0) t_4))
      (* t_2 (/ b t_4)))
     (if (<= (fabs y-scale) 96.0)
       (*
        0.25
        (*
         (fabs a)
         (*
          (pow x-scale 2.0)
          (*
           t_3
           (sqrt
            (*
             8.0
             (/
              (+ (sqrt (/ 1.0 (pow (fabs y-scale) 4.0))) (/ 1.0 t_3))
              (* (pow x-scale 2.0) t_3))))))))
       (*
        0.25
        (*
         (/
          (/
           (sqrt (* 8.0 (* (pow t_1 4.0) (+ (fabs t_9) t_9))))
           (fabs (fabs y-scale)))
          t_1)
         (/ t_7 t_1)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b / (x_45_scale * x_45_scale);
	double t_1 = b * fabs(a);
	double t_2 = -fabs(a);
	double t_3 = pow(fabs(y_45_scale), 2.0);
	double t_4 = x_45_scale * fabs(y_45_scale);
	double t_5 = (t_1 * b) * t_2;
	double t_6 = 0.5 * cos((2.0 * ((((double) M_PI) * angle) * 0.005555555555555556)));
	double t_7 = fabs(y_45_scale) * fabs(y_45_scale);
	double t_8 = fabs(a) / t_7;
	double t_9 = fma((0.5 - t_6), (fabs(a) * fabs(a)), ((0.5 + t_6) * (b * b)));
	double tmp;
	if (fabs(y_45_scale) <= 4.7e-132) {
		tmp = (-sqrt(((t_5 * ((4.0 * (t_5 / ((t_7 * x_45_scale) * x_45_scale))) * 2.0)) * fma(b, t_0, fma(fabs(a), t_8, fabs(((fabs(a) * t_8) - (b * t_0))))))) / ((t_1 * 4.0) / t_4)) / (t_2 * (b / t_4));
	} else if (fabs(y_45_scale) <= 96.0) {
		tmp = 0.25 * (fabs(a) * (pow(x_45_scale, 2.0) * (t_3 * sqrt((8.0 * ((sqrt((1.0 / pow(fabs(y_45_scale), 4.0))) + (1.0 / t_3)) / (pow(x_45_scale, 2.0) * t_3)))))));
	} else {
		tmp = 0.25 * (((sqrt((8.0 * (pow(t_1, 4.0) * (fabs(t_9) + t_9)))) / fabs(fabs(y_45_scale))) / t_1) * (t_7 / t_1));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_1 = Float64(b * abs(a))
	t_2 = Float64(-abs(a))
	t_3 = abs(y_45_scale) ^ 2.0
	t_4 = Float64(x_45_scale * abs(y_45_scale))
	t_5 = Float64(Float64(t_1 * b) * t_2)
	t_6 = Float64(0.5 * cos(Float64(2.0 * Float64(Float64(pi * angle) * 0.005555555555555556))))
	t_7 = Float64(abs(y_45_scale) * abs(y_45_scale))
	t_8 = Float64(abs(a) / t_7)
	t_9 = fma(Float64(0.5 - t_6), Float64(abs(a) * abs(a)), Float64(Float64(0.5 + t_6) * Float64(b * b)))
	tmp = 0.0
	if (abs(y_45_scale) <= 4.7e-132)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(t_5 * Float64(Float64(4.0 * Float64(t_5 / Float64(Float64(t_7 * x_45_scale) * x_45_scale))) * 2.0)) * fma(b, t_0, fma(abs(a), t_8, abs(Float64(Float64(abs(a) * t_8) - Float64(b * t_0)))))))) / Float64(Float64(t_1 * 4.0) / t_4)) / Float64(t_2 * Float64(b / t_4)));
	elseif (abs(y_45_scale) <= 96.0)
		tmp = Float64(0.25 * Float64(abs(a) * Float64((x_45_scale ^ 2.0) * Float64(t_3 * sqrt(Float64(8.0 * Float64(Float64(sqrt(Float64(1.0 / (abs(y_45_scale) ^ 4.0))) + Float64(1.0 / t_3)) / Float64((x_45_scale ^ 2.0) * t_3))))))));
	else
		tmp = Float64(0.25 * Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((t_1 ^ 4.0) * Float64(abs(t_9) + t_9)))) / abs(abs(y_45_scale))) / t_1) * Float64(t_7 / t_1)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$3 = N[Power[N[Abs[y$45$scale], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(x$45$scale * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$1 * b), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(0.5 * N[Cos[N[(2.0 * N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[Abs[y$45$scale], $MachinePrecision] * N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[(N[Abs[a], $MachinePrecision] / t$95$7), $MachinePrecision]}, Block[{t$95$9 = N[(N[(0.5 - t$95$6), $MachinePrecision] * N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 + t$95$6), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 4.7e-132], N[(N[((-N[Sqrt[N[(N[(t$95$5 * N[(N[(4.0 * N[(t$95$5 / N[(N[(t$95$7 * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(b * t$95$0 + N[(N[Abs[a], $MachinePrecision] * t$95$8 + N[Abs[N[(N[(N[Abs[a], $MachinePrecision] * t$95$8), $MachinePrecision] - N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(t$95$1 * 4.0), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[(b / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[y$45$scale], $MachinePrecision], 96.0], N[(0.25 * N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * N[(t$95$3 * N[Sqrt[N[(8.0 * N[(N[(N[Sqrt[N[(1.0 / N[Power[N[Abs[y$45$scale], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision] / N[(N[Power[x$45$scale, 2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[t$95$1, 4.0], $MachinePrecision] * N[(N[Abs[t$95$9], $MachinePrecision] + t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[y$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(t$95$7 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y-scale < 4.7000000000000002e-132

   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 4.1%

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

      1. lift-/.f64 N/A

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

      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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 5.1%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 5.1%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 5.1%

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

   6. Applied rewrites 5.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 7.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 7.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 7.6%

