b from scale-rotated-ellipse

Percentage Accurate: 0.1% → 23.8%

Time: 19.3s

Alternatives: 7

Speedup: 12.3×

• could not determine a ground truth

  1. a = -4.146169337650273e-296
  2. b = 1.0016801374884211e+201
  3. angle = -8.390142697619709e-274
  4. x-scale = -1.486397985321748e-290
  5. y-scale = 7.670626476506938e+292

Specification

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (/ angle 180) PI))
       (t_1 (sin t_0))
       (t_2 (cos t_0))
       (t_3
        (/
         (/ (+ (pow (* a t_2) 2) (pow (* b t_1) 2)) y-scale)
         y-scale))
       (t_4
        (/
         (/ (+ (pow (* a t_1) 2) (pow (* b t_2) 2)) x-scale)
         x-scale))
       (t_5 (* (* b a) (* b (- a))))
       (t_6 (/ (* 4 t_5) (pow (* x-scale y-scale) 2))))
  (/
   (-
    (sqrt
     (*
      (* (* 2 t_6) t_5)
      (-
       (+ t_4 t_3)
       (sqrt
        (+
         (pow (- t_4 t_3) 2)
         (pow
          (/
           (/ (* (* (* 2 (- (pow b 2) (pow a 2))) t_1) t_2) x-scale)
           y-scale)
          2)))))))
   t_6)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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) PI))
       (t_1 (sin t_0))
       (t_2 (cos t_0))
       (t_3
        (/
         (/ (+ (pow (* a t_2) 2) (pow (* b t_1) 2)) y-scale)
         y-scale))
       (t_4
        (/
         (/ (+ (pow (* a t_1) 2) (pow (* b t_2) 2)) x-scale)
         x-scale))
       (t_5 (* (* b a) (* b (- a))))
       (t_6 (/ (* 4 t_5) (pow (* x-scale y-scale) 2))))
  (/
   (-
    (sqrt
     (*
      (* (* 2 t_6) t_5)
      (-
       (+ t_4 t_3)
       (sqrt
        (+
         (pow (- t_4 t_3) 2)
         (pow
          (/
           (/ (* (* (* 2 (- (pow b 2) (pow a 2))) t_1) t_2) x-scale)
           y-scale)
          2)))))))
   t_6)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 23.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|b\right| \cdot \left|a\right|\right) \cdot \left(\left|b\right| \cdot \left(-\left|a\right|\right)\right)\\ t_1 := \frac{\frac{{\left(\left|a\right|\right)}^{2}}{y-scale}}{y-scale}\\ t_2 := {\left(\left|b\right|\right)}^{0}\\ t_3 := \frac{4 \cdot t\_0}{{\left(x-scale \cdot y-scale\right)}^{0}}\\ t_4 := \sin \left(\frac{angle}{180} \cdot \pi\right)\\ t_5 := \frac{\frac{{\left(\left|a\right| \cdot t\_4\right)}^{0} + {\left(\left|b\right| \cdot 1\right)}^{0}}{x-scale}}{x-scale}\\ \mathbf{if}\;\left|b\right| \leq 269999999999999983781351517232651829316791575871472185235871269351981701922816:\\ \;\;\;\;\frac{1}{4} \cdot \frac{\left|a\right| \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{\left(\left|b\right|\right)}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{\left(\left(0 \cdot t\_3\right) \cdot t\_0\right) \cdot \left(\left(t\_5 + t\_1\right) - \sqrt{{\left(t\_5 - t\_1\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left(t\_2 - {\left(\left|a\right|\right)}^{0}\right)\right) \cdot t\_4\right) \cdot 1}{x-scale}}{y-scale}\right)}^{0}}\right)}}{t\_3}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* (* (fabs b) (fabs a)) (* (fabs b) (- (fabs a)))))
       (t_1 (/ (/ (pow (fabs a) 2) y-scale) y-scale))
       (t_2 (pow (fabs b) 0))
       (t_3 (/ (* 4 t_0) (pow (* x-scale y-scale) 0)))
       (t_4 (sin (* (/ angle 180) PI)))
       (t_5
        (/
         (/
          (+ (pow (* (fabs a) t_4) 0) (pow (* (fabs b) 1) 0))
          x-scale)
         x-scale)))
  (if (<=
       (fabs b)
       269999999999999983781351517232651829316791575871472185235871269351981701922816)
    (*
     1/4
     (/
      (*
       (fabs a)
