b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 8.8%

Time: 37.2s

Alternatives: 5

Speedup: 8.0×

• could not determine a ground truth

  1. a = -4.0665594276248405e-302
  2. b = -8.768929651032944e+63
  3. angle = 2.0681948451268632e+183
  4. x-scale = 6.360474675594535e+295
  5. y-scale = 8.233724329197631e-243

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: 0.1% 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: 8.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\\ t_1 := {\left(\left|a\right|\right)}^{4}\\ t_2 := -\left|a\right|\\ t_3 := \left(b \cdot \left|a\right|\right) \cdot \left(b \cdot t\_2\right)\\ t_4 := \frac{4 \cdot t\_3}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ t_5 := 0.5 - 0.5 \cdot t\_0\\ t_6 := \frac{4 \cdot \left(\left|a\right| \cdot b\right)}{y-scale \cdot x-scale} \cdot \frac{t\_2 \cdot b}{y-scale \cdot x-scale}\\ t_7 := {\left(\left|a\right|\right)}^{2}\\ \mathbf{if}\;\left|a\right| \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_6\right) \cdot t\_3\right) \cdot \left(t\_7 \cdot \mathsf{fma}\left(0.5, \frac{t\_0}{{y-scale}^{2}}, 0.5 \cdot \frac{1}{{y-scale}^{2}}\right)\right)}}{t\_6}\\ \mathbf{elif}\;\left|a\right| \leq 2.75 \cdot 10^{+45}:\\ \;\;\;\;0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{t\_1 \cdot \left(t\_7 \cdot t\_5 - \sqrt{t\_1 \cdot {t\_5}^{2}}\right)}{{y-scale}^{2}}}}{t\_7}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_4\right) \cdot t\_3\right) \cdot 0}}{t\_4}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (cos (* 0.011111111111111112 (* angle PI))))
        (t_1 (pow (fabs a) 4.0))
        (t_2 (- (fabs a)))
        (t_3 (* (* b (fabs a)) (* b t_2)))
        (t_4 (/ (* 4.0 t_3) (pow (* x-scale y-scale) 2.0)))
        (t_5 (- 0.5 (* 0.5 t_0)))
        (t_6
         (*
          (/ (* 4.0 (* (fabs a) b)) (* y-scale x-scale))
          (/ (* t_2 b) (* y-scale x-scale))))
        (t_7 (pow (fabs a) 2.0)))
   (if (<= (fabs a) 1.6e-162)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_6) t_3)
         (*
          t_7
          (fma
           0.5
           (/ t_0 (pow y-scale 2.0))
           (* 0.5 (/ 1.0 (pow y-scale 2.0))))))))
      t_6)
     (if (<= (fabs a) 2.75e+45)
       (*
        0.25
        (/
         (*
          (pow y-scale 2.0)
          (sqrt
           (*
            8.0
            (/
             (* t_1 (- (* t_7 t_5) (sqrt (* t_1 (pow t_5 2.0)))))
             (pow y-scale 2.0)))))
         t_7))
       (/ (- (sqrt (* (* (* 2.0 t_4) t_3) 0.0))) t_4)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = cos((0.011111111111111112 * (angle * ((double) M_PI))));
	double t_1 = pow(fabs(a), 4.0);
	double t_2 = -fabs(a);
	double t_3 = (b * fabs(a)) * (b * t_2);
	double t_4 = (4.0 * t_3) / pow((x_45_scale * y_45_scale), 2.0);
	double t_5 = 0.5 - (0.5 * t_0);
	double t_6 = ((4.0 * (fabs(a) * b)) / (y_45_scale * x_45_scale)) * ((t_2 * b) / (y_45_scale * x_45_scale));
	double t_7 = pow(fabs(a), 2.0);
	double tmp;
	if (fabs(a) <= 1.6e-162) {
		tmp = -sqrt((((2.0 * t_6) * t_3) * (t_7 * fma(0.5, (t_0 / pow(y_45_scale, 2.0)), (0.5 * (1.0 / pow(y_45_scale, 2.0))))))) / t_6;
	} else if (fabs(a) <= 2.75e+45) {
		tmp = 0.25 * ((pow(y_45_scale, 2.0) * sqrt((8.0 * ((t_1 * ((t_7 * t_5) - sqrt((t_1 * pow(t_5, 2.0))))) / pow(y_45_scale, 2.0))))) / t_7);
	} else {
		tmp = -sqrt((((2.0 * t_4) * t_3) * 0.0)) / t_4;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = cos(Float64(0.011111111111111112 * Float64(angle * pi)))
	t_1 = abs(a) ^ 4.0
	t_2 = Float64(-abs(a))
	t_3 = Float64(Float64(b * abs(a)) * Float64(b * t_2))
	t_4 = Float64(Float64(4.0 * t_3) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	t_5 = Float64(0.5 - Float64(0.5 * t_0))
	t_6 = Float64(Float64(Float64(4.0 * Float64(abs(a) * b)) / Float64(y_45_scale * x_45_scale)) * Float64(Float64(t_2 * b) / Float64(y_45_scale * x_45_scale)))
	t_7 = abs(a) ^ 2.0
	tmp = 0.0
	if (abs(a) <= 1.6e-162)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_6) * t_3) * Float64(t_7 * fma(0.5, Float64(t_0 / (y_45_scale ^ 2.0)), Float64(0.5 * Float64(1.0 / (y_45_scale ^ 2.0)))))))) / t_6);
	elseif (abs(a) <= 2.75e+45)
		tmp = Float64(0.25 * Float64(Float64((y_45_scale ^ 2.0) * sqrt(Float64(8.0 * Float64(Float64(t_1 * Float64(Float64(t_7 * t_5) - sqrt(Float64(t_1 * (t_5 ^ 2.0))))) / (y_45_scale ^ 2.0))))) / t_7));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_4) * t_3) * 0.0))) / t_4);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Abs[a], $MachinePrecision], 4.0], $MachinePrecision]}, Block[{t$95$2 = (-N[Abs[a], $MachinePrecision])}, Block[{t$95$3 = N[(N[(b * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(b * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(4.0 * t$95$3), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 - N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = 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$2 * b), $MachinePrecision] / N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Power[N[Abs[a], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[a], $MachinePrecision], 1.6e-162], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$6), $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$7 * N[(0.5 * N[(t$95$0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$6), $MachinePrecision], If[LessEqual[N[Abs[a], $MachinePrecision], 2.75e+45], N[(0.25 * N[(N[(N[Power[y$45$scale, 2.0], $MachinePrecision] * N[Sqrt[N[(8.0 * N[(N[(t$95$1 * N[(N[(t$95$7 * t$95$5), $MachinePrecision] - N[Sqrt[N[(t$95$1 * N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$4), $MachinePrecision] * t$95$3), $MachinePrecision] * 0.0), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.5999999999999999e-162