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

   8. Applied rewrites 7.6%

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

   10. Applied rewrites 6.6%

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

   if 4.7000000000000002e-132 < y-scale < 96

   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

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

   4. Applied rewrites 0.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 0.8%

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

   8. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

   10. Applied rewrites 2.6%

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

   if 96 < y-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 x-scale around 0

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

   4. Applied rewrites 1.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   6. Applied rewrites 15.8%

      \[\leadsto 0.25 \cdot \left(\frac{\frac{\sqrt{8 \cdot \left({\left(b \cdot a\right)}^{4} \cdot \left(\left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right| + \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right)\right)}}{\left|y-scale\right|}}{b \cdot a} \cdot \color{blue}{\frac{y-scale \cdot y-scale}{b \cdot a}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 15.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \left|b\right| \cdot \left|a\right|\\ t_2 := \left|a\right| \cdot \left|a\right|\\ t_3 := 0.5 \cdot \cos \left(2 \cdot t\_0\right)\\ t_4 := 0.5 - t\_3\\ t_5 := \mathsf{fma}\left(t\_4, t\_2, \left(0.5 + t\_3\right) \cdot \left(\left|b\right| \cdot \left|b\right|\right)\right)\\ \mathbf{if}\;\left|a\right| \leq 9 \cdot 10^{-42}:\\ \;\;\;\;\left(\left(y-scale \cdot y-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left({t\_1}^{4} \cdot \left(\left|t\_5\right| + t\_5\right)\right)}}{\left|y-scale\right|}}{\left(t\_2 \cdot \left|b\right|\right) \cdot \left|b\right|}\right) \cdot 0.25\\ \mathbf{elif}\;\left|a\right| \leq 4.6 \cdot 10^{+182}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{8 \cdot \left({\left(\left|b\right|\right)}^{4} \cdot {\left(\left|a\right|\right)}^{6}\right)}}{\left|\left|x-scale\right|\right|} \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|}{t\_1} \cdot \frac{0.25}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\left|a\right|} \cdot \frac{\left(\left(\left(\left(y-scale \cdot y-scale\right) \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|\right) \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\frac{t\_4}{y-scale \cdot y-scale} + \sqrt{\frac{{\sin t\_0}^{4}}{\left(y-scale \cdot y-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right) \cdot {\left(\left|a\right|\right)}^{4}\right)}}{\left|y-scale\right|}}{\left|x-scale\right|}\right) \cdot \left|b\right|}{\left|a\right|}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (* (fabs b) (fabs a)))
        (t_2 (* (fabs a) (fabs a)))
        (t_3 (* 0.5 (cos (* 2.0 t_0))))
        (t_4 (- 0.5 t_3))
        (t_5 (fma t_4 t_2 (* (+ 0.5 t_3) (* (fabs b) (fabs b))))))
   (if (<= (fabs a) 9e-42)
     (*
      (*
       (* y-scale y-scale)
       (/
        (/ (sqrt (* 8.0 (* (pow t_1 4.0) (+ (fabs t_5) t_5)))) (fabs y-scale))
        (* (* t_2 (fabs b)) (fabs b))))
      0.25)
     (if (<= (fabs a) 4.6e+182)
       (*
        (/
         (*
          (*
           (/
            (sqrt (* 8.0 (* (pow (fabs b) 4.0) (pow (fabs a) 6.0))))
            (fabs (fabs x-scale)))
           (fabs x-scale))
          (fabs x-scale))
         t_1)
        (/ 0.25 t_1))
       (*
        (/ 0.25 (fabs a))
        (/
         (*
          (*
           (* (* (* y-scale y-scale) (fabs x-scale)) (fabs x-scale))
           (/
            (/
             (sqrt
              (*
               8.0
               (*
                (+
                 (/ t_4 (* y-scale y-scale))
                 (sqrt
                  (/
                   (pow (sin t_0) 4.0)
                   (* (* y-scale y-scale) (* y-scale y-scale)))))
                (pow (fabs a) 4.0))))
             (fabs y-scale))
            (fabs x-scale)))
          (fabs b))
         (fabs a)))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = fabs(b) * fabs(a);
	double t_2 = fabs(a) * fabs(a);
	double t_3 = 0.5 * cos((2.0 * t_0));
	double t_4 = 0.5 - t_3;
	double t_5 = fma(t_4, t_2, ((0.5 + t_3) * (fabs(b) * fabs(b))));
	double tmp;
	if (fabs(a) <= 9e-42) {
		tmp = ((y_45_scale * y_45_scale) * ((sqrt((8.0 * (pow(t_1, 4.0) * (fabs(t_5) + t_5)))) / fabs(y_45_scale)) / ((t_2 * fabs(b)) * fabs(b)))) * 0.25;
	} else if (fabs(a) <= 4.6e+182) {
		tmp = ((((sqrt((8.0 * (pow(fabs(b), 4.0) * pow(fabs(a), 6.0)))) / fabs(fabs(x_45_scale))) * fabs(x_45_scale)) * fabs(x_45_scale)) / t_1) * (0.25 / t_1);
	} else {
		tmp = (0.25 / fabs(a)) * ((((((y_45_scale * y_45_scale) * fabs(x_45_scale)) * fabs(x_45_scale)) * ((sqrt((8.0 * (((t_4 / (y_45_scale * y_45_scale)) + sqrt((pow(sin(t_0), 4.0) / ((y_45_scale * y_45_scale) * (y_45_scale * y_45_scale))))) * pow(fabs(a), 4.0)))) / fabs(y_45_scale)) / fabs(x_45_scale))) * fabs(b)) / fabs(a));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = Float64(abs(b) * abs(a))
	t_2 = Float64(abs(a) * abs(a))
	t_3 = Float64(0.5 * cos(Float64(2.0 * t_0)))
	t_4 = Float64(0.5 - t_3)
	t_5 = fma(t_4, t_2, Float64(Float64(0.5 + t_3) * Float64(abs(b) * abs(b))))
	tmp = 0.0
	if (abs(a) <= 9e-42)
		tmp = Float64(Float64(Float64(y_45_scale * y_45_scale) * Float64(Float64(sqrt(Float64(8.0 * Float64((t_1 ^ 4.0) * Float64(abs(t_5) + t_5)))) / abs(y_45_scale)) / Float64(Float64(t_2 * abs(b)) * abs(b)))) * 0.25);
	elseif (abs(a) <= 4.6e+182)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((abs(b) ^ 4.0) * (abs(a) ^ 6.0)))) / abs(abs(x_45_scale))) * abs(x_45_scale)) * abs(x_45_scale)) / t_1) * Float64(0.25 / t_1));
	else
		tmp = Float64(Float64(0.25 / abs(a)) * Float64(Float64(Float64(Float64(Float64(Float64(y_45_scale * y_45_scale) * abs(x_45_scale)) * abs(x_45_scale)) * Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(t_4 / Float64(y_45_scale * y_45_scale)) + sqrt(Float64((sin(t_0) ^ 4.0) / Float64(Float64(y_45_scale * y_45_scale) * Float64(y_45_scale * y_45_scale))))) * (abs(a) ^ 4.0)))) / abs(y_45_scale)) / abs(x_45_scale))) * abs(b)) / abs(a)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[b], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[a], $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(0.5 - t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * t$95$2 + N[(N[(0.5 + t$95$3), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 9e-42], N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[t$95$1, 4.0], $MachinePrecision] * N[(N[Abs[t$95$5], $MachinePrecision] + t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 4.6e+182], N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(0.25 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(0.25 / N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[N[(8.0 * N[(N[(N[(t$95$4 / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] + N[Sqrt[N[(N[Power[N[Sin[t$95$0], $MachinePrecision], 4.0], $MachinePrecision] / N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] / N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 9.0000000000000002e-42

   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 x-scale around 0

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

   4. Applied rewrites 1.1%

      \[\leadsto \color{blue}{0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({b}^{4} \cdot \left(\sqrt{{\left(\mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)}^{2}} + \mathsf{fma}\left({a}^{2}, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, {b}^{2} \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)\right)\right)}{{y-scale}^{2}}}}{{a}^{2} \cdot {b}^{2}}} \]

   6. Applied rewrites 8.1%

      \[\leadsto \left(\left(y-scale \cdot y-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left({\left(b \cdot a\right)}^{4} \cdot \left(\left|\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right| + \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot b\right)\right)\right)\right)}}{\left|y-scale\right|}}{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}\right) \cdot \color{blue}{0.25} \]

   if 9.0000000000000002e-42 < a < 4.6000000000000001e182

   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

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

   4. Applied rewrites 0.4%

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

   6. Applied rewrites 0.6%

      \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   7. Taylor expanded in y-scale around 0