       (*
        (pow x-scale 0)
        (sqrt
         (*
          8
          (/
           (*
            (pow (fabs b) 4)
            (-
             (+ 1/2 (* 1/2 (cos (* 1/90 (* angle PI)))))
             (sqrt (pow (cos (* 1/180 (* angle PI))) 4))))
           (pow x-scale 0))))))
      t_2))
    (/
     (-
      (sqrt
       (*
        (* (* 0 t_3) t_0)
        (-
         (+ t_5 t_1)
         (sqrt
          (+
           (pow (- t_5 t_1) 0)
           (pow
            (/
             (/ (* (* (* 0 (- t_2 (pow (fabs a) 0))) t_4) 1) x-scale)
             y-scale)
            0)))))))
     t_3))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (fabs(b) * fabs(a)) * (fabs(b) * -fabs(a));
	double t_1 = (pow(fabs(a), 2.0) / y_45_scale) / y_45_scale;
	double t_2 = pow(fabs(b), 0.0);
	double t_3 = (4.0 * t_0) / pow((x_45_scale * y_45_scale), 0.0);
	double t_4 = sin(((angle / 180.0) * ((double) M_PI)));
	double t_5 = ((pow((fabs(a) * t_4), 0.0) + pow((fabs(b) * 1.0), 0.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if (fabs(b) <= 2.7e+77) {
		tmp = 0.25 * ((fabs(a) * (pow(x_45_scale, 0.0) * sqrt((8.0 * ((pow(fabs(b), 4.0) * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) - sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0)))) / pow(x_45_scale, 0.0)))))) / t_2);
	} else {
		tmp = -sqrt((((0.0 * t_3) * t_0) * ((t_5 + t_1) - sqrt((pow((t_5 - t_1), 0.0) + pow((((((0.0 * (t_2 - pow(fabs(a), 0.0))) * t_4) * 1.0) / x_45_scale) / y_45_scale), 0.0)))))) / t_3;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.abs(b) * Math.abs(a)) * (Math.abs(b) * -Math.abs(a));
	double t_1 = (Math.pow(Math.abs(a), 2.0) / y_45_scale) / y_45_scale;
	double t_2 = Math.pow(Math.abs(b), 0.0);
	double t_3 = (4.0 * t_0) / Math.pow((x_45_scale * y_45_scale), 0.0);
	double t_4 = Math.sin(((angle / 180.0) * Math.PI));
	double t_5 = ((Math.pow((Math.abs(a) * t_4), 0.0) + Math.pow((Math.abs(b) * 1.0), 0.0)) / x_45_scale) / x_45_scale;
	double tmp;
	if (Math.abs(b) <= 2.7e+77) {
		tmp = 0.25 * ((Math.abs(a) * (Math.pow(x_45_scale, 0.0) * Math.sqrt((8.0 * ((Math.pow(Math.abs(b), 4.0) * ((0.5 + (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) - Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 4.0)))) / Math.pow(x_45_scale, 0.0)))))) / t_2);
	} else {
		tmp = -Math.sqrt((((0.0 * t_3) * t_0) * ((t_5 + t_1) - Math.sqrt((Math.pow((t_5 - t_1), 0.0) + Math.pow((((((0.0 * (t_2 - Math.pow(Math.abs(a), 0.0))) * t_4) * 1.0) / x_45_scale) / y_45_scale), 0.0)))))) / t_3;
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (math.fabs(b) * math.fabs(a)) * (math.fabs(b) * -math.fabs(a))
	t_1 = (math.pow(math.fabs(a), 2.0) / y_45_scale) / y_45_scale
	t_2 = math.pow(math.fabs(b), 0.0)
	t_3 = (4.0 * t_0) / math.pow((x_45_scale * y_45_scale), 0.0)
	t_4 = math.sin(((angle / 180.0) * math.pi))
	t_5 = ((math.pow((math.fabs(a) * t_4), 0.0) + math.pow((math.fabs(b) * 1.0), 0.0)) / x_45_scale) / x_45_scale
	tmp = 0
	if math.fabs(b) <= 2.7e+77:
		tmp = 0.25 * ((math.fabs(a) * (math.pow(x_45_scale, 0.0) * math.sqrt((8.0 * ((math.pow(math.fabs(b), 4.0) * ((0.5 + (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) - math.sqrt(math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 4.0)))) / math.pow(x_45_scale, 0.0)))))) / t_2)
	else:
		tmp = -math.sqrt((((0.0 * t_3) * t_0) * ((t_5 + t_1) - math.sqrt((math.pow((t_5 - t_1), 0.0) + math.pow((((((0.0 * (t_2 - math.pow(math.fabs(a), 0.0))) * t_4) * 1.0) / x_45_scale) / y_45_scale), 0.0)))))) / t_3
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(abs(b) * abs(a)) * Float64(abs(b) * Float64(-abs(a))))