   1. Initial program 0.1%

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

   2. Taylor expanded in x-scale around inf

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

   4. Applied rewrites 1.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 1.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 1.6%

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

   6. Applied rewrites 1.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 3.2%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 3.2%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 3.2%

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

   8. Applied rewrites 3.2%

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

   10. Applied rewrites 4.1%

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

   11. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-fma.f64 N/A

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

   13. Applied rewrites 5.2%

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

   if 1.5999999999999999e-162 < a < 2.75e45

   1. Initial program 0.1%

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

   3. Applied rewrites 0.2%

      \[\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}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale}, \frac{1}{x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale} - \mathsf{hypot}\left(\frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{y-scale \cdot x-scale}, \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), a \cdot a, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(b \cdot b\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), b \cdot b, \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)}{y-scale \cdot y-scale}\right)\right)}}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   4. Taylor expanded in b around 0

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

   6. Applied rewrites 0.6%

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

   7. Taylor expanded in x-scale around 0

      \[\leadsto 0.25 \cdot \frac{{y-scale}^{2} \cdot \sqrt{8 \cdot \frac{{a}^{4} \cdot \left({a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{a}^{4} \cdot {\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}}\right)}{{y-scale}^{2}}}}{{\color{blue}{a}}^{2}} \]
   8. Step-by-step derivation

