raw-angle from scale-rotated-ellipse

Percentage Accurate: 16.9% → 57.7%

Time: 43.6s

Alternatives: 28

Speedup: 22.2×

• could not determine a ground truth

  1. a = 3.4542475613271794e+138
  2. b = 1.4060044046152488e+65
  3. angle = -7.96815605754228e+61
  4. x-scale = -9.736092845635132e-261
  5. y-scale = -9.688409460997876e+142

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ 180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\pi} \end{array} \end{array} \]

FPCore

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

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

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.cos(t_0);
	double t_2 = Math.sin(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	return 180.0 * (Math.atan((((t_4 - t_5) - Math.sqrt((Math.pow((t_5 - t_4), 2.0) + Math.pow(t_3, 2.0)))) / t_3)) / Math.PI);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 16.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_2\right) \cdot t\_1}{x-scale}}{y-scale}\\ t_4 := \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{y-scale}}{y-scale}\\ t_5 := \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{x-scale}}{x-scale}\\ 180 \cdot \frac{\tan^{-1} \left(\frac{\left(t\_4 - t\_5\right) - \sqrt{{\left(t\_5 - t\_4\right)}^{2} + {t\_3}^{2}}}{t\_3}\right)}{\pi} \end{array} \end{array} \]

FPCore

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

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

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.cos(t_0);
	double t_2 = Math.sin(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_2) * t_1) / x_45_scale) / y_45_scale;
	double t_4 = ((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / y_45_scale) / y_45_scale;
	double t_5 = ((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / x_45_scale) / x_45_scale;
	return 180.0 * (Math.atan((((t_4 - t_5) - Math.sqrt((Math.pow((t_5 - t_4), 2.0) + Math.pow(t_3, 2.0)))) / t_3)) / Math.PI);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 57.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\\ \mathbf{if}\;b\_m \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t\_0}{x-scale \cdot t\_1}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot t\_1}{\sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{t\_2 \cdot t\_2}\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (cos t_0))
        (t_2 (pow (cbrt (sqrt PI)) 3.0)))
   (if (<= b_m 4.2e-114)
     (* 180.0 (/ (atan (/ (* y-scale (sin t_0)) (* x-scale t_1))) PI))
     (if (<= b_m 5.4e+99)
       (*
        180.0
        (/
         (atan
          (/
           (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
           (* (* angle PI) (* (+ b_m a) (- b_m a)))))
         PI))
       (*
        180.0
        (/
         (atan
          (/
           (* y-scale t_1)
           (*
            (sin
             (*
              (* (* 0.005555555555555556 angle) (sqrt PI))
              (sqrt (* t_2 t_2))))
            (- x-scale))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = cos(t_0);
	double t_2 = pow(cbrt(sqrt(((double) M_PI))), 3.0);
	double tmp;
	if (b_m <= 4.2e-114) {
		tmp = 180.0 * (atan(((y_45_scale * sin(t_0)) / (x_45_scale * t_1))) / ((double) M_PI));
	} else if (b_m <= 5.4e+99) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * t_1) / (sin((((0.005555555555555556 * angle) * sqrt(((double) M_PI))) * sqrt((t_2 * t_2)))) * -x_45_scale))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.cos(t_0);
	double t_2 = Math.pow(Math.cbrt(Math.sqrt(Math.PI)), 3.0);
	double tmp;
	if (b_m <= 4.2e-114) {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.sin(t_0)) / (x_45_scale * t_1))) / Math.PI);
	} else if (b_m <= 5.4e+99) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * t_1) / (Math.sin((((0.005555555555555556 * angle) * Math.sqrt(Math.PI)) * Math.sqrt((t_2 * t_2)))) * -x_45_scale))) / Math.PI);
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cos(t_0)
	t_2 = cbrt(sqrt(pi)) ^ 3.0
	tmp = 0.0
	if (b_m <= 4.2e-114)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(t_0)) / Float64(x_45_scale * t_1))) / pi));
	elseif (b_m <= 5.4e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * t_1) / Float64(sin(Float64(Float64(Float64(0.005555555555555556 * angle) * sqrt(pi)) * sqrt(Float64(t_2 * t_2)))) * Float64(-x_45_scale)))) / pi));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[b$95$m, 4.2e-114], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.4e+99], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * t$95$1), $MachinePrecision] / N[(N[Sin[N[(N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$2 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.19999999999999985e-114

   1. Initial program 14.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 31.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 47.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 47.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   if 4.19999999999999985e-114 < b < 5.39999999999999978e99

   1. Initial program 40.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 63.1

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

   8. Applied rewrites 63.1%

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

   if 5.39999999999999978e99 < b

   1. Initial program 9.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 58.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 58.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      2. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. add-sqr-sqrt N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-sqrt.f64 57.4

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\sqrt{\pi}}\right)}\right)}{\pi} \]

   10. Applied rewrites 57.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. add-cube-cbrt N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. pow3 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{3}}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{{\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{3}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. rem-square-sqrt N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{{\left(\sqrt[3]{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right)}^{3}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{{\left(\sqrt[3]{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right)}^{3}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. cbrt-prod N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}}^{3}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. unpow-prod-down N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-pow.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-cbrt.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-pow.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-cbrt.f64 67.8

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{\pi}}\right)}}^{3}}\right)}\right)}{\pi} \]

   12. Applied rewrites 67.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}}\right)}\right)}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{\sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\\ \mathbf{if}\;b\_m \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t\_0}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 3.5 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left(t\_1 \cdot t\_1\right)\right)\right)}\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI)))
        (t_1 (pow (cbrt (sqrt PI)) 3.0)))
   (if (<= b_m 4.2e-114)
     (* 180.0 (/ (atan (/ (* y-scale (sin t_0)) (* x-scale (cos t_0)))) PI))
     (if (<= b_m 3.5e+109)
       (*
        180.0
        (/
         (atan
          (/
           (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
           (* (* angle PI) (* (+ b_m a) (- b_m a)))))
         PI))
       (*
        (atan
         (/
          -1.0
          (*
           (/ x-scale y-scale)
           (tan (* 0.005555555555555556 (* angle (* t_1 t_1)))))))
        (* 180.0 (/ 1.0 PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double t_1 = pow(cbrt(sqrt(((double) M_PI))), 3.0);
	double tmp;
	if (b_m <= 4.2e-114) {
		tmp = 180.0 * (atan(((y_45_scale * sin(t_0)) / (x_45_scale * cos(t_0)))) / ((double) M_PI));
	} else if (b_m <= 3.5e+109) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = atan((-1.0 / ((x_45_scale / y_45_scale) * tan((0.005555555555555556 * (angle * (t_1 * t_1))))))) * (180.0 * (1.0 / ((double) M_PI)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double t_1 = Math.pow(Math.cbrt(Math.sqrt(Math.PI)), 3.0);
	double tmp;
	if (b_m <= 4.2e-114) {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.sin(t_0)) / (x_45_scale * Math.cos(t_0)))) / Math.PI);
	} else if (b_m <= 3.5e+109) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan((0.005555555555555556 * (angle * (t_1 * t_1))))))) * (180.0 * (1.0 / Math.PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	t_1 = cbrt(sqrt(pi)) ^ 3.0
	tmp = 0.0
	if (b_m <= 4.2e-114)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(t_0)) / Float64(x_45_scale * cos(t_0)))) / pi));
	elseif (b_m <= 3.5e+109)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * Float64(t_1 * t_1))))))) * Float64(180.0 * Float64(1.0 / pi)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[b$95$m, 4.2e-114], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.5e+109], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.19999999999999985e-114

   1. Initial program 14.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 31.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 47.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 47.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   if 4.19999999999999985e-114 < b < 3.49999999999999983e109

   1. Initial program 39.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 30.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 63.5

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

   8. Applied rewrites 63.5%

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

   if 3.49999999999999983e109 < b

   1. Initial program 7.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 7.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 56.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 56.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. add-cube-cbrt N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. pow3 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{3}}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. rem-square-sqrt N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot {\left(\sqrt[3]{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right)}^{3}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       5. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot {\left(\sqrt[3]{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right)}^{3}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       7. cbrt-prod N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}}^{3}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       8. unpow-prod-down N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       10. lower-pow.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left(\color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       11. lower-cbrt.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left({\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}}^{3} \cdot {\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       12. lower-pow.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left({\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3} \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}^{3}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       13. lower-cbrt.f64 67.0