	t_1 = Float64(Float64((abs(a) ^ 2.0) / y_45_scale) / y_45_scale)
	t_2 = abs(b) ^ 0.0
	t_3 = Float64(Float64(4.0 * t_0) / (Float64(x_45_scale * y_45_scale) ^ 0.0))
	t_4 = sin(Float64(Float64(angle / 180.0) * pi))
	t_5 = Float64(Float64(Float64((Float64(abs(a) * t_4) ^ 0.0) + (Float64(abs(b) * 1.0) ^ 0.0)) / x_45_scale) / x_45_scale)
	tmp = 0.0
	if (abs(b) <= 2.7e+77)
		tmp = Float64(0.25 * Float64(Float64(abs(a) * Float64((x_45_scale ^ 0.0) * sqrt(Float64(8.0 * Float64(Float64((abs(b) ^ 4.0) * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) - sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0)))) / (x_45_scale ^ 0.0)))))) / t_2));
	else
		tmp = Float64(Float64(-sqrt(Float64(Float64(Float64(0.0 * t_3) * t_0) * Float64(Float64(t_5 + t_1) - sqrt(Float64((Float64(t_5 - t_1) ^ 0.0) + (Float64(Float64(Float64(Float64(Float64(0.0 * Float64(t_2 - (abs(a) ^ 0.0))) * t_4) * 1.0) / x_45_scale) / y_45_scale) ^ 0.0))))))) / t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (abs(b) * abs(a)) * (abs(b) * -abs(a));
	t_1 = ((abs(a) ^ 2.0) / y_45_scale) / y_45_scale;
	t_2 = abs(b) ^ 0.0;
	t_3 = (4.0 * t_0) / ((x_45_scale * y_45_scale) ^ 0.0);
	t_4 = sin(((angle / 180.0) * pi));
	t_5 = ((((abs(a) * t_4) ^ 0.0) + ((abs(b) * 1.0) ^ 0.0)) / x_45_scale) / x_45_scale;
	tmp = 0.0;
	if (abs(b) <= 2.7e+77)
		tmp = 0.25 * ((abs(a) * ((x_45_scale ^ 0.0) * sqrt((8.0 * (((abs(b) ^ 4.0) * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * pi))))) - sqrt((cos((0.005555555555555556 * (angle * pi))) ^ 4.0)))) / (x_45_scale ^ 0.0)))))) / t_2);
	else
		tmp = -sqrt((((0.0 * t_3) * t_0) * ((t_5 + t_1) - sqrt((((t_5 - t_1) ^ 0.0) + ((((((0.0 * (t_2 - (abs(a) ^ 0.0))) * t_4) * 1.0) / x_45_scale) / y_45_scale) ^ 0.0)))))) / t_3;
	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] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * (-N[Abs[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[Abs[a], $MachinePrecision], 2], $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Abs[b], $MachinePrecision], 0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(4 * t$95$0), $MachinePrecision] / N[Power[N[(x$45$scale * y$45$scale), $MachinePrecision], 0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(angle / 180), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[Power[N[(N[Abs[a], $MachinePrecision] * t$95$4), $MachinePrecision], 0], $MachinePrecision] + N[Power[N[(N[Abs[b], $MachinePrecision] * 1), $MachinePrecision], 0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]}, If[LessEqual[N[Abs[b], $MachinePrecision], 269999999999999983781351517232651829316791575871472185235871269351981701922816], N[(1/4 * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 0], $MachinePrecision] * N[Sqrt[N[(8 * N[(N[(N[Power[N[Abs[b], $MachinePrecision], 4], $MachinePrecision] * N[(N[(1/2 + N[(1/2 * N[Cos[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(1/180 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[((-N[Sqrt[N[(N[(N[(0 * t$95$3), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(t$95$5 + t$95$1), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(t$95$5 - t$95$1), $MachinePrecision], 0], $MachinePrecision] + N[Power[N[(N[(N[(N[(N[(0 * N[(t$95$2 - N[Power[N[Abs[a], $MachinePrecision], 0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] * 1), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision], 0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$3), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6999999999999998e77