   9. Applied rewrites 4.0%

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

   if 2.75e45 < a

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 2.1%

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

   7. Applied rewrites 2.1%

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

   8. Taylor expanded in y-scale around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 5.3%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 6.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|b\right| \cdot a\right) \cdot \left(\left|b\right| \cdot \left(-a\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ t_2 := \frac{4 \cdot \left(a \cdot \left|b\right|\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot \left|b\right|}{y-scale \cdot x-scale}\\ \mathbf{if}\;\left|b\right| \leq 1.56 \cdot 10^{-143}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot 0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot t\_0\right) \cdot \left({a}^{2} \cdot \mathsf{fma}\left(0.5, \frac{\cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)}{{y-scale}^{2}}, 0.5 \cdot \frac{1}{{y-scale}^{2}}\right)\right)}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (fabs b) a) (* (fabs b) (- a))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale y-scale) 2.0)))
        (t_2
         (*
          (/ (* 4.0 (* a (fabs b))) (* y-scale x-scale))
          (/ (* (- a) (fabs b)) (* y-scale x-scale)))))
   (if (<= (fabs b) 1.56e-143)
     (/ (- (sqrt (* (* (* 2.0 t_1) t_0) 0.0))) t_1)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_2) t_0)
         (*
          (pow a 2.0)
          (fma
           0.5
           (/ (cos (* 0.011111111111111112 (* angle PI))) (pow y-scale 2.0))
           (* 0.5 (/ 1.0 (pow y-scale 2.0))))))))
      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) * (fabs(b) * -a);
	double t_1 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double t_2 = ((4.0 * (a * fabs(b))) / (y_45_scale * x_45_scale)) * ((-a * fabs(b)) / (y_45_scale * x_45_scale));
	double tmp;
	if (fabs(b) <= 1.56e-143) {
		tmp = -sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	} else {
		tmp = -sqrt((((2.0 * t_2) * t_0) * (pow(a, 2.0) * fma(0.5, (cos((0.011111111111111112 * (angle * ((double) M_PI)))) / pow(y_45_scale, 2.0)), (0.5 * (1.0 / pow(y_45_scale, 2.0))))))) / t_2;
	}
	return tmp;
}

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(abs(b) * a) * Float64(abs(b) * Float64(-a)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	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)))
	tmp = 0.0
	if (abs(b) <= 1.56e-143)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * 0.0))) / t_1);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * t_0) * Float64((a ^ 2.0) * fma(0.5, Float64(cos(Float64(0.011111111111111112 * Float64(angle * pi))) / (y_45_scale ^ 2.0)), Float64(0.5 * Float64(1.0 / (y_45_scale ^ 2.0)))))))) / t_2);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $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]}, If[LessEqual[N[Abs[b], $MachinePrecision], 1.56e-143], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.0), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Power[a, 2.0], $MachinePrecision] * N[(0.5 * N[(N[Cos[N[(0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.5600000000000001e-143

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 2.1%

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

   7. Applied rewrites 2.1%

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

   8. Taylor expanded in y-scale around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 5.3%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   if 1.5600000000000001e-143 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in x-scale around inf

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

   4. Applied rewrites 1.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 1.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 1.6%

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

   6. Applied rewrites 1.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 3.2%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 3.2%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 3.2%