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{\pi}}\right)}}^{3}\right)\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 67.0%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left({\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\right)\right)\right)}\right) \cdot \left(180 \cdot \frac{1}{\pi}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t\_0}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}{\sin \left(\sqrt{\pi} \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 4.2e-114)
     (* 180.0 (/ (atan (/ (* y-scale (sin t_0)) (* x-scale (cos t_0)))) PI))
     (if (<= b_m 6.5e+99)
       (*
        180.0
        (/
         (atan
          (/
           (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
           (* (* angle PI) (* (+ b_m a) (- b_m a)))))
         PI))
       (*
        180.0
        (/
         (atan
          (/
           (*
            y-scale
            (cos
             (*
              0.005555555555555556
              (* angle (* (sqrt (* PI (sqrt PI))) (sqrt (sqrt PI)))))))
           (*
            (sin (* (sqrt PI) (* (* 0.005555555555555556 angle) (sqrt PI))))
            (- x-scale))))
         PI))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 4.2e-114) {
		tmp = 180.0 * (atan(((y_45_scale * sin(t_0)) / (x_45_scale * cos(t_0)))) / ((double) M_PI));
	} else if (b_m <= 6.5e+99) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * cos((0.005555555555555556 * (angle * (sqrt((((double) M_PI) * sqrt(((double) M_PI)))) * sqrt(sqrt(((double) M_PI)))))))) / (sin((sqrt(((double) M_PI)) * ((0.005555555555555556 * angle) * sqrt(((double) M_PI))))) * -x_45_scale))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (b_m <= 4.2e-114) {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.sin(t_0)) / (x_45_scale * Math.cos(t_0)))) / Math.PI);
	} else if (b_m <= 6.5e+99) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.cos((0.005555555555555556 * (angle * (Math.sqrt((Math.PI * Math.sqrt(Math.PI))) * Math.sqrt(Math.sqrt(Math.PI))))))) / (Math.sin((Math.sqrt(Math.PI) * ((0.005555555555555556 * angle) * Math.sqrt(Math.PI)))) * -x_45_scale))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if b_m <= 4.2e-114:
		tmp = 180.0 * (math.atan(((y_45_scale * math.sin(t_0)) / (x_45_scale * math.cos(t_0)))) / math.pi)
	elif b_m <= 6.5e+99:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * math.cos((0.005555555555555556 * (angle * (math.sqrt((math.pi * math.sqrt(math.pi))) * math.sqrt(math.sqrt(math.pi))))))) / (math.sin((math.sqrt(math.pi) * ((0.005555555555555556 * angle) * math.sqrt(math.pi)))) * -x_45_scale))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 4.2e-114)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(t_0)) / Float64(x_45_scale * cos(t_0)))) / pi));
	elseif (b_m <= 6.5e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * cos(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(Float64(pi * sqrt(pi))) * sqrt(sqrt(pi))))))) / Float64(sin(Float64(sqrt(pi) * Float64(Float64(0.005555555555555556 * angle) * sqrt(pi)))) * Float64(-x_45_scale)))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (b_m <= 4.2e-114)
		tmp = 180.0 * (atan(((y_45_scale * sin(t_0)) / (x_45_scale * cos(t_0)))) / pi);
	elseif (b_m <= 6.5e+99)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * cos((0.005555555555555556 * (angle * (sqrt((pi * sqrt(pi))) * sqrt(sqrt(pi))))))) / (sin((sqrt(pi) * ((0.005555555555555556 * angle) * sqrt(pi)))) * -x_45_scale))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 4.2e-114], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 6.5e+99], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Cos[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.19999999999999985e-114

   1. Initial program 14.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 31.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 47.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 47.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   if 4.19999999999999985e-114 < b < 6.5000000000000004e99

   1. Initial program 40.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 63.1

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

   8. Applied rewrites 63.1%

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

   if 6.5000000000000004e99 < b

   1. Initial program 9.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 58.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 58.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      2. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. add-sqr-sqrt N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-sqrt.f64 57.4

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\sqrt{\pi}}\right)}\right)}{\pi} \]

   10. Applied rewrites 57.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-PI.f64 57.4

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}\right)}{\pi} \]

       2. rem-square-sqrt N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. sqrt-unprod N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. rem-square-sqrt N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. sqrt-prod N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. pow1/2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}}\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)}\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-sqrt.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. pow1/2 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       14. lower-sqrt.f64 64.8

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \color{blue}{\sqrt{\sqrt{\pi}}}\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}\right)}{\pi} \]

   12. Applied rewrites 64.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)}\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}\right)}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}{\sin \left(\sqrt{\pi} \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;b\_m \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin t\_0}{x-scale \cdot \cos t\_0}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 4.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* angle PI))))
   (if (<= b_m 4.2e-114)
     (* 180.0 (/ (atan (/ (* y-scale (sin t_0)) (* x-scale (cos t_0)))) PI))
     (if (<= b_m 4.4e+99)
       (*
        180.0
        (/
         (atan
          (/
           (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
           (* (* angle PI) (* (+ b_m a) (- b_m a)))))
         PI))
       (*
        (* 180.0 (/ 1.0 PI))
        (atan
         (/
          -1.0
          (*
           (/ x-scale y-scale)
           (tan
            (*
             0.005555555555555556
             (* angle (* (sqrt (* PI (sqrt PI))) (sqrt (sqrt PI))))))))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	double tmp;
	if (b_m <= 4.2e-114) {
		tmp = 180.0 * (atan(((y_45_scale * sin(t_0)) / (x_45_scale * cos(t_0)))) / ((double) M_PI));
	} else if (b_m <= 4.4e+99) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * (1.0 / ((double) M_PI))) * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((0.005555555555555556 * (angle * (sqrt((((double) M_PI) * sqrt(((double) M_PI)))) * sqrt(sqrt(((double) M_PI))))))))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 0.005555555555555556 * (angle * Math.PI);
	double tmp;
	if (b_m <= 4.2e-114) {
		tmp = 180.0 * (Math.atan(((y_45_scale * Math.sin(t_0)) / (x_45_scale * Math.cos(t_0)))) / Math.PI);
	} else if (b_m <= 4.4e+99) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = (180.0 * (1.0 / Math.PI)) * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan((0.005555555555555556 * (angle * (Math.sqrt((Math.PI * Math.sqrt(Math.PI))) * Math.sqrt(Math.sqrt(Math.PI)))))))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 0.005555555555555556 * (angle * math.pi)
	tmp = 0
	if b_m <= 4.2e-114:
		tmp = 180.0 * (math.atan(((y_45_scale * math.sin(t_0)) / (x_45_scale * math.cos(t_0)))) / math.pi)
	elif b_m <= 4.4e+99:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = (180.0 * (1.0 / math.pi)) * math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan((0.005555555555555556 * (angle * (math.sqrt((math.pi * math.sqrt(math.pi))) * math.sqrt(math.sqrt(math.pi)))))))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(0.005555555555555556 * Float64(angle * pi))
	tmp = 0.0
	if (b_m <= 4.2e-114)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * sin(t_0)) / Float64(x_45_scale * cos(t_0)))) / pi));
	elseif (b_m <= 4.4e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(Float64(pi * sqrt(pi))) * sqrt(sqrt(pi))))))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 0.005555555555555556 * (angle * pi);
	tmp = 0.0;
	if (b_m <= 4.2e-114)
		tmp = 180.0 * (atan(((y_45_scale * sin(t_0)) / (x_45_scale * cos(t_0)))) / pi);
	elseif (b_m <= 4.4e+99)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = (180.0 * (1.0 / pi)) * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((0.005555555555555556 * (angle * (sqrt((pi * sqrt(pi))) * sqrt(sqrt(pi)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 4.2e-114], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.4e+99], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 4.19999999999999985e-114

   1. Initial program 14.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 31.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-PI.f64 47.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 47.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   if 4.19999999999999985e-114 < b < 4.39999999999999956e99

   1. Initial program 40.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 63.1

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

   8. Applied rewrites 63.1%

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

   if 4.39999999999999956e99 < b

   1. Initial program 9.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 58.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 58.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 58.1

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

       2. rem-square-sqrt N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. sqrt-unprod N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. rem-square-sqrt N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       5. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       7. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       8. sqrt-prod N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       9. pow1/2 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       13. pow1/2 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       14. lower-sqrt.f64 63.9