   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 0.1%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 0.4%

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

   7. Taylor expanded in y-scale around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

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

   9. Applied rewrites 3.4%

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

   10. Taylor expanded in undef-var around zero

       \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
   11. Step-by-step derivation

   12. Applied rewrites 4.9%

       \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]

   13. Taylor expanded in undef-var around zero

       \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{{b}^{2}} \]
   14. Step-by-step derivation

   15. Applied rewrites 8.5%

       \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{{b}^{2}} \]

   16. Taylor expanded in undef-var around zero

       \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{{b}^{0}} \]
   17. Step-by-step derivation

   18. Applied rewrites 20.6%

       \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{{b}^{0}} \]

   if 2.6999999999999998e77 < 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 undef-var around zero

      \[\leadsto \frac{-\sqrt{\left(\left(\color{blue}{0} \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 0.1%

      \[\leadsto \frac{-\sqrt{\left(\left(\color{blue}{0} \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   5. Taylor expanded in undef-var around zero

      \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{\color{blue}{0}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 0.1%

      \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{\color{blue}{0}}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   8. Taylor expanded in undef-var around zero

      \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 0.1%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   11. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   12. Step-by-step derivation

   13. Applied rewrites 0.1%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   14. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   15. Step-by-step derivation

   16. Applied rewrites 0.1%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   17. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   18. Step-by-step derivation

   19. Applied rewrites 0.1%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   20. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   21. Step-by-step derivation

   22. Applied rewrites 0.0%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   23. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   24. Step-by-step derivation

   25. Applied rewrites 0.1%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   26. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   27. Step-by-step derivation

   28. Applied rewrites 0.1%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   29. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   30. Step-by-step derivation

   31. Applied rewrites 0.1%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{\color{blue}{0}}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   32. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{\color{blue}{0}} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   33. Step-by-step derivation

   34. Applied rewrites 0.2%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{\color{blue}{0}} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   35. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(\color{blue}{0} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   36. Step-by-step derivation

   37. Applied rewrites 0.8%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(\color{blue}{0} \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   38. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{\color{blue}{0}} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   39. Step-by-step derivation

   40. Applied rewrites 1.9%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{\color{blue}{0}} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   41. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{\color{blue}{0}}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   42. Step-by-step derivation

   43. Applied rewrites 2.7%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{\color{blue}{0}}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   44. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{\color{blue}{0}}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
   45. Step-by-step derivation

   46. Applied rewrites 2.7%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{\color{blue}{0}}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

   47. Taylor expanded in undef-var around zero

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{\color{blue}{0}}}} \]
   48. Step-by-step derivation

   49. Applied rewrites 3.9%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{\color{blue}{0}}}} \]

   50. Taylor expanded in angle around 0

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \color{blue}{1}\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]
   51. Step-by-step derivation

   52. Applied rewrites 3.9%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \color{blue}{1}\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

   53. Taylor expanded in angle around 0

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \color{blue}{1}\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]
   54. Step-by-step derivation

   55. Applied rewrites 3.9%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot \color{blue}{1}\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

   56. Taylor expanded in angle around 0

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \color{blue}{1}\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]
   57. Step-by-step derivation

   58. Applied rewrites 3.9%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \color{blue}{1}\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

   59. Taylor expanded in angle around 0

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \color{blue}{1}\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]
   60. Step-by-step derivation

   61. Applied rewrites 3.9%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \color{blue}{1}\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

   62. Taylor expanded in angle around 0

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \color{blue}{1}}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]
   63. Step-by-step derivation

   64. Applied rewrites 3.9%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \color{blue}{1}}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

   65. Taylor expanded in angle around 0

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot 1}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]
   66. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{a}^{2}}{\color{blue}{y-scale}}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot 1}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

       2. lower-pow.f64 4.3%

          \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot 1}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

   67. Applied rewrites 4.3%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot 1\right)}^{0} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot 1}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

   68. Taylor expanded in angle around 0

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot 1}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]
   69. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{a}^{2}}{\color{blue}{y-scale}}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot 1}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

       2. lower-pow.f64 4.3%

          \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot 1}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]