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      7. lower-pow.f64 0.9%

         \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   9. Applied rewrites 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   11. Applied rewrites 10.7%

       \[\leadsto \frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{b \cdot a} \cdot \color{blue}{\frac{0.25}{b \cdot a}} \]

   if 4.6000000000000001e182 < a

   1. Initial program 2.7%

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

   2. Taylor expanded in b around inf

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

   4. Applied rewrites 0.8%

      \[\leadsto \color{blue}{0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\mathsf{fma}\left(4, \frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}, {\left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} - \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}^{2}\right)} + \left(\frac{{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}}\right)\right)}{{a}^{2}}} \]

   5. Taylor expanded in x-scale around inf

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \frac{\sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}{{y-scale}^{2}}}}{x-scale}\right)\right)}{{a}^{2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

   7. Applied rewrites 1.4%

      \[\leadsto 0.25 \cdot \frac{b \cdot \left({x-scale}^{2} \cdot \left({y-scale}^{2} \cdot \frac{\sqrt{8 \cdot \frac{{a}^{4} \cdot \left(\sqrt{\frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{4}}{{y-scale}^{4}}} + \frac{{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)}{{y-scale}^{2}}}}{x-scale}\right)\right)}{{a}^{2}} \]

   9. Applied rewrites 3.5%

      \[\leadsto \frac{0.25}{a} \cdot \color{blue}{\frac{\left(\left(\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale\right) \cdot \frac{\frac{\sqrt{8 \cdot \left(\left(\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)}{y-scale \cdot y-scale} + \sqrt{\frac{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{4}}{\left(y-scale \cdot y-scale\right) \cdot \left(y-scale \cdot y-scale\right)}}\right) \cdot {a}^{4}\right)}}{\left|y-scale\right|}}{x-scale}\right) \cdot b}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 12.1% accurate, 5.3× speedup

Math

\[\begin{array}{l} t_0 := \left|b\right| \cdot a\\ t_1 := \frac{a}{y-scale \cdot y-scale}\\ t_2 := \frac{4 \cdot \left(a \cdot \left|b\right|\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot \left|b\right|}{y-scale \cdot x-scale}\\ t_3 := \left(t\_0 \cdot \left|b\right|\right) \cdot \left(-a\right)\\ t_4 := \frac{\left|b\right|}{x-scale \cdot x-scale}\\ t_5 := \mathsf{fma}\left(\left|b\right|, t\_4, \mathsf{fma}\left(a, t\_1, \left|a \cdot t\_1 - \left|b\right| \cdot t\_4\right|\right)\right)\\ \mathbf{if}\;\left|b\right| \leq 2.45 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{-\sqrt{\left(t\_3 \cdot \left(\left(4 \cdot \frac{t\_3}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 2\right)\right) \cdot t\_5}}{\frac{t\_0 \cdot 4}{x-scale \cdot y-scale}}}{\left(-a\right) \cdot \frac{\left|b\right|}{x-scale \cdot y-scale}}\\ \mathbf{elif}\;\left|b\right| \leq 9.8 \cdot 10^{+213}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{8 \cdot \left({\left(\left|b\right|\right)}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{t\_0} \cdot \frac{0.25}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot \left(t\_0 \cdot \left(\left|b\right| \cdot \left(-a\right)\right)\right)\right) \cdot t\_5}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (fabs b) a))
        (t_1 (/ a (* y-scale y-scale)))
        (t_2
         (*
          (/ (* 4.0 (* a (fabs b))) (* y-scale x-scale))
          (/ (* (- a) (fabs b)) (* y-scale x-scale))))
        (t_3 (* (* t_0 (fabs b)) (- a)))
        (t_4 (/ (fabs b) (* x-scale x-scale)))
        (t_5
         (fma (fabs b) t_4 (fma a t_1 (fabs (- (* a t_1) (* (fabs b) t_4)))))))
   (if (<= (fabs b) 2.45e-145)
     (/
      (/
       (-
        (sqrt
         (*
          (*
           t_3
           (* (* 4.0 (/ t_3 (* (* (* y-scale y-scale) x-scale) x-scale))) 2.0))
          t_5)))
       (/ (* t_0 4.0) (* x-scale y-scale)))
      (* (- a) (/ (fabs b) (* x-scale y-scale))))
     (if (<= (fabs b) 9.8e+213)
       (*
        (/
         (*
          (*
           (/ (sqrt (* 8.0 (* (pow (fabs b) 4.0) (pow a 6.0)))) (fabs x-scale))
           x-scale)
          x-scale)
         t_0)
        (/ 0.25 t_0))
       (/
        (- (sqrt (* (* (* 2.0 t_2) (* t_0 (* (fabs b) (- a)))) t_5)))
        t_2)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(b) * a;
	double t_1 = a / (y_45_scale * y_45_scale);
	double t_2 = ((4.0 * (a * fabs(b))) / (y_45_scale * x_45_scale)) * ((-a * fabs(b)) / (y_45_scale * x_45_scale));
	double t_3 = (t_0 * fabs(b)) * -a;
	double t_4 = fabs(b) / (x_45_scale * x_45_scale);
	double t_5 = fma(fabs(b), t_4, fma(a, t_1, fabs(((a * t_1) - (fabs(b) * t_4)))));
	double tmp;
	if (fabs(b) <= 2.45e-145) {
		tmp = (-sqrt(((t_3 * ((4.0 * (t_3 / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 2.0)) * t_5)) / ((t_0 * 4.0) / (x_45_scale * y_45_scale))) / (-a * (fabs(b) / (x_45_scale * y_45_scale)));
	} else if (fabs(b) <= 9.8e+213) {
		tmp = ((((sqrt((8.0 * (pow(fabs(b), 4.0) * pow(a, 6.0)))) / fabs(x_45_scale)) * x_45_scale) * x_45_scale) / t_0) * (0.25 / t_0);
	} else {
		tmp = -sqrt((((2.0 * t_2) * (t_0 * (fabs(b) * -a))) * t_5)) / t_2;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(b) * a)
	t_1 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_2 = Float64(Float64(Float64(4.0 * Float64(a * abs(b))) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(Float64(-a) * abs(b)) / Float64(y_45_scale * x_45_scale)))
	t_3 = Float64(Float64(t_0 * abs(b)) * Float64(-a))
	t_4 = Float64(abs(b) / Float64(x_45_scale * x_45_scale))
	t_5 = fma(abs(b), t_4, fma(a, t_1, abs(Float64(Float64(a * t_1) - Float64(abs(b) * t_4)))))
	tmp = 0.0
	if (abs(b) <= 2.45e-145)
		tmp = Float64(Float64(Float64(-sqrt(Float64(Float64(t_3 * Float64(Float64(4.0 * Float64(t_3 / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 2.0)) * t_5))) / Float64(Float64(t_0 * 4.0) / Float64(x_45_scale * y_45_scale))) / Float64(Float64(-a) * Float64(abs(b) / Float64(x_45_scale * y_45_scale))));
	elseif (abs(b) <= 9.8e+213)
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((abs(b) ^ 4.0) * (a ^ 6.0)))) / abs(x_45_scale)) * x_45_scale) * x_45_scale) / t_0) * Float64(0.25 / t_0));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * Float64(t_0 * Float64(abs(b) * Float64(-a)))) * t_5))) / t_2);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$1 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(4.0 * N[(a * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[((-a) * N[Abs[b], $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[Abs[b], $MachinePrecision]), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[b], $MachinePrecision] / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Abs[b], $MachinePrecision] * t$95$4 + N[(a * t$95$1 + N[Abs[N[(N[(a * t$95$1), $MachinePrecision] - N[(N[Abs[b], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2.45e-145], N[(N[((-N[Sqrt[N[(N[(t$95$3 * N[(N[(4.0 * N[(t$95$3 / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision]) / N[(N[(t$95$0 * 4.0), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-a) * N[(N[Abs[b], $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[b], $MachinePrecision], 9.8e+213], N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[N[Abs[b], $MachinePrecision], 4.0], $MachinePrecision] * N[Power[a, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(0.25 / t$95$0), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(t$95$0 * N[(N[Abs[b], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.4499999999999998e-145