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \color{blue}{\sqrt{\sqrt{\pi}}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 63.9%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-114}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 180 \cdot \frac{1}{\pi}\\ \mathbf{if}\;b\_m \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{0.005555555555555556 \cdot angle}{\frac{1}{\pi}}\right)}\right)\\ \mathbf{elif}\;b\_m \leq 4.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 180.0 (/ 1.0 PI))))
   (if (<= b_m 9.5e-207)
     (*
      180.0
      (/
       (atan
        (*
         (*
          (* b_m -180.0)
          (/
           b_m
           (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
         (* y-scale x-scale)))
       PI))
     (if (<= b_m 1.22e-124)
       (*
        t_0
        (atan
         (/
          -1.0
          (*
           (/ x-scale y-scale)
           (tan (/ (* 0.005555555555555556 angle) (/ 1.0 PI)))))))
       (if (<= b_m 4.4e+99)
         (*
          180.0
          (/
           (atan
            (/
             (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
             (* (* angle PI) (* (+ b_m a) (- b_m a)))))
           PI))
         (*
          t_0
          (atan
           (/
            -1.0
            (*
             (/ x-scale y-scale)
             (tan
              (*
               0.005555555555555556
               (*
                angle
                (* (sqrt (* PI (sqrt PI))) (sqrt (sqrt PI)))))))))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 180.0 * (1.0 / ((double) M_PI));
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.22e-124) {
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan(((0.005555555555555556 * angle) / (1.0 / ((double) M_PI)))))));
	} else if (b_m <= 4.4e+99) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((0.005555555555555556 * (angle * (sqrt((((double) M_PI) * sqrt(((double) M_PI)))) * sqrt(sqrt(((double) M_PI))))))))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 180.0 * (1.0 / Math.PI);
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.22e-124) {
		tmp = t_0 * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan(((0.005555555555555556 * angle) / (1.0 / Math.PI))))));
	} else if (b_m <= 4.4e+99) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = t_0 * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan((0.005555555555555556 * (angle * (Math.sqrt((Math.PI * Math.sqrt(Math.PI))) * Math.sqrt(Math.sqrt(Math.PI)))))))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 180.0 * (1.0 / math.pi)
	tmp = 0
	if b_m <= 9.5e-207:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.22e-124:
		tmp = t_0 * math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan(((0.005555555555555556 * angle) / (1.0 / math.pi))))))
	elif b_m <= 4.4e+99:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = t_0 * math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan((0.005555555555555556 * (angle * (math.sqrt((math.pi * math.sqrt(math.pi))) * math.sqrt(math.sqrt(math.pi)))))))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(180.0 * Float64(1.0 / pi))
	tmp = 0.0
	if (b_m <= 9.5e-207)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.22e-124)
		tmp = Float64(t_0 * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(Float64(0.005555555555555556 * angle) / Float64(1.0 / pi)))))));
	elseif (b_m <= 4.4e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(t_0 * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * Float64(sqrt(Float64(pi * sqrt(pi))) * sqrt(sqrt(pi))))))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 180.0 * (1.0 / pi);
	tmp = 0.0;
	if (b_m <= 9.5e-207)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.22e-124)
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan(((0.005555555555555556 * angle) / (1.0 / pi))))));
	elseif (b_m <= 4.4e+99)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((0.005555555555555556 * (angle * (sqrt((pi * sqrt(pi))) * sqrt(sqrt(pi)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 9.5e-207], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.22e-124], N[(t$95$0 * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] / N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.4e+99], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * N[(N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 9.50000000000000007e-207

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 9.50000000000000007e-207 < b < 1.22000000000000001e-124

   1. Initial program 10.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 39.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.5%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. remove-double-div N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{1}{\frac{1}{\mathsf{PI}\left(\right)}}}\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       5. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\left(\frac{1}{180} \cdot angle\right) \cdot \frac{1}{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       6. un-div-inv N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{1}{\mathsf{PI}\left(\right)}}\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       7. lower-/.f64 48.4

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\frac{0.005555555555555556 \cdot angle}{\frac{1}{\pi}}\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 48.4%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\frac{0.005555555555555556 \cdot angle}{\frac{1}{\pi}}\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   if 1.22000000000000001e-124 < b < 4.39999999999999956e99

   1. Initial program 38.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 4.39999999999999956e99 < b

   1. Initial program 9.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 58.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 58.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 58.1

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

       2. rem-square-sqrt N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. sqrt-unprod N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. rem-square-sqrt N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       5. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       6. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       7. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \sqrt{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       8. sqrt-prod N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       9. pow1/2 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{{\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       11. lower-sqrt.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left(\color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left(\sqrt{\color{blue}{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot {\left(\sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{2}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       13. pow1/2 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\sqrt{\sqrt{\mathsf{PI}\left(\right)}}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       14. lower-sqrt.f64 63.9

           \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \color{blue}{\sqrt{\sqrt{\pi}}}\right)\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 63.9%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)}\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{0.005555555555555556 \cdot angle}{\frac{1}{\pi}}\right)}\right)\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 47.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 180 \cdot \frac{1}{\pi}\\ \mathbf{if}\;b\_m \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{0.005555555555555556 \cdot angle}{\frac{1}{\pi}}\right)}\right)\\ \mathbf{elif}\;b\_m \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 180.0 (/ 1.0 PI))))
   (if (<= b_m 9.5e-207)
     (*
      180.0
      (/
       (atan
        (*
         (*
          (* b_m -180.0)
          (/
           b_m
           (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
         (* y-scale x-scale)))
       PI))
     (if (<= b_m 1.22e-124)
       (*
        t_0
        (atan
         (/
          -1.0
          (*
           (/ x-scale y-scale)
           (tan (/ (* 0.005555555555555556 angle) (/ 1.0 PI)))))))
       (if (<= b_m 1.1e+102)
         (*
          180.0
          (/
           (atan
            (/
             (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
             (* (* angle PI) (* (+ b_m a) (- b_m a)))))
           PI))
         (*
          t_0
          (atan
           (/
            -1.0
            (*
             (/ x-scale y-scale)
             (tan (* PI (* 0.005555555555555556 angle))))))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 180.0 * (1.0 / ((double) M_PI));
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.22e-124) {
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan(((0.005555555555555556 * angle) / (1.0 / ((double) M_PI)))))));
	} else if (b_m <= 1.1e+102) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((((double) M_PI) * (0.005555555555555556 * angle))))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 180.0 * (1.0 / Math.PI);
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.22e-124) {
		tmp = t_0 * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan(((0.005555555555555556 * angle) / (1.0 / Math.PI))))));
	} else if (b_m <= 1.1e+102) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = t_0 * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan((Math.PI * (0.005555555555555556 * angle))))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 180.0 * (1.0 / math.pi)
	tmp = 0
	if b_m <= 9.5e-207:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.22e-124:
		tmp = t_0 * math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan(((0.005555555555555556 * angle) / (1.0 / math.pi))))))
	elif b_m <= 1.1e+102:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = t_0 * math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan((math.pi * (0.005555555555555556 * angle))))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(180.0 * Float64(1.0 / pi))
	tmp = 0.0
	if (b_m <= 9.5e-207)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.22e-124)
		tmp = Float64(t_0 * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(Float64(0.005555555555555556 * angle) / Float64(1.0 / pi)))))));
	elseif (b_m <= 1.1e+102)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(t_0 * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(pi * Float64(0.005555555555555556 * angle)))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 180.0 * (1.0 / pi);
	tmp = 0.0;
	if (b_m <= 9.5e-207)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.22e-124)
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan(((0.005555555555555556 * angle) / (1.0 / pi))))));
	elseif (b_m <= 1.1e+102)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((pi * (0.005555555555555556 * angle))))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 9.5e-207], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.22e-124], N[(t$95$0 * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] / N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.1e+102], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 9.50000000000000007e-207

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 9.50000000000000007e-207 < b < 1.22000000000000001e-124

   1. Initial program 10.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 39.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.5%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. remove-double-div N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\frac{1}{\frac{1}{\mathsf{PI}\left(\right)}}}\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       5. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\left(\frac{1}{180} \cdot angle\right) \cdot \frac{1}{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       6. un-div-inv N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\frac{\frac{1}{180} \cdot angle}{\frac{1}{\mathsf{PI}\left(\right)}}\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       7. lower-/.f64 48.4

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\frac{0.005555555555555556 \cdot angle}{\frac{1}{\pi}}\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 48.4%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\frac{0.005555555555555556 \cdot angle}{\frac{1}{\pi}}\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   if 1.22000000000000001e-124 < b < 1.10000000000000004e102