   70. Applied rewrites 4.3%

       \[\leadsto \frac{-\sqrt{\left(\left(0 \cdot \frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}\right) \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)\right) \cdot \left(\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} + \frac{\frac{{a}^{2}}{y-scale}}{y-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{0} + {\left(b \cdot 1\right)}^{0}}{x-scale}}{x-scale} - \frac{\color{blue}{\frac{{a}^{2}}{y-scale}}}{y-scale}\right)}^{0} + {\left(\frac{\frac{\left(\left(0 \cdot \left({b}^{0} - {a}^{0}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot 1}{x-scale}}{y-scale}\right)}^{0}}\right)}}{\frac{4 \cdot \left(\left(b \cdot a\right) \cdot \left(b \cdot \left(-a\right)\right)\right)}{{\left(x-scale \cdot y-scale\right)}^{0}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 20.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(\frac{1}{180} \cdot angle\right)\\ t_1 := \cos \left(\frac{1}{90} \cdot \left(\pi \cdot angle\right)\right)\\ \mathbf{if}\;\left|x-scale\right| \leq \frac{8373182103885391}{93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968}:\\ \;\;\;\;\frac{\frac{1}{4}}{b} \cdot \frac{\left(\left|a\right| \cdot \left(\left|x-scale\right| \cdot \left|x-scale\right|\right)\right) \cdot \frac{\sqrt{8 \cdot \left(\left(\left(\frac{1}{2} - t\_1 \cdot \frac{-1}{2}\right) - \sqrt{{\cos t\_0}^{4}}\right) \cdot {b}^{4}\right)}}{\left|\left|x-scale\right|\right|}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4}}{b} \cdot \frac{\left(\frac{\sqrt{-8 \cdot \left(\left(\left(-\left(\frac{1}{2} - t\_1 \cdot \frac{1}{2}\right)\right) + \sqrt{{\sin t\_0}^{4}}\right) \cdot {b}^{4}\right)}}{\left|y-scale\right|} \cdot \left(y-scale \cdot y-scale\right)\right) \cdot \left|a\right|}{b}\\ \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (let* ((t_0 (* PI (* 1/180 angle))) (t_1 (cos (* 1/90 (* PI angle)))))
  (if (<=
       (fabs x-scale)
       8373182103885391/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968)
    (*
     (/ 1/4 b)
     (/
      (*
       (* (fabs a) (* (fabs x-scale) (fabs x-scale)))
       (/
        (sqrt
         (*
          8
          (*
           (- (- 1/2 (* t_1 -1/2)) (sqrt (pow (cos t_0) 4)))
           (pow b 4))))
        (fabs (fabs x-scale))))
      b))
    (*
     (/ 1/4 b)
     (/
      (*
       (*
        (/
         (sqrt
          (*
           -8
           (*
            (+ (- (- 1/2 (* t_1 1/2))) (sqrt (pow (sin t_0) 4)))
            (pow b 4))))
         (fabs y-scale))
        (* y-scale y-scale))
       (fabs a))
      b)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_1 = cos((0.011111111111111112 * (((double) M_PI) * angle)));
	double tmp;
	if (fabs(x_45_scale) <= 9e-122) {
		tmp = (0.25 / b) * (((fabs(a) * (fabs(x_45_scale) * fabs(x_45_scale))) * (sqrt((8.0 * (((0.5 - (t_1 * -0.5)) - sqrt(pow(cos(t_0), 4.0))) * pow(b, 4.0)))) / fabs(fabs(x_45_scale)))) / b);
	} else {
		tmp = (0.25 / b) * ((((sqrt((-8.0 * ((-(0.5 - (t_1 * 0.5)) + sqrt(pow(sin(t_0), 4.0))) * pow(b, 4.0)))) / fabs(y_45_scale)) * (y_45_scale * y_45_scale)) * fabs(a)) / b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	double t_1 = Math.cos((0.011111111111111112 * (Math.PI * angle)));
	double tmp;
	if (Math.abs(x_45_scale) <= 9e-122) {
		tmp = (0.25 / b) * (((Math.abs(a) * (Math.abs(x_45_scale) * Math.abs(x_45_scale))) * (Math.sqrt((8.0 * (((0.5 - (t_1 * -0.5)) - Math.sqrt(Math.pow(Math.cos(t_0), 4.0))) * Math.pow(b, 4.0)))) / Math.abs(Math.abs(x_45_scale)))) / b);
	} else {
		tmp = (0.25 / b) * ((((Math.sqrt((-8.0 * ((-(0.5 - (t_1 * 0.5)) + Math.sqrt(Math.pow(Math.sin(t_0), 4.0))) * Math.pow(b, 4.0)))) / Math.abs(y_45_scale)) * (y_45_scale * y_45_scale)) * Math.abs(a)) / b);
	}
	return tmp;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = math.pi * (0.005555555555555556 * angle)
	t_1 = math.cos((0.011111111111111112 * (math.pi * angle)))
	tmp = 0
	if math.fabs(x_45_scale) <= 9e-122:
		tmp = (0.25 / b) * (((math.fabs(a) * (math.fabs(x_45_scale) * math.fabs(x_45_scale))) * (math.sqrt((8.0 * (((0.5 - (t_1 * -0.5)) - math.sqrt(math.pow(math.cos(t_0), 4.0))) * math.pow(b, 4.0)))) / math.fabs(math.fabs(x_45_scale)))) / b)
	else:
		tmp = (0.25 / b) * ((((math.sqrt((-8.0 * ((-(0.5 - (t_1 * 0.5)) + math.sqrt(math.pow(math.sin(t_0), 4.0))) * math.pow(b, 4.0)))) / math.fabs(y_45_scale)) * (y_45_scale * y_45_scale)) * math.fabs(a)) / b)
	return tmp