   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 4.1%

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

      1. lift-/.f64 N/A

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

      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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 5.1%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 5.1%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 5.1%

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

   6. Applied rewrites 5.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 7.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 7.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 7.6%

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

   8. Applied rewrites 7.6%

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

   10. Applied rewrites 6.6%

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

   if 2.4499999999999998e-145 < b < 9.7999999999999999e213

   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

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

   4. Applied rewrites 0.4%

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

   6. Applied rewrites 0.6%

      \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   7. Taylor expanded in y-scale around 0

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      7. lower-pow.f64 0.9%

         \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   9. Applied rewrites 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   11. Applied rewrites 10.7%

       \[\leadsto \frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{b \cdot a} \cdot \color{blue}{\frac{0.25}{b \cdot a}} \]

   if 9.7999999999999999e213 < b

   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 4.1%

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

      1. lift-/.f64 N/A

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

      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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 5.1%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 5.1%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 5.1%

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

   6. Applied rewrites 5.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 7.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 7.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 7.6%

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

   8. Applied rewrites 7.6%

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

   10. Applied rewrites 9.7%

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

Alternative 5: 12.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{b \cdot b}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ t_1 := \frac{\frac{a \cdot a}{y-scale}}{y-scale}\\ t_2 := y-scale \cdot \left|x-scale\right|\\ t_3 := \frac{4 \cdot \left(a \cdot b\right)}{t\_2} \cdot \frac{\left(-a\right) \cdot b}{t\_2}\\ \mathbf{if}\;\left|x-scale\right| \leq 1.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(t\_0 - t\_1\right)}^{2}} + \left(t\_1 + t\_0\right)\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|\left|x-scale\right|\right|} \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|}{b \cdot a} \cdot \frac{0.25}{b \cdot a}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (* b b) (* (fabs x-scale) (fabs x-scale))))
        (t_1 (/ (/ (* a a) y-scale) y-scale))
        (t_2 (* y-scale (fabs x-scale)))
        (t_3 (* (/ (* 4.0 (* a b)) t_2) (/ (* (- a) b) t_2))))
   (if (<= (fabs x-scale) 1.5e+169)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_3) (* (* b a) (* b (- a))))
         (+ (sqrt (pow (- t_0 t_1) 2.0)) (+ t_1 t_0)))))
      t_3)
     (*
      (/
       (*
        (*
         (/ (sqrt (* 8.0 (* (pow b 4.0) (pow a 6.0)))) (fabs (fabs x-scale)))
         (fabs x-scale))
        (fabs x-scale))
       (* b a))
      (/ 0.25 (* b a))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * b) / (fabs(x_45_scale) * fabs(x_45_scale));
	double t_1 = ((a * a) / y_45_scale) / y_45_scale;
	double t_2 = y_45_scale * fabs(x_45_scale);
	double t_3 = ((4.0 * (a * b)) / t_2) * ((-a * b) / t_2);
	double tmp;
	if (fabs(x_45_scale) <= 1.5e+169) {
		tmp = -sqrt((((2.0 * t_3) * ((b * a) * (b * -a))) * (sqrt(pow((t_0 - t_1), 2.0)) + (t_1 + t_0)))) / t_3;
	} else {
		tmp = ((((sqrt((8.0 * (pow(b, 4.0) * pow(a, 6.0)))) / fabs(fabs(x_45_scale))) * fabs(x_45_scale)) * fabs(x_45_scale)) / (b * a)) * (0.25 / (b * a));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (b * b) / (abs(x_45scale) * abs(x_45scale))
    t_1 = ((a * a) / y_45scale) / y_45scale
    t_2 = y_45scale * abs(x_45scale)
    t_3 = ((4.0d0 * (a * b)) / t_2) * ((-a * b) / t_2)
    if (abs(x_45scale) <= 1.5d+169) then
        tmp = -sqrt((((2.0d0 * t_3) * ((b * a) * (b * -a))) * (sqrt(((t_0 - t_1) ** 2.0d0)) + (t_1 + t_0)))) / t_3
    else
        tmp = ((((sqrt((8.0d0 * ((b ** 4.0d0) * (a ** 6.0d0)))) / abs(abs(x_45scale))) * abs(x_45scale)) * abs(x_45scale)) / (b * a)) * (0.25d0 / (b * a))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * b) / (Math.abs(x_45_scale) * Math.abs(x_45_scale));
	double t_1 = ((a * a) / y_45_scale) / y_45_scale;
	double t_2 = y_45_scale * Math.abs(x_45_scale);
	double t_3 = ((4.0 * (a * b)) / t_2) * ((-a * b) / t_2);
	double tmp;
	if (Math.abs(x_45_scale) <= 1.5e+169) {
		tmp = -Math.sqrt((((2.0 * t_3) * ((b * a) * (b * -a))) * (Math.sqrt(Math.pow((t_0 - t_1), 2.0)) + (t_1 + t_0)))) / t_3;
	} else {
		tmp = ((((Math.sqrt((8.0 * (Math.pow(b, 4.0) * Math.pow(a, 6.0)))) / Math.abs(Math.abs(x_45_scale))) * Math.abs(x_45_scale)) * Math.abs(x_45_scale)) / (b * a)) * (0.25 / (b * a));
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * b) / (math.fabs(x_45_scale) * math.fabs(x_45_scale))
	t_1 = ((a * a) / y_45_scale) / y_45_scale
	t_2 = y_45_scale * math.fabs(x_45_scale)
	t_3 = ((4.0 * (a * b)) / t_2) * ((-a * b) / t_2)
	tmp = 0
	if math.fabs(x_45_scale) <= 1.5e+169:
		tmp = -math.sqrt((((2.0 * t_3) * ((b * a) * (b * -a))) * (math.sqrt(math.pow((t_0 - t_1), 2.0)) + (t_1 + t_0)))) / t_3
	else:
		tmp = ((((math.sqrt((8.0 * (math.pow(b, 4.0) * math.pow(a, 6.0)))) / math.fabs(math.fabs(x_45_scale))) * math.fabs(x_45_scale)) * math.fabs(x_45_scale)) / (b * a)) * (0.25 / (b * a))
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * b) / Float64(abs(x_45_scale) * abs(x_45_scale)))
	t_1 = Float64(Float64(Float64(a * a) / y_45_scale) / y_45_scale)
	t_2 = Float64(y_45_scale * abs(x_45_scale))
	t_3 = Float64(Float64(Float64(4.0 * Float64(a * b)) / t_2) * Float64(Float64(Float64(-a) * b) / t_2))
	tmp = 0.0
	if (abs(x_45_scale) <= 1.5e+169)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * Float64(sqrt((Float64(t_0 - t_1) ^ 2.0)) + Float64(t_1 + t_0))))) / t_3);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((b ^ 4.0) * (a ^ 6.0)))) / abs(abs(x_45_scale))) * abs(x_45_scale)) * abs(x_45_scale)) / Float64(b * a)) * Float64(0.25 / Float64(b * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * b) / (abs(x_45_scale) * abs(x_45_scale));
	t_1 = ((a * a) / y_45_scale) / y_45_scale;
	t_2 = y_45_scale * abs(x_45_scale);
	t_3 = ((4.0 * (a * b)) / t_2) * ((-a * b) / t_2);
	tmp = 0.0;
	if (abs(x_45_scale) <= 1.5e+169)
		tmp = -sqrt((((2.0 * t_3) * ((b * a) * (b * -a))) * (sqrt(((t_0 - t_1) ^ 2.0)) + (t_1 + t_0)))) / t_3;
	else
		tmp = ((((sqrt((8.0 * ((b ^ 4.0) * (a ^ 6.0)))) / abs(abs(x_45_scale))) * abs(x_45_scale)) * abs(x_45_scale)) / (b * a)) * (0.25 / (b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * a), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$2 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 1.5e+169], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$3), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[Power[N[(t$95$0 - t$95$1), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[(t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[a, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.25 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.5e169