   1. Initial program 38.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 1.10000000000000004e102 < b

   1. Initial program 9.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 58.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 58.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. lower-*.f64 61.3

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 61.3%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{0.005555555555555556 \cdot angle}{\frac{1}{\pi}}\right)}\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.3% accurate, 11.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 180 \cdot \frac{1}{\pi}\\ \mathbf{if}\;b\_m \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\\ \mathbf{elif}\;b\_m \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* 180.0 (/ 1.0 PI))))
   (if (<= b_m 9.5e-207)
     (*
      180.0
      (/
       (atan
        (*
         (*
          (* b_m -180.0)
          (/
           b_m
           (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
         (* y-scale x-scale)))
       PI))
     (if (<= b_m 1.65e-130)
       (*
        t_0
        (atan
         (/
          -1.0
          (*
           (/ x-scale y-scale)
           (tan (* 0.005555555555555556 (* angle PI)))))))
       (if (<= b_m 1.1e+102)
         (*
          180.0
          (/
           (atan
            (/
             (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
             (* (* angle PI) (* (+ b_m a) (- b_m a)))))
           PI))
         (*
          t_0
          (atan
           (/
            -1.0
            (*
             (/ x-scale y-scale)
             (tan (* PI (* 0.005555555555555556 angle))))))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 180.0 * (1.0 / ((double) M_PI));
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.65e-130) {
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((0.005555555555555556 * (angle * ((double) M_PI)))))));
	} else if (b_m <= 1.1e+102) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((((double) M_PI) * (0.005555555555555556 * angle))))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = 180.0 * (1.0 / Math.PI);
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.65e-130) {
		tmp = t_0 * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan((0.005555555555555556 * (angle * Math.PI))))));
	} else if (b_m <= 1.1e+102) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = t_0 * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan((Math.PI * (0.005555555555555556 * angle))))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = 180.0 * (1.0 / math.pi)
	tmp = 0
	if b_m <= 9.5e-207:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.65e-130:
		tmp = t_0 * math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan((0.005555555555555556 * (angle * math.pi))))))
	elif b_m <= 1.1e+102:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = t_0 * math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan((math.pi * (0.005555555555555556 * angle))))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(180.0 * Float64(1.0 / pi))
	tmp = 0.0
	if (b_m <= 9.5e-207)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.65e-130)
		tmp = Float64(t_0 * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(0.005555555555555556 * Float64(angle * pi)))))));
	elseif (b_m <= 1.1e+102)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(t_0 * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(pi * Float64(0.005555555555555556 * angle)))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = 180.0 * (1.0 / pi);
	tmp = 0.0;
	if (b_m <= 9.5e-207)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.65e-130)
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((0.005555555555555556 * (angle * pi))))));
	elseif (b_m <= 1.1e+102)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = t_0 * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((pi * (0.005555555555555556 * angle))))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 9.5e-207], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.65e-130], N[(t$95$0 * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.1e+102], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 9.50000000000000007e-207

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 9.50000000000000007e-207 < b < 1.6499999999999999e-130

   1. Initial program 10.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 39.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.5%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{\frac{x-scale}{y-scale}} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       5. lift-tan.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       6. lift-*.f64 39.5

          \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

       7. remove-double-neg N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       8. neg-sub0 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{0 - \left(\mathsf{neg}\left(\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       9. lower--.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{0 - \left(\mathsf{neg}\left(\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       10. lift-*.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{0 - \left(\mathsf{neg}\left(\color{blue}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       11. *-commutative N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{0 - \left(\mathsf{neg}\left(\color{blue}{\tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{x-scale}{y-scale}}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       12. distribute-rgt-neg-in N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{0 - \color{blue}{\tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\frac{x-scale}{y-scale}\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{0 - \color{blue}{\tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\frac{x-scale}{y-scale}\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       14. lift-/.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{0 - \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{x-scale}{y-scale}}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       15. distribute-frac-neg2 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{0 - \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{x-scale}{\mathsf{neg}\left(y-scale\right)}}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       16. lower-/.f64 N/A

           \[\leadsto \tan^{-1} \left(\frac{-1}{0 - \tan \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\frac{x-scale}{\mathsf{neg}\left(y-scale\right)}}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       17. lower-neg.f64 39.5

           \[\leadsto \tan^{-1} \left(\frac{-1}{0 - \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{\color{blue}{-y-scale}}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 39.5%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{0 - \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{-y-scale}}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   if 1.6499999999999999e-130 < b < 1.10000000000000004e102

   1. Initial program 38.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 1.10000000000000004e102 < b

   1. Initial program 9.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 58.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 58.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. lower-*.f64 61.3

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 61.3%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.5% accurate, 11.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;\frac{180}{\frac{\pi}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}}\\ \mathbf{elif}\;b\_m \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 9.5e-207)
   (*
    180.0
    (/
     (atan
      (*
       (*
        (* b_m -180.0)
        (/
         b_m
         (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
       (* y-scale x-scale)))
     PI))
   (if (<= b_m 1.22e-124)
     (/
      180.0
      (/
       PI
       (atan
        (/
         -1.0
         (*
          (/ x-scale y-scale)
          (tan (* angle (* 0.005555555555555556 PI))))))))
     (if (<= b_m 1.1e+102)
       (*
        180.0
        (/
         (atan
          (/
           (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
           (* (* angle PI) (* (+ b_m a) (- b_m a)))))
         PI))
       (*
        (* 180.0 (/ 1.0 PI))
        (atan
         (/
          -1.0
          (*
           (/ x-scale y-scale)
           (tan (* PI (* 0.005555555555555556 angle)))))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.22e-124) {
		tmp = 180.0 / (((double) M_PI) / atan((-1.0 / ((x_45_scale / y_45_scale) * tan((angle * (0.005555555555555556 * ((double) M_PI))))))));
	} else if (b_m <= 1.1e+102) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * (1.0 / ((double) M_PI))) * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((((double) M_PI) * (0.005555555555555556 * angle))))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.22e-124) {
		tmp = 180.0 / (Math.PI / Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan((angle * (0.005555555555555556 * Math.PI)))))));
	} else if (b_m <= 1.1e+102) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = (180.0 * (1.0 / Math.PI)) * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan((Math.PI * (0.005555555555555556 * angle))))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 9.5e-207:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.22e-124:
		tmp = 180.0 / (math.pi / math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan((angle * (0.005555555555555556 * math.pi)))))))
	elif b_m <= 1.1e+102:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = (180.0 * (1.0 / math.pi)) * math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan((math.pi * (0.005555555555555556 * angle))))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 9.5e-207)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.22e-124)
		tmp = Float64(180.0 / Float64(pi / atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(angle * Float64(0.005555555555555556 * pi))))))));
	elseif (b_m <= 1.1e+102)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(pi * Float64(0.005555555555555556 * angle)))))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 9.5e-207)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.22e-124)
		tmp = 180.0 / (pi / atan((-1.0 / ((x_45_scale / y_45_scale) * tan((angle * (0.005555555555555556 * pi)))))));
	elseif (b_m <= 1.1e+102)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = (180.0 * (1.0 / pi)) * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((pi * (0.005555555555555556 * angle))))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 9.5e-207], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.22e-124], N[(180.0 / N[(Pi / N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.1e+102], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < 9.50000000000000007e-207

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 9.50000000000000007e-207 < b < 1.22000000000000001e-124

   1. Initial program 10.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 39.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.5%

       \[\leadsto \color{blue}{\frac{180}{\frac{\pi}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right)}} \]

       2. *-commutative N/A

          \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)}\right)}} \]

       3. associate-*r* N/A

          \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}}\right)}} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}}\right)}} \]

       5. lower-*.f64 39.2

          \[\leadsto \frac{180}{\frac{\pi}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\color{blue}{\left(0.005555555555555556 \cdot \pi\right)} \cdot angle\right)}\right)}} \]

   12. Applied rewrites 39.2%

       \[\leadsto \frac{180}{\frac{\pi}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(0.005555555555555556 \cdot \pi\right) \cdot angle\right)}}\right)}} \]

   if 1.22000000000000001e-124 < b < 1.10000000000000004e102

   1. Initial program 38.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 1.10000000000000004e102 < b

   1. Initial program 9.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 58.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 58.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. lower-*.f64 61.3