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_1 = cos(Float64(0.011111111111111112 * Float64(pi * angle)))
	tmp = 0.0
	if (abs(x_45_scale) <= 9e-122)
		tmp = Float64(Float64(0.25 / b) * Float64(Float64(Float64(abs(a) * Float64(abs(x_45_scale) * abs(x_45_scale))) * Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(t_1 * -0.5)) - sqrt((cos(t_0) ^ 4.0))) * (b ^ 4.0)))) / abs(abs(x_45_scale)))) / b));
	else
		tmp = Float64(Float64(0.25 / b) * Float64(Float64(Float64(Float64(sqrt(Float64(-8.0 * Float64(Float64(Float64(-Float64(0.5 - Float64(t_1 * 0.5))) + sqrt((sin(t_0) ^ 4.0))) * (b ^ 4.0)))) / abs(y_45_scale)) * Float64(y_45_scale * y_45_scale)) * abs(a)) / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = pi * (0.005555555555555556 * angle);
	t_1 = cos((0.011111111111111112 * (pi * angle)));
	tmp = 0.0;
	if (abs(x_45_scale) <= 9e-122)
		tmp = (0.25 / b) * (((abs(a) * (abs(x_45_scale) * abs(x_45_scale))) * (sqrt((8.0 * (((0.5 - (t_1 * -0.5)) - sqrt((cos(t_0) ^ 4.0))) * (b ^ 4.0)))) / abs(abs(x_45_scale)))) / b);
	else
		tmp = (0.25 / b) * ((((sqrt((-8.0 * ((-(0.5 - (t_1 * 0.5)) + sqrt((sin(t_0) ^ 4.0))) * (b ^ 4.0)))) / abs(y_45_scale)) * (y_45_scale * y_45_scale)) * abs(a)) / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(Pi * N[(1/180 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(1/90 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x$45$scale], $MachinePrecision], 8373182103885391/93035356709837681990313447409664580397266094167976711716030745495121828878514934185752454491361736391777602765602070775492429008462675968], N[(N[(1/4 / b), $MachinePrecision] * N[(N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Abs[x$45$scale], $MachinePrecision] * N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8 * N[(N[(N[(1/2 - N[(t$95$1 * -1/2), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[t$95$0], $MachinePrecision], 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[Abs[x$45$scale], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], N[(N[(1/4 / b), $MachinePrecision] * N[(N[(N[(N[(N[Sqrt[N[(-8 * N[(N[((-N[(1/2 - N[(t$95$1 * 1/2), $MachinePrecision]), $MachinePrecision]) + N[Sqrt[N[Power[N[Sin[t$95$0], $MachinePrecision], 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[y$45$scale], $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x-scale < 8.9999999999999996e-122

   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 0.1%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 0.4%

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

   7. Taylor expanded in y-scale around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
   8. Step-by-step derivation

      1. lower-sqrt.f64 N/A

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

   9. Applied rewrites 3.4%

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

   11. Applied rewrites 16.6%

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

   if 8.9999999999999996e-122 < x-scale

   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 0.1%

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

   4. Taylor expanded in a around inf

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

   6. Applied rewrites 0.4%

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

   7. Taylor expanded in x-scale around 0

      \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{-8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}} + -1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({y-scale}^{2} \cdot \sqrt{-8 \cdot \frac{{b}^{4} \cdot \left(\sqrt{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{4}} + -1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}{{y-scale}^{2}}}\right)}{{b}^{2}} \]

   9. Applied rewrites 2.9%

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

   11. Applied rewrites 17.0%

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

Alternative 3: 20.0% accurate, 3.6× speedup

Math

\[\frac{1}{4} \cdot \frac{\left|a\right| \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{{b}^{0}} \]

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 1/4
 (/
  (*
   (fabs a)
   (*
    (pow x-scale 0)
    (sqrt
     (*
      8
      (/
       (*
        (pow b 4)
        (-
         (+ 1/2 (* 1/2 (cos (* 1/90 (* angle PI)))))
         (sqrt (pow (cos (* 1/180 (* angle PI))) 4))))
       (pow x-scale 0))))))
  (pow b 0))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((fabs(a) * (pow(x_45_scale, 0.0) * sqrt((8.0 * ((pow(b, 4.0) * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) - sqrt(pow(cos((0.005555555555555556 * (angle * ((double) M_PI)))), 4.0)))) / pow(x_45_scale, 0.0)))))) / pow(b, 0.0));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * ((Math.abs(a) * (Math.pow(x_45_scale, 0.0) * Math.sqrt((8.0 * ((Math.pow(b, 4.0) * ((0.5 + (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) - Math.sqrt(Math.pow(Math.cos((0.005555555555555556 * (angle * Math.PI))), 4.0)))) / Math.pow(x_45_scale, 0.0)))))) / Math.pow(b, 0.0));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.25 * ((math.fabs(a) * (math.pow(x_45_scale, 0.0) * math.sqrt((8.0 * ((math.pow(b, 4.0) * ((0.5 + (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) - math.sqrt(math.pow(math.cos((0.005555555555555556 * (angle * math.pi))), 4.0)))) / math.pow(x_45_scale, 0.0)))))) / math.pow(b, 0.0))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(Float64(abs(a) * Float64((x_45_scale ^ 0.0) * sqrt(Float64(8.0 * Float64(Float64((b ^ 4.0) * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) - sqrt((cos(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 4.0)))) / (x_45_scale ^ 0.0)))))) / (b ^ 0.0)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * ((abs(a) * ((x_45_scale ^ 0.0) * sqrt((8.0 * (((b ^ 4.0) * ((0.5 + (0.5 * cos((0.011111111111111112 * (angle * pi))))) - sqrt((cos((0.005555555555555556 * (angle * pi))) ^ 4.0)))) / (x_45_scale ^ 0.0)))))) / (b ^ 0.0));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(1/4 * N[(N[(N[Abs[a], $MachinePrecision] * N[(N[Power[x$45$scale, 0], $MachinePrecision] * N[Sqrt[N[(8 * N[(N[(N[Power[b, 4], $MachinePrecision] * N[(N[(1/2 + N[(1/2 * N[Cos[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(1/180 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x$45$scale, 0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