   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 4.1%

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

      1. lift-/.f64 N/A

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

      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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 5.1%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 5.1%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 5.1%

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

   6. Applied rewrites 5.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 7.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 7.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 7.6%

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

   8. Applied rewrites 7.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 7.6%

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 7.6%

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

   10. Applied rewrites 7.6%

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

       1. lift-/.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. pow2 N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 8.3%

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 8.3%

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

   12. Applied rewrites 8.3%

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

       1. lift-pow.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 8.3%

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

       4. lift-pow.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 8.3%

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

   14. Applied rewrites 8.3%

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

       1. lift-pow.f64 N/A

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

       2. pow2 N/A

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

       3. lift-*.f64 8.3%

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

       4. lift-pow.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 8.3%

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

   16. Applied rewrites 8.3%

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

   if 1.5e169 < 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

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

   4. Applied rewrites 0.4%

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

   6. Applied rewrites 0.6%

      \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   7. Taylor expanded in y-scale around 0

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      7. lower-pow.f64 0.9%

         \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   9. Applied rewrites 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   11. Applied rewrites 10.7%

       \[\leadsto \frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{b \cdot a} \cdot \color{blue}{\frac{0.25}{b \cdot a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 11.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{a}{y-scale \cdot y-scale}\\ t_1 := y-scale \cdot \left|x-scale\right|\\ t_2 := \frac{4 \cdot \left(a \cdot b\right)}{t\_1} \cdot \frac{\left(-a\right) \cdot b}{t\_1}\\ t_3 := \frac{b}{\left|x-scale\right| \cdot \left|x-scale\right|}\\ \mathbf{if}\;\left|x-scale\right| \leq 1.5 \cdot 10^{+169}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \mathsf{fma}\left(b, t\_3, \mathsf{fma}\left(a, t\_0, \left|a \cdot t\_0 - b \cdot t\_3\right|\right)\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|\left|x-scale\right|\right|} \cdot \left|x-scale\right|\right) \cdot \left|x-scale\right|}{b \cdot a} \cdot \frac{0.25}{b \cdot a}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ a (* y-scale y-scale)))
        (t_1 (* y-scale (fabs x-scale)))
        (t_2 (* (/ (* 4.0 (* a b)) t_1) (/ (* (- a) b) t_1)))
        (t_3 (/ b (* (fabs x-scale) (fabs x-scale)))))
   (if (<= (fabs x-scale) 1.5e+169)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_2) (* (* b a) (* b (- a))))
         (fma b t_3 (fma a t_0 (fabs (- (* a t_0) (* b t_3))))))))
      t_2)
     (*
      (/
       (*
        (*
         (/ (sqrt (* 8.0 (* (pow b 4.0) (pow a 6.0)))) (fabs (fabs x-scale)))
         (fabs x-scale))
        (fabs x-scale))
       (* b a))
      (/ 0.25 (* b a))))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = a / (y_45_scale * y_45_scale);
	double t_1 = y_45_scale * fabs(x_45_scale);
	double t_2 = ((4.0 * (a * b)) / t_1) * ((-a * b) / t_1);
	double t_3 = b / (fabs(x_45_scale) * fabs(x_45_scale));
	double tmp;
	if (fabs(x_45_scale) <= 1.5e+169) {
		tmp = -sqrt((((2.0 * t_2) * ((b * a) * (b * -a))) * fma(b, t_3, fma(a, t_0, fabs(((a * t_0) - (b * t_3))))))) / t_2;
	} else {
		tmp = ((((sqrt((8.0 * (pow(b, 4.0) * pow(a, 6.0)))) / fabs(fabs(x_45_scale))) * fabs(x_45_scale)) * fabs(x_45_scale)) / (b * a)) * (0.25 / (b * a));
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(a / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(y_45_scale * abs(x_45_scale))
	t_2 = Float64(Float64(Float64(4.0 * Float64(a * b)) / t_1) * Float64(Float64(Float64(-a) * b) / t_1))
	t_3 = Float64(b / Float64(abs(x_45_scale) * abs(x_45_scale)))
	tmp = 0.0
	if (abs(x_45_scale) <= 1.5e+169)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * Float64(Float64(b * a) * Float64(b * Float64(-a)))) * fma(b, t_3, fma(a, t_0, abs(Float64(Float64(a * t_0) - Float64(b * t_3)))))))) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((b ^ 4.0) * (a ^ 6.0)))) / abs(abs(x_45_scale))) * abs(x_45_scale)) * abs(x_45_scale)) / Float64(b * a)) * Float64(0.25 / Float64(b * a)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$45$scale * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(4.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(N[((-a) * b), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(b / N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 1.5e+169], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * t$95$3 + N[(a * t$95$0 + N[Abs[N[(N[(a * t$95$0), $MachinePrecision] - N[(b * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[a, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.25 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 1.5e169

   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 4.1%

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

      1. lift-/.f64 N/A

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

      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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 5.1%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 5.1%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 5.1%

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

   6. Applied rewrites 5.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 7.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 7.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 7.6%

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

   8. Applied rewrites 7.6%

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

   10. Applied rewrites 9.7%

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

   if 1.5e169 < 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

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

   4. Applied rewrites 0.4%

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

   6. Applied rewrites 0.6%

      \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   7. Taylor expanded in y-scale around 0