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 61.3%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;\frac{180}{\frac{\pi}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.3% accurate, 11.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\\ \mathbf{if}\;b\_m \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_m \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          (* 180.0 (/ 1.0 PI))
          (atan
           (/
            -1.0
            (*
             (/ x-scale y-scale)
             (tan (* PI (* 0.005555555555555556 angle)))))))))
   (if (<= b_m 9.5e-207)
     (*
      180.0
      (/
       (atan
        (*
         (*
          (* b_m -180.0)
          (/
           b_m
           (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
         (* y-scale x-scale)))
       PI))
     (if (<= b_m 1.65e-130)
       t_0
       (if (<= b_m 1.1e+102)
         (*
          180.0
          (/
           (atan
            (/
             (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
             (* (* angle PI) (* (+ b_m a) (- b_m a)))))
           PI))
         t_0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (180.0 * (1.0 / ((double) M_PI))) * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((((double) M_PI) * (0.005555555555555556 * angle))))));
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.65e-130) {
		tmp = t_0;
	} else if (b_m <= 1.1e+102) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (180.0 * (1.0 / Math.PI)) * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * Math.tan((Math.PI * (0.005555555555555556 * angle))))));
	double tmp;
	if (b_m <= 9.5e-207) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.65e-130) {
		tmp = t_0;
	} else if (b_m <= 1.1e+102) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (180.0 * (1.0 / math.pi)) * math.atan((-1.0 / ((x_45_scale / y_45_scale) * math.tan((math.pi * (0.005555555555555556 * angle))))))
	tmp = 0
	if b_m <= 9.5e-207:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.65e-130:
		tmp = t_0
	elif b_m <= 1.1e+102:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = t_0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * tan(Float64(pi * Float64(0.005555555555555556 * angle)))))))
	tmp = 0.0
	if (b_m <= 9.5e-207)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.65e-130)
		tmp = t_0;
	elseif (b_m <= 1.1e+102)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (180.0 * (1.0 / pi)) * atan((-1.0 / ((x_45_scale / y_45_scale) * tan((pi * (0.005555555555555556 * angle))))));
	tmp = 0.0;
	if (b_m <= 9.5e-207)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.65e-130)
		tmp = t_0;
	elseif (b_m <= 1.1e+102)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 9.5e-207], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.65e-130], t$95$0, If[LessEqual[b$95$m, 1.1e+102], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.50000000000000007e-207

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 9.50000000000000007e-207 < b < 1.6499999999999999e-130 or 1.10000000000000004e102 < b

   1. Initial program 10.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 22.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 53.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 53.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.2%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. associate-*r* N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lift-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. lower-*.f64 55.6

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   12. Applied rewrites 55.6%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \color{blue}{\left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   if 1.6499999999999999e-130 < b < 1.10000000000000004e102

   1. Initial program 38.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{\left(b \cdot b\right) \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.5 \cdot 10^{-207}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+102}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.7% accurate, 11.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-1}{x-scale \cdot t\_0}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 3.45 \cdot 10^{+145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot t\_0}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (tan (* 0.005555555555555556 (* angle PI)))))
   (if (<= b_m 6.4e-206)
     (*
      180.0
      (/
       (atan
        (*
         (*
          (* b_m -180.0)
          (/
           b_m
           (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
         (* y-scale x-scale)))
       PI))
     (if (<= b_m 1.6e-130)
       (* 180.0 (/ (atan (* y-scale (/ -1.0 (* x-scale t_0)))) PI))
       (if (<= b_m 3.45e+145)
         (*
          180.0
          (/
           (atan
            (/
             (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
             (* (* angle PI) (* (+ b_m a) (- b_m a)))))
           PI))
         (*
          (* 180.0 (/ 1.0 PI))
          (atan (/ -1.0 (* (/ x-scale y-scale) t_0)))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = tan((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.6e-130) {
		tmp = 180.0 * (atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / ((double) M_PI));
	} else if (b_m <= 3.45e+145) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * (1.0 / ((double) M_PI))) * atan((-1.0 / ((x_45_scale / y_45_scale) * t_0)));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.tan((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.6e-130) {
		tmp = 180.0 * (Math.atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / Math.PI);
	} else if (b_m <= 3.45e+145) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = (180.0 * (1.0 / Math.PI)) * Math.atan((-1.0 / ((x_45_scale / y_45_scale) * t_0)));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.tan((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if b_m <= 6.4e-206:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.6e-130:
		tmp = 180.0 * (math.atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / math.pi)
	elif b_m <= 3.45e+145:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = (180.0 * (1.0 / math.pi)) * math.atan((-1.0 / ((x_45_scale / y_45_scale) * t_0)))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = tan(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (b_m <= 6.4e-206)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.6e-130)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(-1.0 / Float64(x_45_scale * t_0)))) / pi));
	elseif (b_m <= 3.45e+145)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(-1.0 / Float64(Float64(x_45_scale / y_45_scale) * t_0))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = tan((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (b_m <= 6.4e-206)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.6e-130)
		tmp = 180.0 * (atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / pi);
	elseif (b_m <= 3.45e+145)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = (180.0 * (1.0 / pi)) * atan((-1.0 / ((x_45_scale / y_45_scale) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 6.4e-206], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.6e-130], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(-1.0 / N[(x$45$scale * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.45e+145], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(-1.0 / N[(N[(x$45$scale / y$45$scale), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 6.39999999999999952e-206

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 6.39999999999999952e-206 < b < 1.6e-130

   1. Initial program 10.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 39.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      2. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. add-sqr-sqrt N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-sqrt.f64 47.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\sqrt{\pi}}\right)}\right)}{\pi} \]

   10. Applied rewrites 47.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}}\right)}{\pi} \]

   12. Applied rewrites 39.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\color{blue}{\frac{1}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot y-scale}\right)}{\pi} \]

   if 1.6e-130 < b < 3.45e145

   1. Initial program 36.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if 3.45e145 < b

   1. Initial program 0.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 65.2

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 65.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.4%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-1}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 3.45 \cdot 10^{+145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.6% accurate, 11.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 10^{-30}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{\sin \left(\sqrt{\pi} \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a 1e-30)
   (*
    180.0
    (/
     (atan
      (/
       y-scale
       (*
        (sin (* (sqrt PI) (* (* 0.005555555555555556 angle) (sqrt PI))))
        (- x-scale))))
     PI))
   (*
    180.0
    (/
     (atan
      (*
       (*
        (* b_m -180.0)
        (/
         b_m
         (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
       (* y-scale x-scale)))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 1e-30) {
		tmp = 180.0 * (atan((y_45_scale / (sin((sqrt(((double) M_PI)) * ((0.005555555555555556 * angle) * sqrt(((double) M_PI))))) * -x_45_scale))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 1e-30) {
		tmp = 180.0 * (Math.atan((y_45_scale / (Math.sin((Math.sqrt(Math.PI) * ((0.005555555555555556 * angle) * Math.sqrt(Math.PI)))) * -x_45_scale))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 1e-30:
		tmp = 180.0 * (math.atan((y_45_scale / (math.sin((math.sqrt(math.pi) * ((0.005555555555555556 * angle) * math.sqrt(math.pi)))) * -x_45_scale))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 1e-30)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale / Float64(sin(Float64(sqrt(pi) * Float64(Float64(0.005555555555555556 * angle) * sqrt(pi)))) * Float64(-x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 1e-30)
		tmp = 180.0 * (atan((y_45_scale / (sin((sqrt(pi) * ((0.005555555555555556 * angle) * sqrt(pi)))) * -x_45_scale))) / pi);
	else
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 1e-30], N[(180.0 * N[(N[ArcTan[N[(y$45$scale / N[(N[Sin[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1e-30

   1. Initial program 19.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 34.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 47.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      2. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. add-sqr-sqrt N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-sqrt.f64 48.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\sqrt{\pi}}\right)}\right)}{\pi} \]

   10. Applied rewrites 48.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}}\right)}{\pi} \]

   11. Taylor expanded in angle around 0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{1}}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 49.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \color{blue}{1}}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}\right)}{\pi} \]

   if 1e-30 < a

   1. Initial program 11.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 14.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 14.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 23.2