4. Taylor expanded in a around inf

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

6. Applied rewrites 0.4%

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

7. Taylor expanded in y-scale around 0

   \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

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

9. Applied rewrites 3.4%

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

10. Taylor expanded in undef-var around zero

    \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
11. Step-by-step derivation

12. Applied rewrites 4.9%

    \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]

13. Taylor expanded in undef-var around zero

    \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{{b}^{2}} \]
14. Step-by-step derivation

15. Applied rewrites 8.5%

    \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{{b}^{2}} \]

16. Taylor expanded in undef-var around zero

    \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{{b}^{0}} \]
17. Step-by-step derivation

18. Applied rewrites 20.6%

    \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{0} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{0}}}\right)}{{b}^{0}} \]
19. Add Preprocessing

Alternative 4: 16.9% accurate, 5.6× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 (/ 1/4 b)
 (/
  (*
   (* (fabs a) (* x-scale x-scale))
   (/
    (sqrt
     (*
      8
      (*
       (-
        (- 1/2 (* (cos (* 1/90 (* PI angle))) -1/2))
        (sqrt (pow (cos (* PI (* 1/180 angle))) 4)))
       (pow b 4))))
    (fabs x-scale)))
  b)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / b) * (((fabs(a) * (x_45_scale * x_45_scale)) * (sqrt((8.0 * (((0.5 - (cos((0.011111111111111112 * (((double) M_PI) * angle))) * -0.5)) - sqrt(pow(cos((((double) M_PI) * (0.005555555555555556 * angle))), 4.0))) * pow(b, 4.0)))) / fabs(x_45_scale))) / b);
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (0.25 / b) * (((Math.abs(a) * (x_45_scale * x_45_scale)) * (Math.sqrt((8.0 * (((0.5 - (Math.cos((0.011111111111111112 * (Math.PI * angle))) * -0.5)) - Math.sqrt(Math.pow(Math.cos((Math.PI * (0.005555555555555556 * angle))), 4.0))) * Math.pow(b, 4.0)))) / Math.abs(x_45_scale))) / b);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (0.25 / b) * (((math.fabs(a) * (x_45_scale * x_45_scale)) * (math.sqrt((8.0 * (((0.5 - (math.cos((0.011111111111111112 * (math.pi * angle))) * -0.5)) - math.sqrt(math.pow(math.cos((math.pi * (0.005555555555555556 * angle))), 4.0))) * math.pow(b, 4.0)))) / math.fabs(x_45_scale))) / b)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(0.25 / b) * Float64(Float64(Float64(abs(a) * Float64(x_45_scale * x_45_scale)) * Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(pi * angle))) * -0.5)) - sqrt((cos(Float64(pi * Float64(0.005555555555555556 * angle))) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale))) / b))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (0.25 / b) * (((abs(a) * (x_45_scale * x_45_scale)) * (sqrt((8.0 * (((0.5 - (cos((0.011111111111111112 * (pi * angle))) * -0.5)) - sqrt((cos((pi * (0.005555555555555556 * angle))) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale))) / b);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(1/4 / b), $MachinePrecision] * N[(N[(N[(N[Abs[a], $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(8 * N[(N[(N[(1/2 - N[(N[Cos[N[(1/90 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -1/2), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(Pi * N[(1/180 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