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      7. lower-pow.f64 0.9%

         \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   9. Applied rewrites 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   11. Applied rewrites 10.7%

       \[\leadsto \frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{b \cdot a} \cdot \color{blue}{\frac{0.25}{b \cdot a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 11.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|a\right|}{y-scale \cdot y-scale}\\ t_1 := -\left|a\right|\\ t_2 := b \cdot \left|a\right|\\ t_3 := \frac{4 \cdot \left(\left|a\right| \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{t\_1 \cdot b}{y-scale \cdot x-scale}\\ t_4 := \frac{b}{x-scale \cdot x-scale}\\ \mathbf{if}\;\left|a\right| \leq 1.45 \cdot 10^{-42}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_3\right) \cdot \left(t\_2 \cdot \left(b \cdot t\_1\right)\right)\right) \cdot \mathsf{fma}\left(t\_4, b, \mathsf{fma}\left(t\_0, \left|a\right|, t\_4 \cdot b - t\_0 \cdot \left|a\right|\right)\right)}}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {\left(\left|a\right|\right)}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{t\_2} \cdot \frac{0.25}{t\_2}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (/ (fabs a) (* y-scale y-scale)))
        (t_1 (- (fabs a)))
        (t_2 (* b (fabs a)))
        (t_3
         (*
          (/ (* 4.0 (* (fabs a) b)) (* y-scale x-scale))
          (/ (* t_1 b) (* y-scale x-scale))))
        (t_4 (/ b (* x-scale x-scale))))
   (if (<= (fabs a) 1.45e-42)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_3) (* t_2 (* b t_1)))
         (fma t_4 b (fma t_0 (fabs a) (- (* t_4 b) (* t_0 (fabs a))))))))
      t_3)
     (*
      (/
       (*
        (*
         (/ (sqrt (* 8.0 (* (pow b 4.0) (pow (fabs a) 6.0)))) (fabs x-scale))
         x-scale)
        x-scale)
       t_2)
      (/ 0.25 t_2)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = fabs(a) / (y_45_scale * y_45_scale);
	double t_1 = -fabs(a);
	double t_2 = b * fabs(a);
	double t_3 = ((4.0 * (fabs(a) * b)) / (y_45_scale * x_45_scale)) * ((t_1 * b) / (y_45_scale * x_45_scale));
	double t_4 = b / (x_45_scale * x_45_scale);
	double tmp;
	if (fabs(a) <= 1.45e-42) {
		tmp = -sqrt((((2.0 * t_3) * (t_2 * (b * t_1))) * fma(t_4, b, fma(t_0, fabs(a), ((t_4 * b) - (t_0 * fabs(a))))))) / t_3;
	} else {
		tmp = ((((sqrt((8.0 * (pow(b, 4.0) * pow(fabs(a), 6.0)))) / fabs(x_45_scale)) * x_45_scale) * x_45_scale) / t_2) * (0.25 / t_2);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(abs(a) / Float64(y_45_scale * y_45_scale))
	t_1 = Float64(-abs(a))
	t_2 = Float64(b * abs(a))
	t_3 = Float64(Float64(Float64(4.0 * Float64(abs(a) * b)) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(t_1 * b) / Float64(y_45_scale * x_45_scale)))
	t_4 = Float64(b / Float64(x_45_scale * x_45_scale))
	tmp = 0.0
	if (abs(a) <= 1.45e-42)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_3) * Float64(t_2 * Float64(b * t_1))) * fma(t_4, b, fma(t_0, abs(a), Float64(Float64(t_4 * b) - Float64(t_0 * abs(a)))))))) / t_3);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((b ^ 4.0) * (abs(a) ^ 6.0)))) / abs(x_45_scale)) * x_45_scale) * x_45_scale) / t_2) * Float64(0.25 / t_2));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$2 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(4.0 * N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.45e-42], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$3), $MachinePrecision] * N[(t$95$2 * N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * b + N[(t$95$0 * N[Abs[a], $MachinePrecision] + N[(N[(t$95$4 * b), $MachinePrecision] - N[(t$95$0 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(0.25 / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4500000000000001e-42

   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 4.1%

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

      1. lift-/.f64 N/A

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

      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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 5.1%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 5.1%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 5.1%

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

   6. Applied rewrites 5.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 7.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 7.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 7.6%

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

   8. Applied rewrites 7.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 7.6%

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 7.6%

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

   10. Applied rewrites 7.6%

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

       1. lift-/.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. pow2 N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 8.3%

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 8.3%

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

   12. Applied rewrites 8.3%

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

   14. Applied rewrites 5.4%

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

   if 1.4500000000000001e-42 < a

   1. Initial program 2.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.4%

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

   6. Applied rewrites 0.6%

      \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   7. Taylor expanded in y-scale around 0

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      7. lower-pow.f64 0.9%

         \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   9. Applied rewrites 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   11. Applied rewrites 10.7%

       \[\leadsto \frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{b \cdot a} \cdot \color{blue}{\frac{0.25}{b \cdot a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 10.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} t_0 := b \cdot \left|a\right|\\ t_1 := -\left|a\right|\\ t_2 := \frac{b}{x-scale \cdot x-scale}\\ t_3 := \left(t\_0 \cdot b\right) \cdot t\_1\\ t_4 := \frac{\left|a\right|}{y-scale \cdot y-scale}\\ \mathbf{if}\;\left|a\right| \leq 1.06 \cdot 10^{-184}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(4 \cdot \frac{t\_3}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}\right) \cdot 2\right) \cdot \left(t\_3 \cdot \mathsf{fma}\left(t\_2, b, \mathsf{fma}\left(t\_4, \left|a\right|, t\_2 \cdot b - t\_4 \cdot \left|a\right|\right)\right)\right)}}{\frac{4 \cdot \left(\left|a\right| \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{t\_1 \cdot b}{y-scale \cdot x-scale}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {\left(\left|a\right|\right)}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{t\_0} \cdot \frac{0.25}{t\_0}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* b (fabs a)))
        (t_1 (- (fabs a)))
        (t_2 (/ b (* x-scale x-scale)))
        (t_3 (* (* t_0 b) t_1))
        (t_4 (/ (fabs a) (* y-scale y-scale))))
   (if (<= (fabs a) 1.06e-184)
     (/
      (-
       (sqrt
        (*
         (* (* 4.0 (/ t_3 (* (* (* y-scale y-scale) x-scale) x-scale))) 2.0)
         (*
          t_3
          (fma t_2 b (fma t_4 (fabs a) (- (* t_2 b) (* t_4 (fabs a)))))))))
      (*
       (/ (* 4.0 (* (fabs a) b)) (* y-scale x-scale))
       (/ (* t_1 b) (* y-scale x-scale))))
     (*
      (/
       (*
        (*
         (/ (sqrt (* 8.0 (* (pow b 4.0) (pow (fabs a) 6.0)))) (fabs x-scale))
         x-scale)
        x-scale)
       t_0)
      (/ 0.25 t_0)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = b * fabs(a);
	double t_1 = -fabs(a);
	double t_2 = b / (x_45_scale * x_45_scale);
	double t_3 = (t_0 * b) * t_1;
	double t_4 = fabs(a) / (y_45_scale * y_45_scale);
	double tmp;
	if (fabs(a) <= 1.06e-184) {
		tmp = -sqrt((((4.0 * (t_3 / (((y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 2.0) * (t_3 * fma(t_2, b, fma(t_4, fabs(a), ((t_2 * b) - (t_4 * fabs(a)))))))) / (((4.0 * (fabs(a) * b)) / (y_45_scale * x_45_scale)) * ((t_1 * b) / (y_45_scale * x_45_scale)));
	} else {
		tmp = ((((sqrt((8.0 * (pow(b, 4.0) * pow(fabs(a), 6.0)))) / fabs(x_45_scale)) * x_45_scale) * x_45_scale) / t_0) * (0.25 / t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(b * abs(a))
	t_1 = Float64(-abs(a))
	t_2 = Float64(b / Float64(x_45_scale * x_45_scale))
	t_3 = Float64(Float64(t_0 * b) * t_1)
	t_4 = Float64(abs(a) / Float64(y_45_scale * y_45_scale))
	tmp = 0.0
	if (abs(a) <= 1.06e-184)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(4.0 * Float64(t_3 / Float64(Float64(Float64(y_45_scale * y_45_scale) * x_45_scale) * x_45_scale))) * 2.0) * Float64(t_3 * fma(t_2, b, fma(t_4, abs(a), Float64(Float64(t_2 * b) - Float64(t_4 * abs(a))))))))) / Float64(Float64(Float64(4.0 * Float64(abs(a) * b)) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(t_1 * b) / Float64(y_45_scale * x_45_scale))));
	else
		tmp = Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((b ^ 4.0) * (abs(a) ^ 6.0)))) / abs(x_45_scale)) * x_45_scale) * x_45_scale) / t_0) * Float64(0.25 / t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$2 = N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * b), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[Abs[a], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.06e-184], N[((-N[Sqrt[N[(N[(N[(4.0 * N[(t$95$3 / N[(N[(N[(y$45$scale * y$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] * N[(t$95$3 * N[(t$95$2 * b + N[(t$95$4 * N[Abs[a], $MachinePrecision] + N[(N[(t$95$2 * b), $MachinePrecision] - N[(t$95$4 * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(N[(4.0 * N[(N[Abs[a], $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[N[Abs[a], $MachinePrecision], 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(0.25 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.0599999999999999e-184