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

   10. Applied rewrites 23.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.6

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 39.8%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{-30}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{\sin \left(\sqrt{\pi} \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-1}{x-scale \cdot t\_0}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 3.5 \cdot 10^{+145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{-x-scale}}{t\_0}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (tan (* 0.005555555555555556 (* angle PI)))))
   (if (<= b_m 6.4e-206)
     (*
      180.0
      (/
       (atan
        (*
         (*
          (* b_m -180.0)
          (/
           b_m
           (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
         (* y-scale x-scale)))
       PI))
     (if (<= b_m 1.6e-130)
       (* 180.0 (/ (atan (* y-scale (/ -1.0 (* x-scale t_0)))) PI))
       (if (<= b_m 3.5e+145)
         (*
          180.0
          (/
           (atan
            (/
             (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
             (* (* angle PI) (* (+ b_m a) (- b_m a)))))
           PI))
         (* 180.0 (/ (atan (/ (/ y-scale (- x-scale)) t_0)) PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = tan((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.6e-130) {
		tmp = 180.0 * (atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / ((double) M_PI));
	} else if (b_m <= 3.5e+145) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale / -x_45_scale) / t_0)) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.tan((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.6e-130) {
		tmp = 180.0 * (Math.atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / Math.PI);
	} else if (b_m <= 3.5e+145) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale / -x_45_scale) / t_0)) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.tan((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if b_m <= 6.4e-206:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.6e-130:
		tmp = 180.0 * (math.atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / math.pi)
	elif b_m <= 3.5e+145:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale / -x_45_scale) / t_0)) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = tan(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (b_m <= 6.4e-206)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.6e-130)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(-1.0 / Float64(x_45_scale * t_0)))) / pi));
	elseif (b_m <= 3.5e+145)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / Float64(-x_45_scale)) / t_0)) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = tan((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (b_m <= 6.4e-206)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.6e-130)
		tmp = 180.0 * (atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / pi);
	elseif (b_m <= 3.5e+145)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale / -x_45_scale) / t_0)) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 6.4e-206], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.6e-130], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(-1.0 / N[(x$45$scale * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3.5e+145], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / (-x$45$scale)), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 6.39999999999999952e-206

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 6.39999999999999952e-206 < b < 1.6e-130

   1. Initial program 10.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 39.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      2. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. add-sqr-sqrt N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-sqrt.f64 47.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\sqrt{\pi}}\right)}\right)}{\pi} \]

   10. Applied rewrites 47.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}}\right)}{\pi} \]

   12. Applied rewrites 39.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\color{blue}{\frac{1}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot y-scale}\right)}{\pi} \]

   if 1.6e-130 < b < 3.5000000000000001e145

   1. Initial program 36.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if 3.5000000000000001e145 < b

   1. Initial program 0.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 65.2

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 65.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      2. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. add-sqr-sqrt N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-sqrt.f64 63.7

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\sqrt{\pi}}\right)}\right)}{\pi} \]

   10. Applied rewrites 63.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}}\right)}{\pi} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-cos.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lift-sqrt.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lift-sin.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 65.4%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\color{blue}{\frac{\frac{y-scale}{x-scale}}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-1}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{+145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{y-scale}{-x-scale}}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 47.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-1}{x-scale \cdot t\_0}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{y-scale}{t\_0 \cdot \left(-x-scale\right)}\right)}{0.005555555555555556 \cdot \pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (tan (* 0.005555555555555556 (* angle PI)))))
   (if (<= b_m 6.4e-206)
     (*
      180.0
      (/
       (atan
        (*
         (*
          (* b_m -180.0)
          (/
           b_m
           (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
         (* y-scale x-scale)))
       PI))
     (if (<= b_m 1.6e-130)
       (* 180.0 (/ (atan (* y-scale (/ -1.0 (* x-scale t_0)))) PI))
       (if (<= b_m 5.4e+99)
         (*
          180.0
          (/
           (atan
            (/
             (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
             (* (* angle PI) (* (+ b_m a) (- b_m a)))))
           PI))
         (/
          (atan (/ y-scale (* t_0 (- x-scale))))
          (* 0.005555555555555556 PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = tan((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.6e-130) {
		tmp = 180.0 * (atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / ((double) M_PI));
	} else if (b_m <= 5.4e+99) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = atan((y_45_scale / (t_0 * -x_45_scale))) / (0.005555555555555556 * ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.tan((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.6e-130) {
		tmp = 180.0 * (Math.atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / Math.PI);
	} else if (b_m <= 5.4e+99) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = Math.atan((y_45_scale / (t_0 * -x_45_scale))) / (0.005555555555555556 * Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.tan((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if b_m <= 6.4e-206:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.6e-130:
		tmp = 180.0 * (math.atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / math.pi)
	elif b_m <= 5.4e+99:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = math.atan((y_45_scale / (t_0 * -x_45_scale))) / (0.005555555555555556 * math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = tan(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (b_m <= 6.4e-206)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.6e-130)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(-1.0 / Float64(x_45_scale * t_0)))) / pi));
	elseif (b_m <= 5.4e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(atan(Float64(y_45_scale / Float64(t_0 * Float64(-x_45_scale)))) / Float64(0.005555555555555556 * pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = tan((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (b_m <= 6.4e-206)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.6e-130)
		tmp = 180.0 * (atan((y_45_scale * (-1.0 / (x_45_scale * t_0)))) / pi);
	elseif (b_m <= 5.4e+99)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = atan((y_45_scale / (t_0 * -x_45_scale))) / (0.005555555555555556 * pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 6.4e-206], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.6e-130], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(-1.0 / N[(x$45$scale * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.4e+99], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[(y$45$scale / N[(t$95$0 * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 6.39999999999999952e-206

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 6.39999999999999952e-206 < b < 1.6e-130

   1. Initial program 10.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 39.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      2. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \mathsf{PI}\left(\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      3. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. add-sqr-sqrt N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\color{blue}{\left(\frac{1}{180} \cdot angle\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lift-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\left(\left(\frac{1}{180} \cdot angle\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-sqrt.f64 47.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \color{blue}{\sqrt{\pi}}\right)}\right)}{\pi} \]

   10. Applied rewrites 47.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}}\right)}{\pi} \]

   12. Applied rewrites 39.3%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\color{blue}{\frac{1}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot y-scale}\right)}{\pi} \]

   if 1.6e-130 < b < 5.39999999999999978e99

   1. Initial program 38.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 5.39999999999999978e99 < b

   1. Initial program 9.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 58.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 58.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   12. Applied rewrites 58.0%

       \[\leadsto \color{blue}{\frac{\tan^{-1} \left(-\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi \cdot 0.005555555555555556}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{-1}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{y-scale}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{0.005555555555555556 \cdot \pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;-180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot t\_0}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{y-scale}{t\_0 \cdot \left(-x-scale\right)}\right)}{0.005555555555555556 \cdot \pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (tan (* 0.005555555555555556 (* angle PI)))))
   (if (<= b_m 6.4e-206)
     (*
      180.0
      (/
       (atan
        (*
         (*
          (* b_m -180.0)
          (/
           b_m
           (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
         (* y-scale x-scale)))
       PI))
     (if (<= b_m 1.6e-130)
       (* -180.0 (/ (atan (/ y-scale (* x-scale t_0))) PI))
       (if (<= b_m 5.4e+99)
         (*
          180.0
          (/
           (atan
            (/
             (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
             (* (* angle PI) (* (+ b_m a) (- b_m a)))))
           PI))
         (/
          (atan (/ y-scale (* t_0 (- x-scale))))
          (* 0.005555555555555556 PI)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = tan((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.6e-130) {
		tmp = -180.0 * (atan((y_45_scale / (x_45_scale * t_0))) / ((double) M_PI));
	} else if (b_m <= 5.4e+99) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = atan((y_45_scale / (t_0 * -x_45_scale))) / (0.005555555555555556 * ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.tan((0.005555555555555556 * (angle * Math.PI)));
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.6e-130) {
		tmp = -180.0 * (Math.atan((y_45_scale / (x_45_scale * t_0))) / Math.PI);
	} else if (b_m <= 5.4e+99) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = Math.atan((y_45_scale / (t_0 * -x_45_scale))) / (0.005555555555555556 * Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.tan((0.005555555555555556 * (angle * math.pi)))
	tmp = 0
	if b_m <= 6.4e-206:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.6e-130:
		tmp = -180.0 * (math.atan((y_45_scale / (x_45_scale * t_0))) / math.pi)
	elif b_m <= 5.4e+99:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = math.atan((y_45_scale / (t_0 * -x_45_scale))) / (0.005555555555555556 * math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = tan(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (b_m <= 6.4e-206)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.6e-130)
		tmp = Float64(-180.0 * Float64(atan(Float64(y_45_scale / Float64(x_45_scale * t_0))) / pi));
	elseif (b_m <= 5.4e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(atan(Float64(y_45_scale / Float64(t_0 * Float64(-x_45_scale)))) / Float64(0.005555555555555556 * pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = tan((0.005555555555555556 * (angle * pi)));
	tmp = 0.0;
	if (b_m <= 6.4e-206)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.6e-130)
		tmp = -180.0 * (atan((y_45_scale / (x_45_scale * t_0))) / pi);
	elseif (b_m <= 5.4e+99)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = atan((y_45_scale / (t_0 * -x_45_scale))) / (0.005555555555555556 * pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 6.4e-206], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.6e-130], N[(-180.0 * N[(N[ArcTan[N[(y$45$scale / N[(x$45$scale * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.4e+99], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[(y$45$scale / N[(t$95$0 * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 6.39999999999999952e-206