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

   8. Applied rewrites 3.2%

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

   10. Applied rewrites 4.1%

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

   11. Taylor expanded in a around inf

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

       1. lower-*.f64 N/A

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

       2. lower-pow.f64 N/A

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

       3. lower-fma.f64 N/A

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

   13. Applied rewrites 5.2%

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

Alternative 3: 5.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|b\right| \cdot a\right) \cdot \left(\left|b\right| \cdot \left(-a\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ t_2 := \frac{4 \cdot \left(a \cdot \left|b\right|\right)}{y-scale \cdot x-scale} \cdot \frac{\left(-a\right) \cdot \left|b\right|}{y-scale \cdot x-scale}\\ \mathbf{if}\;\left|b\right| \leq 2 \cdot 10^{-74}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot 0}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(2 \cdot t\_2\right) \cdot t\_0\right) \cdot \left(\frac{{a}^{2}}{{y-scale}^{2}} - \sqrt{\frac{{a}^{4}}{{y-scale}^{4}}}\right)}}{t\_2}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (fabs b) a) (* (fabs b) (- a))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale y-scale) 2.0)))
        (t_2
         (*
          (/ (* 4.0 (* a (fabs b))) (* y-scale x-scale))
          (/ (* (- a) (fabs b)) (* y-scale x-scale)))))
   (if (<= (fabs b) 2e-74)
     (/ (- (sqrt (* (* (* 2.0 t_1) t_0) 0.0))) t_1)
     (/
      (-
       (sqrt
        (*
         (* (* 2.0 t_2) t_0)
         (-
          (/ (pow a 2.0) (pow y-scale 2.0))
          (sqrt (/ (pow a 4.0) (pow y-scale 4.0)))))))
      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) * (fabs(b) * -a);
	double t_1 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	double t_2 = ((4.0 * (a * fabs(b))) / (y_45_scale * x_45_scale)) * ((-a * fabs(b)) / (y_45_scale * x_45_scale));
	double tmp;
	if (fabs(b) <= 2e-74) {
		tmp = -sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	} else {
		tmp = -sqrt((((2.0 * t_2) * t_0) * ((pow(a, 2.0) / pow(y_45_scale, 2.0)) - sqrt((pow(a, 4.0) / pow(y_45_scale, 4.0)))))) / t_2;
	}
	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) :: tmp
    t_0 = (abs(b) * a) * (abs(b) * -a)
    t_1 = (4.0d0 * t_0) / ((x_45scale * y_45scale) ** 2.0d0)
    t_2 = ((4.0d0 * (a * abs(b))) / (y_45scale * x_45scale)) * ((-a * abs(b)) / (y_45scale * x_45scale))
    if (abs(b) <= 2d-74) then
        tmp = -sqrt((((2.0d0 * t_1) * t_0) * 0.0d0)) / t_1
    else
        tmp = -sqrt((((2.0d0 * t_2) * t_0) * (((a ** 2.0d0) / (y_45scale ** 2.0d0)) - sqrt(((a ** 4.0d0) / (y_45scale ** 4.0d0)))))) / t_2
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.abs(b) * a) * (Math.abs(b) * -a);
	double t_1 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 2.0);
	double t_2 = ((4.0 * (a * Math.abs(b))) / (y_45_scale * x_45_scale)) * ((-a * Math.abs(b)) / (y_45_scale * x_45_scale));
	double tmp;
	if (Math.abs(b) <= 2e-74) {
		tmp = -Math.sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	} else {
		tmp = -Math.sqrt((((2.0 * t_2) * t_0) * ((Math.pow(a, 2.0) / Math.pow(y_45_scale, 2.0)) - Math.sqrt((Math.pow(a, 4.0) / Math.pow(y_45_scale, 4.0)))))) / t_2;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (math.fabs(b) * a) * (math.fabs(b) * -a)
	t_1 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 2.0)
	t_2 = ((4.0 * (a * math.fabs(b))) / (y_45_scale * x_45_scale)) * ((-a * math.fabs(b)) / (y_45_scale * x_45_scale))
	tmp = 0
	if math.fabs(b) <= 2e-74:
		tmp = -math.sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1
	else:
		tmp = -math.sqrt((((2.0 * t_2) * t_0) * ((math.pow(a, 2.0) / math.pow(y_45_scale, 2.0)) - math.sqrt((math.pow(a, 4.0) / math.pow(y_45_scale, 4.0)))))) / t_2
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(abs(b) * a) * Float64(abs(b) * Float64(-a)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	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)))
	tmp = 0.0
	if (abs(b) <= 2e-74)
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * 0.0))) / t_1);
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_2) * t_0) * Float64(Float64((a ^ 2.0) / (y_45_scale ^ 2.0)) - sqrt(Float64((a ^ 4.0) / (y_45_scale ^ 4.0))))))) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (abs(b) * a) * (abs(b) * -a);
	t_1 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 2.0);
	t_2 = ((4.0 * (a * abs(b))) / (y_45_scale * x_45_scale)) * ((-a * abs(b)) / (y_45_scale * x_45_scale));
	tmp = 0.0;
	if (abs(b) <= 2e-74)
		tmp = -sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
	else
		tmp = -sqrt((((2.0 * t_2) * t_0) * (((a ^ 2.0) / (y_45_scale ^ 2.0)) - sqrt(((a ^ 4.0) / (y_45_scale ^ 4.0)))))) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[Abs[b], $MachinePrecision] * a), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 2.0], $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]}, If[LessEqual[N[Abs[b], $MachinePrecision], 2e-74], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.0), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(2.0 * t$95$2), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[Power[a, 2.0], $MachinePrecision] / N[Power[y$45$scale, 2.0], $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[(N[Power[a, 4.0], $MachinePrecision] / N[Power[y$45$scale, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.9999999999999999e-74