   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 4.1%

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

      1. lift-/.f64 N/A

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

      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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 5.1%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 5.1%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 5.1%

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

   6. Applied rewrites 5.1%

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \color{blue}{\left(\frac{4 \cdot \left(a \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot b}{y-scale \cdot x-scale}\right)}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\sqrt{{\left(\frac{{b}^{2}}{{x-scale}^{2}} - \frac{{a}^{2}}{{y-scale}^{2}}\right)}^{2}} + \left(\frac{{a}^{2}}{{y-scale}^{2}} + \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   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. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 7.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 7.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 7.6%

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

   8. Applied rewrites 7.6%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 7.6%

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 7.6%

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

   10. Applied rewrites 7.6%

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

       1. lift-/.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. pow2 N/A

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

       4. associate-/r* N/A

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

       5. lower-/.f64 N/A

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

       6. lower-/.f64 8.3%

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

       7. lift-pow.f64 N/A

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

       8. pow2 N/A

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

       9. lift-*.f64 8.3%

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

   12. Applied rewrites 8.3%

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

   14. Applied rewrites 1.7%

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

   if 1.0599999999999999e-184 < a

   1. Initial program 2.7%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.4%

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

   6. Applied rewrites 0.6%

      \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

   7. Taylor expanded in y-scale around 0

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      6. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

      7. lower-pow.f64 0.9%

         \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   9. Applied rewrites 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   11. Applied rewrites 10.7%

       \[\leadsto \frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{b \cdot a} \cdot \color{blue}{\frac{0.25}{b \cdot a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 10.4% accurate, 11.4× speedup

Math

\[\frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{b \cdot a} \cdot \frac{0.25}{b \cdot a} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (/
   (*
    (* (/ (sqrt (* 8.0 (* (pow b 4.0) (pow a 6.0)))) (fabs x-scale)) x-scale)
    x-scale)
   (* b a))
  (/ 0.25 (* b a))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((sqrt((8.0 * (pow(b, 4.0) * pow(a, 6.0)))) / fabs(x_45_scale)) * x_45_scale) * x_45_scale) / (b * a)) * (0.25 / (b * a));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((((sqrt((8.0d0 * ((b ** 4.0d0) * (a ** 6.0d0)))) / abs(x_45scale)) * x_45scale) * x_45scale) / (b * a)) * (0.25d0 / (b * a))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((Math.sqrt((8.0 * (Math.pow(b, 4.0) * Math.pow(a, 6.0)))) / Math.abs(x_45_scale)) * x_45_scale) * x_45_scale) / (b * a)) * (0.25 / (b * a));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((((math.sqrt((8.0 * (math.pow(b, 4.0) * math.pow(a, 6.0)))) / math.fabs(x_45_scale)) * x_45_scale) * x_45_scale) / (b * a)) * (0.25 / (b * a))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((b ^ 4.0) * (a ^ 6.0)))) / abs(x_45_scale)) * x_45_scale) * x_45_scale) / Float64(b * a)) * Float64(0.25 / Float64(b * a)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = ((((sqrt((8.0 * ((b ^ 4.0) * (a ^ 6.0)))) / abs(x_45_scale)) * x_45_scale) * x_45_scale) / (b * a)) * (0.25 / (b * a));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[a, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(b * a), $MachinePrecision]), $MachinePrecision] * N[(0.25 / N[(b * a), $MachinePrecision]), $MachinePrecision]), $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

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

4. Applied rewrites 0.4%

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

6. Applied rewrites 0.6%

   \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

7. Taylor expanded in y-scale around 0

   \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   7. lower-pow.f64 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

9. Applied rewrites 0.9%

   \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

11. Applied rewrites 10.7%

    \[\leadsto \frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{b \cdot a} \cdot \color{blue}{\frac{0.25}{b \cdot a}} \]
12. Add Preprocessing

Alternative 10: 5.8% accurate, 11.4× speedup

Math

\[\frac{0.25}{a} \cdot \frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{a \cdot \left(b \cdot b\right)} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (*
  (/ 0.25 a)
  (/
   (*
    (* (/ (sqrt (* 8.0 (* (pow b 4.0) (pow a 6.0)))) (fabs x-scale)) x-scale)
    x-scale)
   (* a (* b b)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / a) * ((((sqrt((8.0 * (pow(b, 4.0) * pow(a, 6.0)))) / fabs(x_45_scale)) * x_45_scale) * x_45_scale) / (a * (b * b)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (0.25d0 / a) * ((((sqrt((8.0d0 * ((b ** 4.0d0) * (a ** 6.0d0)))) / abs(x_45scale)) * x_45scale) * x_45scale) / (a * (b * b)))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / a) * ((((Math.sqrt((8.0 * (Math.pow(b, 4.0) * Math.pow(a, 6.0)))) / Math.abs(x_45_scale)) * x_45_scale) * x_45_scale) / (a * (b * b)));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (0.25 / a) * ((((math.sqrt((8.0 * (math.pow(b, 4.0) * math.pow(a, 6.0)))) / math.fabs(x_45_scale)) * x_45_scale) * x_45_scale) / (a * (b * b)))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 / a) * Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((b ^ 4.0) * (a ^ 6.0)))) / abs(x_45_scale)) * x_45_scale) * x_45_scale) / Float64(a * Float64(b * b))))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (0.25 / a) * ((((sqrt((8.0 * ((b ^ 4.0) * (a ^ 6.0)))) / abs(x_45_scale)) * x_45_scale) * x_45_scale) / (a * (b * b)));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 / a), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[a, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] / N[(a * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

4. Applied rewrites 0.4%

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

6. Applied rewrites 0.6%

   \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

7. Taylor expanded in y-scale around 0

   \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   7. lower-pow.f64 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

9. Applied rewrites 0.9%

   \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

11. Applied rewrites 5.8%

    \[\leadsto \frac{0.25}{a} \cdot \color{blue}{\frac{\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale}{a \cdot \left(b \cdot b\right)}} \]
12. Add Preprocessing