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 6.39999999999999952e-206 < b < 1.6e-130

   1. Initial program 10.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 39.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 39.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.5%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   12. Applied rewrites 39.3%

       \[\leadsto \color{blue}{-180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}} \]

   if 1.6e-130 < b < 5.39999999999999978e99

   1. Initial program 38.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

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

   if 5.39999999999999978e99 < b

   1. Initial program 9.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 13.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 58.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 58.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 58.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   12. Applied rewrites 58.0%

       \[\leadsto \color{blue}{\frac{\tan^{-1} \left(-\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi \cdot 0.005555555555555556}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;-180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{y-scale}{\tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{0.005555555555555556 \cdot \pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := -180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{if}\;b\_m \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b\_m \leq 5.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (*
          -180.0
          (/
           (atan
            (/
             y-scale
             (* x-scale (tan (* 0.005555555555555556 (* angle PI))))))
           PI))))
   (if (<= b_m 6.4e-206)
     (*
      180.0
      (/
       (atan
        (*
         (*
          (* b_m -180.0)
          (/
           b_m
           (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
         (* y-scale x-scale)))
       PI))
     (if (<= b_m 1.6e-130)
       t_0
       (if (<= b_m 5.4e+99)
         (*
          180.0
          (/
           (atan
            (/
             (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
             (* (* angle PI) (* (+ b_m a) (- b_m a)))))
           PI))
         t_0)))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = -180.0 * (atan((y_45_scale / (x_45_scale * tan((0.005555555555555556 * (angle * ((double) M_PI))))))) / ((double) M_PI));
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.6e-130) {
		tmp = t_0;
	} else if (b_m <= 5.4e+99) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = -180.0 * (Math.atan((y_45_scale / (x_45_scale * Math.tan((0.005555555555555556 * (angle * Math.PI)))))) / Math.PI);
	double tmp;
	if (b_m <= 6.4e-206) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 1.6e-130) {
		tmp = t_0;
	} else if (b_m <= 5.4e+99) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = -180.0 * (math.atan((y_45_scale / (x_45_scale * math.tan((0.005555555555555556 * (angle * math.pi)))))) / math.pi)
	tmp = 0
	if b_m <= 6.4e-206:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 1.6e-130:
		tmp = t_0
	elif b_m <= 5.4e+99:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = t_0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(-180.0 * Float64(atan(Float64(y_45_scale / Float64(x_45_scale * tan(Float64(0.005555555555555556 * Float64(angle * pi)))))) / pi))
	tmp = 0.0
	if (b_m <= 6.4e-206)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.6e-130)
		tmp = t_0;
	elseif (b_m <= 5.4e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = -180.0 * (atan((y_45_scale / (x_45_scale * tan((0.005555555555555556 * (angle * pi)))))) / pi);
	tmp = 0.0;
	if (b_m <= 6.4e-206)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 1.6e-130)
		tmp = t_0;
	elseif (b_m <= 5.4e+99)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(-180.0 * N[(N[ArcTan[N[(y$45$scale / N[(x$45$scale * N[Tan[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$m, 6.4e-206], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.6e-130], t$95$0, If[LessEqual[b$95$m, 5.4e+99], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if b < 6.39999999999999952e-206

   1. Initial program 15.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 24.2

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

   10. Applied rewrites 24.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.1

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 37.1%

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

   if 6.39999999999999952e-206 < b < 1.6e-130 or 5.39999999999999978e99 < b

   1. Initial program 10.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 22.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 53.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 53.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.2%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   12. Applied rewrites 53.1%

       \[\leadsto \color{blue}{-180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}} \]

   if 1.6e-130 < b < 5.39999999999999978e99

   1. Initial program 38.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 58.2

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

   8. Applied rewrites 58.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{\left(b \cdot b\right) \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.4 \cdot 10^{-206}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-130}:\\ \;\;\;\;-180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;-180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.7% accurate, 13.9× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(angle \cdot angle, \frac{y-scale \cdot \pi}{x-scale} \cdot 0.001851851851851852, -180 \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{angle}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 6.4 \cdot 10^{+145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot 32400, \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(y-scale \cdot 5.7155921353452215 \cdot 10^{-8}\right)}{\pi \cdot \left(\pi \cdot x-scale\right)}, \frac{y-scale \cdot -180}{\pi \cdot x-scale}\right)}{angle}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 5.5e-158)
   (*
    180.0
    (/
     (atan
      (*
       (*
        (* b_m -180.0)
        (/
         b_m
         (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
       (* y-scale x-scale)))
     PI))
   (if (<= b_m 1.22e-124)
     (*
      180.0
      (/
       (atan
        (/
         (fma
          (* angle angle)
          (* (/ (* y-scale PI) x-scale) 0.001851851851851852)
          (* -180.0 (/ y-scale (* PI x-scale))))
         angle))
       PI))
     (if (<= b_m 6.4e+145)
       (*
        180.0
        (/
         (atan
          (/
           (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
           (* (* angle PI) (* (+ b_m a) (- b_m a)))))
         PI))
       (*
        (* 180.0 (/ 1.0 PI))
        (atan
         (/
          (fma
           (* (* angle angle) 32400.0)
           (/
            (* (* PI (* PI PI)) (* y-scale 5.7155921353452215e-8))
            (* PI (* PI x-scale)))
           (/ (* y-scale -180.0) (* PI x-scale)))
          angle)))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 5.5e-158) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.22e-124) {
		tmp = 180.0 * (atan((fma((angle * angle), (((y_45_scale * ((double) M_PI)) / x_45_scale) * 0.001851851851851852), (-180.0 * (y_45_scale / (((double) M_PI) * x_45_scale)))) / angle)) / ((double) M_PI));
	} else if (b_m <= 6.4e+145) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * (1.0 / ((double) M_PI))) * atan((fma(((angle * angle) * 32400.0), (((((double) M_PI) * (((double) M_PI) * ((double) M_PI))) * (y_45_scale * 5.7155921353452215e-8)) / (((double) M_PI) * (((double) M_PI) * x_45_scale))), ((y_45_scale * -180.0) / (((double) M_PI) * x_45_scale))) / angle));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 5.5e-158)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.22e-124)
		tmp = Float64(180.0 * Float64(atan(Float64(fma(Float64(angle * angle), Float64(Float64(Float64(y_45_scale * pi) / x_45_scale) * 0.001851851851851852), Float64(-180.0 * Float64(y_45_scale / Float64(pi * x_45_scale)))) / angle)) / pi));
	elseif (b_m <= 6.4e+145)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(fma(Float64(Float64(angle * angle) * 32400.0), Float64(Float64(Float64(pi * Float64(pi * pi)) * Float64(y_45_scale * 5.7155921353452215e-8)) / Float64(pi * Float64(pi * x_45_scale))), Float64(Float64(y_45_scale * -180.0) / Float64(pi * x_45_scale))) / angle)));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 5.5e-158], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.22e-124], N[(180.0 * N[(N[ArcTan[N[(N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision] * 0.001851851851851852), $MachinePrecision] + N[(-180.0 * N[(y$45$scale / N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / angle), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 6.4e+145], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(N[(N[(N[(angle * angle), $MachinePrecision] * 32400.0), $MachinePrecision] * N[(N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * 5.7155921353452215e-8), $MachinePrecision]), $MachinePrecision] / N[(Pi * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$45$scale * -180.0), $MachinePrecision] / N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < 5.50000000000000025e-158