   1. Initial program 0.1%

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

   2. Taylor expanded in angle around 0

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 2.1%

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

   7. Applied rewrites 2.1%

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

   8. Taylor expanded in y-scale around 0

      \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 5.3%

       \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   if 1.9999999999999999e-74 < b

   1. Initial program 0.1%

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

   2. Taylor expanded in x-scale around inf

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

   4. Applied rewrites 1.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 1.6%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 1.6%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 1.6%

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

   6. Applied rewrites 1.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. lower-/.f64 3.2%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 3.2%

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

      21. lift-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lift-*.f64 3.2%

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in angle around 0

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

       1. lower--.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-pow.f64 N/A

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

       8. lower-pow.f64 3.7%

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

   11. Applied rewrites 3.7%

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

Alternative 4: 5.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} t_0 := \left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\\ t_1 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{2}}\\ \frac{-\sqrt{\left(\left(2 \cdot t\_1\right) \cdot t\_0\right) \cdot 0}}{t\_1} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* b a) (* b (- a))))
        (t_1 (/ (* 4.0 t_0) (pow (* x-scale y-scale) 2.0))))
   (/ (- (sqrt (* (* (* 2.0 t_1) t_0) 0.0))) t_1)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (b * a) * (b * -a);
	double t_1 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 2.0);
	return -sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
}

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
    t_0 = (b * a) * (b * -a)
    t_1 = (4.0d0 * t_0) / ((x_45scale * y_45scale) ** 2.0d0)
    code = -sqrt((((2.0d0 * t_1) * t_0) * 0.0d0)) / t_1
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 * a) * (b * -a);
	double t_1 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 2.0);
	return -Math.sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (b * a) * (b * -a)
	t_1 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 2.0)
	return -math.sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(b * a) * Float64(b * Float64(-a)))
	t_1 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 2.0))
	return Float64(Float64(-sqrt(Float64(Float64(Float64(2.0 * t_1) * t_0) * 0.0))) / t_1)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (b * a) * (b * -a);
	t_1 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 2.0);
	tmp = -sqrt((((2.0 * t_1) * t_0) * 0.0)) / t_1;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b * a), $MachinePrecision] * N[(b * (-a)), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * t$95$0), $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$1), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.0), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision]]]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.1%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 2.1%

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

7. Applied rewrites 2.1%

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

8. Taylor expanded in y-scale around 0

   \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 5.3%

    \[\leadsto \frac{-\sqrt{\left(\left(2 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot 0}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
11. Add Preprocessing

Alternative 5: 1.1% accurate, 8.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale\\ t_1 := \left(b \cdot a\right) \cdot b\\ \frac{-\sqrt{\left(\left(\frac{1}{y-scale \cdot y-scale} - \frac{\sqrt{1}}{y-scale \cdot y-scale}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\left(\frac{-a}{t\_0} \cdot t\_1\right) \cdot 8\right) \cdot \left(t\_1 \cdot \left(-a\right)\right)\right)}}{\left(\left(-a\right) \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot b\right)} \cdot t\_0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* (* x-scale x-scale) y-scale) y-scale)) (t_1 (* (* b a) b)))
   (*
    (/
     (-
      (sqrt
       (*
        (*
         (- (/ 1.0 (* y-scale y-scale)) (/ (sqrt 1.0) (* y-scale y-scale)))
         (* a a))
        (* (* (* (/ (- a) t_0) t_1) 8.0) (* t_1 (- a))))))
     (* (* (- a) b) (* (* 4.0 a) b)))
    t_0)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale;
	double t_1 = (b * a) * b;
	return (-sqrt(((((1.0 / (y_45_scale * y_45_scale)) - (sqrt(1.0) / (y_45_scale * y_45_scale))) * (a * a)) * ((((-a / t_0) * t_1) * 8.0) * (t_1 * -a)))) / ((-a * b) * ((4.0 * a) * b))) * t_0;
}