Alternative 11: 2.6% accurate, 11.5× speedup

Math

\[\frac{\left(\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale\right) \cdot 0.25}{\left(\left(a \cdot a\right) \cdot b\right) \cdot b} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (*
   (*
    (* (/ (sqrt (* 8.0 (* (pow b 4.0) (pow a 6.0)))) (fabs x-scale)) x-scale)
    x-scale)
   0.25)
  (* (* (* a a) b) b)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((sqrt((8.0 * (pow(b, 4.0) * pow(a, 6.0)))) / fabs(x_45_scale)) * x_45_scale) * x_45_scale) * 0.25) / (((a * a) * b) * b);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = ((((sqrt((8.0d0 * ((b ** 4.0d0) * (a ** 6.0d0)))) / abs(x_45scale)) * x_45scale) * x_45scale) * 0.25d0) / (((a * a) * b) * b)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return ((((Math.sqrt((8.0 * (Math.pow(b, 4.0) * Math.pow(a, 6.0)))) / Math.abs(x_45_scale)) * x_45_scale) * x_45_scale) * 0.25) / (((a * a) * b) * b);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return ((((math.sqrt((8.0 * (math.pow(b, 4.0) * math.pow(a, 6.0)))) / math.fabs(x_45_scale)) * x_45_scale) * x_45_scale) * 0.25) / (((a * a) * b) * b)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(Float64(sqrt(Float64(8.0 * Float64((b ^ 4.0) * (a ^ 6.0)))) / abs(x_45_scale)) * x_45_scale) * x_45_scale) * 0.25) / Float64(Float64(Float64(a * a) * b) * b))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = ((((sqrt((8.0 * ((b ^ 4.0) * (a ^ 6.0)))) / abs(x_45_scale)) * x_45_scale) * x_45_scale) * 0.25) / (((a * a) * b) * b);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[b, 4.0], $MachinePrecision] * N[Power[a, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * x$45$scale), $MachinePrecision] * x$45$scale), $MachinePrecision] * 0.25), $MachinePrecision] / N[(N[(N[(a * a), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision]), $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

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

4. Applied rewrites 0.4%

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

6. Applied rewrites 0.6%

   \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

7. Taylor expanded in y-scale around 0

   \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   7. lower-pow.f64 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

9. Applied rewrites 0.9%

   \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

11. Applied rewrites 2.6%

    \[\leadsto \frac{\left(\left(\frac{\sqrt{8 \cdot \left({b}^{4} \cdot {a}^{6}\right)}}{\left|x-scale\right|} \cdot x-scale\right) \cdot x-scale\right) \cdot 0.25}{\color{blue}{\left(\left(a \cdot a\right) \cdot b\right) \cdot b}} \]
12. Add Preprocessing

Alternative 12: 2.0% accurate, 11.1× speedup

Math

\[\frac{0.25 \cdot \left(\frac{\sqrt{8 \cdot \left({a}^{6} \cdot {b}^{4}\right)}}{\left|x-scale\right|} \cdot \left(\left|x-scale\right| \cdot \left|x-scale\right|\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (/
  (*
   0.25
   (*
    (/ (sqrt (* 8.0 (* (pow a 6.0) (pow b 4.0)))) (fabs x-scale))
    (* (fabs x-scale) (fabs x-scale))))
  (* (* a a) (* b b))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 * ((sqrt((8.0 * (pow(a, 6.0) * pow(b, 4.0)))) / fabs(x_45_scale)) * (fabs(x_45_scale) * fabs(x_45_scale)))) / ((a * a) * (b * b));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle, x_45scale, y_45scale)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (0.25d0 * ((sqrt((8.0d0 * ((a ** 6.0d0) * (b ** 4.0d0)))) / abs(x_45scale)) * (abs(x_45scale) * abs(x_45scale)))) / ((a * a) * (b * b))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 * ((Math.sqrt((8.0 * (Math.pow(a, 6.0) * Math.pow(b, 4.0)))) / Math.abs(x_45_scale)) * (Math.abs(x_45_scale) * Math.abs(x_45_scale)))) / ((a * a) * (b * b));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (0.25 * ((math.sqrt((8.0 * (math.pow(a, 6.0) * math.pow(b, 4.0)))) / math.fabs(x_45_scale)) * (math.fabs(x_45_scale) * math.fabs(x_45_scale)))) / ((a * a) * (b * b))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 * Float64(Float64(sqrt(Float64(8.0 * Float64((a ^ 6.0) * (b ^ 4.0)))) / abs(x_45_scale)) * Float64(abs(x_45_scale) * abs(x_45_scale)))) / Float64(Float64(a * a) * Float64(b * b)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (0.25 * ((sqrt((8.0 * ((a ^ 6.0) * (b ^ 4.0)))) / abs(x_45_scale)) * (abs(x_45_scale) * abs(x_45_scale)))) / ((a * a) * (b * b));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(0.25 * N[(N[(N[Sqrt[N[(8.0 * N[(N[Power[a, 6.0], $MachinePrecision] * N[Power[b, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(a * a), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $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

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

4. Applied rewrites 0.4%

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

6. Applied rewrites 0.6%

   \[\leadsto \frac{0.25 \cdot \left(\left(\sqrt{\frac{{\left(a \cdot b\right)}^{4} \cdot \left(\mathsf{fma}\left(\frac{b}{x-scale}, \frac{b}{x-scale}, \frac{a \cdot a}{y-scale \cdot y-scale}\right) + \left|\frac{b \cdot b}{x-scale \cdot x-scale} - \frac{a \cdot a}{y-scale \cdot y-scale}\right|\right)}{\left(y-scale \cdot y-scale\right) \cdot \left(x-scale \cdot x-scale\right)} \cdot 8} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left(x-scale \cdot x-scale\right)\right)}{\color{blue}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)}} \]

7. Taylor expanded in y-scale around 0

   \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   5. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

   7. lower-pow.f64 0.9%

      \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

9. Applied rewrites 0.9%

   \[\leadsto \frac{0.25 \cdot \left(\sqrt{8 \cdot \frac{{a}^{6} \cdot {b}^{4}}{{x-scale}^{2}}} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

10. Taylor expanded in x-scale around 0

    \[\leadsto \frac{0.25 \cdot \left(\frac{\sqrt{8 \cdot \left({a}^{6} \cdot {b}^{4}\right)}}{x-scale} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
11. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\frac{\sqrt{8 \cdot \left({a}^{6} \cdot {b}^{4}\right)}}{x-scale} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

    2. lower-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\frac{\sqrt{8 \cdot \left({a}^{6} \cdot {b}^{4}\right)}}{x-scale} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\frac{\sqrt{8 \cdot \left({a}^{6} \cdot {b}^{4}\right)}}{x-scale} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\frac{\sqrt{8 \cdot \left({a}^{6} \cdot {b}^{4}\right)}}{x-scale} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

    5. lower-pow.f64 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \left(\frac{\sqrt{8 \cdot \left({a}^{6} \cdot {b}^{4}\right)}}{x-scale} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

    6. lower-pow.f64 1.1%

       \[\leadsto \frac{0.25 \cdot \left(\frac{\sqrt{8 \cdot \left({a}^{6} \cdot {b}^{4}\right)}}{x-scale} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]

12. Applied rewrites 1.1%

    \[\leadsto \frac{0.25 \cdot \left(\frac{\sqrt{8 \cdot \left({a}^{6} \cdot {b}^{4}\right)}}{x-scale} \cdot \left(x-scale \cdot x-scale\right)\right)}{\left(a \cdot a\right) \cdot \left(b \cdot b\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025191 
(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))))