4. Taylor expanded in a around inf

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

6. Applied rewrites 0.4%

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

7. Taylor expanded in y-scale around 0

   \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

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

9. Applied rewrites 3.4%

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

11. Applied rewrites 16.6%

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

Alternative 5: 7.3% accurate, 5.6× speedup

Math

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

FPCore

(FPCore (a b angle x-scale y-scale)
  :precision binary64
  (*
 1/4
 (*
  (fabs a)
  (/
   (*
    (/
     (sqrt
      (*
       8
       (*
        (-
         (- 1/2 (* (cos (* 1/90 (* PI angle))) -1/2))
         (sqrt (pow (cos (* PI (* 1/180 angle))) 4)))
        (pow b 4))))
     (fabs x-scale))
    (* x-scale x-scale))
   (* b b)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (fabs(a) * (((sqrt((8.0 * (((0.5 - (cos((0.011111111111111112 * (((double) M_PI) * angle))) * -0.5)) - sqrt(pow(cos((((double) M_PI) * (0.005555555555555556 * angle))), 4.0))) * pow(b, 4.0)))) / fabs(x_45_scale)) * (x_45_scale * x_45_scale)) / (b * b)));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.25 * (Math.abs(a) * (((Math.sqrt((8.0 * (((0.5 - (Math.cos((0.011111111111111112 * (Math.PI * angle))) * -0.5)) - Math.sqrt(Math.pow(Math.cos((Math.PI * (0.005555555555555556 * angle))), 4.0))) * Math.pow(b, 4.0)))) / Math.abs(x_45_scale)) * (x_45_scale * x_45_scale)) / (b * b)));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.25 * (math.fabs(a) * (((math.sqrt((8.0 * (((0.5 - (math.cos((0.011111111111111112 * (math.pi * angle))) * -0.5)) - math.sqrt(math.pow(math.cos((math.pi * (0.005555555555555556 * angle))), 4.0))) * math.pow(b, 4.0)))) / math.fabs(x_45_scale)) * (x_45_scale * x_45_scale)) / (b * b)))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(0.25 * Float64(abs(a) * Float64(Float64(Float64(sqrt(Float64(8.0 * Float64(Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(pi * angle))) * -0.5)) - sqrt((cos(Float64(pi * Float64(0.005555555555555556 * angle))) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale)) * Float64(x_45_scale * x_45_scale)) / Float64(b * b))))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.25 * (abs(a) * (((sqrt((8.0 * (((0.5 - (cos((0.011111111111111112 * (pi * angle))) * -0.5)) - sqrt((cos((pi * (0.005555555555555556 * angle))) ^ 4.0))) * (b ^ 4.0)))) / abs(x_45_scale)) * (x_45_scale * x_45_scale)) / (b * b)));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(1/4 * N[(N[Abs[a], $MachinePrecision] * N[(N[(N[(N[Sqrt[N[(8 * N[(N[(N[(1/2 - N[(N[Cos[N[(1/90 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -1/2), $MachinePrecision]), $MachinePrecision] - N[Sqrt[N[Power[N[Cos[N[(Pi * N[(1/180 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[b, 4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[x$45$scale], $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1%

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

3. Applied rewrites 0.1%

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

4. Taylor expanded in a around inf

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

6. Applied rewrites 0.4%

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

7. Taylor expanded in y-scale around 0

   \[\leadsto \frac{1}{4} \cdot \frac{a \cdot \left({x-scale}^{2} \cdot \sqrt{8 \cdot \frac{{b}^{4} \cdot \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) - \sqrt{{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{4}}\right)}{{x-scale}^{2}}}\right)}{{b}^{2}} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

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

9. Applied rewrites 3.4%

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

11. Applied rewrites 7.2%

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

Alternative 6: 1.2% accurate, 12.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, N[(N[(N[(N[((-N[Sqrt[N[(N[(N[(t$95$2 + t$95$0), $MachinePrecision] - N[Abs[N[(t$95$0 - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$1 / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(a * b), $MachinePrecision] * 4), $MachinePrecision]), $MachinePrecision] / N[((-a) * b), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $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 0.3%

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

6. Applied rewrites 0.4%

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

8. Applied rewrites 0.6%

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

10. Applied rewrites 1.2%

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

Alternative 7: 0.9% accurate, 12.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(b / N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(a * b), $MachinePrecision] * b), $MachinePrecision] * (-a)), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, N[(N[(N[(N[((-N[Sqrt[N[(N[(N[(t$95$2 + t$95$0), $MachinePrecision] - N[Abs[N[(t$95$0 - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$1 / N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(y$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[(N[(a * b), $MachinePrecision] * 4), $MachinePrecision] * N[((-a) * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * y$45$scale), $MachinePrecision] * N[(x$45$scale * x$45$scale), $MachinePrecision]), $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 0.3%

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

6. Applied rewrites 0.4%

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

8. Applied rewrites 0.6%

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

10. Applied rewrites 0.9%

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

Reproduce

herbie shell --seed 2025285 -o generate:evaluate
(FPCore (a b angle x-scale y-scale)
  :name "b from scale-rotated-ellipse"
  :precision binary64
  (/ (- (sqrt (* (* (* 2 (/ (* 4 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2))) (* (* b a) (* b (- a)))) (- (+ (/ (/ (+ (pow (* a (sin (* (/ angle 180) PI))) 2) (pow (* b (cos (* (/ angle 180) PI))) 2)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180) PI))) 2) (pow (* b (sin (* (/ angle 180) PI))) 2)) y-scale) y-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180) PI))) 2) (pow (* b (cos (* (/ angle 180) PI))) 2)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180) PI))) 2) (pow (* b (sin (* (/ angle 180) PI))) 2)) y-scale) y-scale)) 2) (pow (/ (/ (* (* (* 2 (- (pow b 2) (pow a 2))) (sin (* (/ angle 180) PI))) (cos (* (/ angle 180) PI))) x-scale) y-scale) 2))))))) (/ (* 4 (* (* b a) (* b (- a)))) (pow (* x-scale y-scale) 2))))