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
    t_0 = ((x_45scale * x_45scale) * y_45scale) * y_45scale
    t_1 = (b * a) * b
    code = (-sqrt(((((1.0d0 / (y_45scale * y_45scale)) - (sqrt(1.0d0) / (y_45scale * y_45scale))) * (a * a)) * ((((-a / t_0) * t_1) * 8.0d0) * (t_1 * -a)))) / ((-a * b) * ((4.0d0 * a) * b))) * t_0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale;
	double t_1 = (b * a) * b;
	return (-Math.sqrt(((((1.0 / (y_45_scale * y_45_scale)) - (Math.sqrt(1.0) / (y_45_scale * y_45_scale))) * (a * a)) * ((((-a / t_0) * t_1) * 8.0) * (t_1 * -a)))) / ((-a * b) * ((4.0 * a) * b))) * t_0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = ((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale
	t_1 = (b * a) * b
	return (-math.sqrt(((((1.0 / (y_45_scale * y_45_scale)) - (math.sqrt(1.0) / (y_45_scale * y_45_scale))) * (a * a)) * ((((-a / t_0) * t_1) * 8.0) * (t_1 * -a)))) / ((-a * b) * ((4.0 * a) * b))) * t_0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(Float64(x_45_scale * x_45_scale) * y_45_scale) * y_45_scale)
	t_1 = Float64(Float64(b * a) * b)
	return Float64(Float64(Float64(-sqrt(Float64(Float64(Float64(Float64(1.0 / Float64(y_45_scale * y_45_scale)) - Float64(sqrt(1.0) / Float64(y_45_scale * y_45_scale))) * Float64(a * a)) * Float64(Float64(Float64(Float64(Float64(-a) / t_0) * t_1) * 8.0) * Float64(t_1 * Float64(-a)))))) / Float64(Float64(Float64(-a) * b) * Float64(Float64(4.0 * a) * b))) * t_0)
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = ((x_45_scale * x_45_scale) * y_45_scale) * y_45_scale;
	t_1 = (b * a) * b;
	tmp = (-sqrt(((((1.0 / (y_45_scale * y_45_scale)) - (sqrt(1.0) / (y_45_scale * y_45_scale))) * (a * a)) * ((((-a / t_0) * t_1) * 8.0) * (t_1 * -a)))) / ((-a * b) * ((4.0 * a) * b))) * t_0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * a), $MachinePrecision] * b), $MachinePrecision]}, N[(N[((-N[Sqrt[N[(N[(N[(N[(1.0 / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] - N[(N[Sqrt[1.0], $MachinePrecision] / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[((-a) / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] * 8.0), $MachinePrecision] * N[(t$95$1 * (-a)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[((-a) * b), $MachinePrecision] * N[(N[(4.0 * a), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Initial program 0.1%

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

2. Taylor expanded in angle around 0

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

4. Applied rewrites 0.1%

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

5. Taylor expanded in a around inf

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

   1. lower-*.f64 N/A

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

   2. lower-pow.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-pow.f64 2.1%

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

7. Applied rewrites 2.1%

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

9. Applied rewrites 1.1%

   \[\leadsto \color{blue}{\frac{-\sqrt{\left(\left(\frac{1}{y-scale \cdot y-scale} - \frac{\sqrt{1}}{y-scale \cdot y-scale}\right) \cdot \left(a \cdot a\right)\right) \cdot \left(\left(\left(\frac{-a}{\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale} \cdot \left(\left(b \cdot a\right) \cdot b\right)\right) \cdot 8\right) \cdot \left(\left(\left(b \cdot a\right) \cdot b\right) \cdot \left(-a\right)\right)\right)}}{\left(\left(-a\right) \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot b\right)} \cdot \left(\left(\left(x-scale \cdot x-scale\right) \cdot y-scale\right) \cdot y-scale\right)} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025187 
(FPCore (a b angle x-scale y-scale)
  :name "b 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))))