   1. Initial program 14.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 23.7

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

   10. Applied rewrites 23.7%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 30.4

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 36.3%

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

   if 5.50000000000000025e-158 < b < 1.22000000000000001e-124

   1. Initial program 15.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 44.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 46.8

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 46.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{{angle}^{2} \cdot \left(\frac{-1}{1080} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} - \frac{-1}{360} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right) - 180 \cdot \frac{y-scale}{x-scale \cdot \mathsf{PI}\left(\right)}}{angle}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{{angle}^{2} \cdot \left(\frac{-1}{1080} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} - \frac{-1}{360} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right) - 180 \cdot \frac{y-scale}{x-scale \cdot \mathsf{PI}\left(\right)}}{angle}\right)}}{\mathsf{PI}\left(\right)} \]

   11. Applied rewrites 44.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(angle \cdot angle, \frac{y-scale \cdot \pi}{x-scale} \cdot 0.001851851851851852, -180 \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{angle}\right)}}{\pi} \]

   if 1.22000000000000001e-124 < b < 6.40000000000000015e145

   1. Initial program 36.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if 6.40000000000000015e145 < b

   1. Initial program 0.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 65.2

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 65.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.4%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \tan^{-1} \color{blue}{\left(\frac{-180 \cdot \frac{y-scale}{x-scale \cdot \mathsf{PI}\left(\right)} + 32400 \cdot \frac{{angle}^{2} \cdot \left(y-scale \cdot \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3} - \frac{-1}{11664000} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}{x-scale \cdot {\mathsf{PI}\left(\right)}^{2}}}{angle}\right)} \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

   13. Applied rewrites 58.0%

       \[\leadsto \tan^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(32400 \cdot \left(angle \cdot angle\right), \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(5.7155921353452215 \cdot 10^{-8} \cdot y-scale\right)}{\pi \cdot \left(\pi \cdot x-scale\right)}, \frac{-180 \cdot y-scale}{\pi \cdot x-scale}\right)}{angle}\right)} \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(angle \cdot angle, \frac{y-scale \cdot \pi}{x-scale} \cdot 0.001851851851851852, -180 \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{angle}\right)}{\pi}\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{+145}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{\mathsf{fma}\left(\left(angle \cdot angle\right) \cdot 32400, \frac{\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(y-scale \cdot 5.7155921353452215 \cdot 10^{-8}\right)}{\pi \cdot \left(\pi \cdot x-scale\right)}, \frac{y-scale \cdot -180}{\pi \cdot x-scale}\right)}{angle}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 46.0% accurate, 16.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(angle \cdot angle, \frac{y-scale \cdot \pi}{x-scale} \cdot 0.001851851851851852, -180 \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{angle}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.6 \cdot 10^{+146}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{y-scale}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 5.5e-158)
   (*
    180.0
    (/
     (atan
      (*
       (*
        (* b_m -180.0)
        (/
         b_m
         (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
       (* y-scale x-scale)))
     PI))
   (if (<= b_m 1.22e-124)
     (*
      180.0
      (/
       (atan
        (/
         (fma
          (* angle angle)
          (* (/ (* y-scale PI) x-scale) 0.001851851851851852)
          (* -180.0 (/ y-scale (* PI x-scale))))
         angle))
       PI))
     (if (<= b_m 5.6e+146)
       (*
        180.0
        (/
         (atan
          (/
           (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
           (* (* angle PI) (* (+ b_m a) (- b_m a)))))
         PI))
       (*
        (* 180.0 (/ 1.0 PI))
        (atan
         (/
          -1.0
          (* (* 0.005555555555555556 (* angle PI)) (/ x-scale y-scale)))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 5.5e-158) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 1.22e-124) {
		tmp = 180.0 * (atan((fma((angle * angle), (((y_45_scale * ((double) M_PI)) / x_45_scale) * 0.001851851851851852), (-180.0 * (y_45_scale / (((double) M_PI) * x_45_scale)))) / angle)) / ((double) M_PI));
	} else if (b_m <= 5.6e+146) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * (1.0 / ((double) M_PI))) * atan((-1.0 / ((0.005555555555555556 * (angle * ((double) M_PI))) * (x_45_scale / y_45_scale))));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 5.5e-158)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 1.22e-124)
		tmp = Float64(180.0 * Float64(atan(Float64(fma(Float64(angle * angle), Float64(Float64(Float64(y_45_scale * pi) / x_45_scale) * 0.001851851851851852), Float64(-180.0 * Float64(y_45_scale / Float64(pi * x_45_scale)))) / angle)) / pi));
	elseif (b_m <= 5.6e+146)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(-1.0 / Float64(Float64(0.005555555555555556 * Float64(angle * pi)) * Float64(x_45_scale / y_45_scale)))));
	end
	return tmp
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 5.5e-158], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.22e-124], N[(180.0 * N[(N[ArcTan[N[(N[(N[(angle * angle), $MachinePrecision] * N[(N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision] * 0.001851851851851852), $MachinePrecision] + N[(-180.0 * N[(y$45$scale / N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / angle), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.6e+146], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(-1.0 / N[(N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < 5.50000000000000025e-158

   1. Initial program 14.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 23.7

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

   10. Applied rewrites 23.7%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 30.4

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 36.3%

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

   if 5.50000000000000025e-158 < b < 1.22000000000000001e-124

   1. Initial program 15.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 44.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 46.8

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 46.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{{angle}^{2} \cdot \left(\frac{-1}{1080} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} - \frac{-1}{360} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right) - 180 \cdot \frac{y-scale}{x-scale \cdot \mathsf{PI}\left(\right)}}{angle}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{{angle}^{2} \cdot \left(\frac{-1}{1080} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale} - \frac{-1}{360} \cdot \frac{y-scale \cdot \mathsf{PI}\left(\right)}{x-scale}\right) - 180 \cdot \frac{y-scale}{x-scale \cdot \mathsf{PI}\left(\right)}}{angle}\right)}}{\mathsf{PI}\left(\right)} \]

   11. Applied rewrites 44.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(angle \cdot angle, \frac{y-scale \cdot \pi}{x-scale} \cdot 0.001851851851851852, -180 \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{angle}\right)}}{\pi} \]

   if 1.22000000000000001e-124 < b < 5.6000000000000002e146

   1. Initial program 36.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 29.0%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

   if 5.6000000000000002e146 < b

   1. Initial program 0.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 65.2

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 65.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.4%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lower-PI.f64 63.6

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   13. Applied rewrites 63.6%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5 \cdot 10^{-158}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-124}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\mathsf{fma}\left(angle \cdot angle, \frac{y-scale \cdot \pi}{x-scale} \cdot 0.001851851851851852, -180 \cdot \frac{y-scale}{\pi \cdot x-scale}\right)}{angle}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{+146}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{y-scale}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 46.2% accurate, 16.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.4 \cdot 10^{-138}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.6 \cdot 10^{+146}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{y-scale}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.4e-138)
   (*
    180.0
    (/
     (atan
      (*
       (*
        (* b_m -180.0)
        (/
         b_m
         (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
       (* y-scale x-scale)))
     PI))
   (if (<= b_m 5.6e+146)
     (*
      180.0
      (/
       (atan
        (/
         (* 90.0 (* -2.0 (/ (* y-scale (* b_m b_m)) x-scale)))
         (* (* angle PI) (* (+ b_m a) (- b_m a)))))
       PI))
     (*
      (* 180.0 (/ 1.0 PI))
      (atan
       (/
        -1.0
        (* (* 0.005555555555555556 (* angle PI)) (/ x-scale y-scale))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2.4e-138) {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	} else if (b_m <= 5.6e+146) {
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * ((double) M_PI)) * ((b_m + a) * (b_m - a))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * (1.0 / ((double) M_PI))) * atan((-1.0 / ((0.005555555555555556 * (angle * ((double) M_PI))) * (x_45_scale / y_45_scale))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 2.4e-138) {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	} else if (b_m <= 5.6e+146) {
		tmp = 180.0 * (Math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * Math.PI) * ((b_m + a) * (b_m - a))))) / Math.PI);
	} else {
		tmp = (180.0 * (1.0 / Math.PI)) * Math.atan((-1.0 / ((0.005555555555555556 * (angle * Math.PI)) * (x_45_scale / y_45_scale))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 2.4e-138:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	elif b_m <= 5.6e+146:
		tmp = 180.0 * (math.atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * math.pi) * ((b_m + a) * (b_m - a))))) / math.pi)
	else:
		tmp = (180.0 * (1.0 / math.pi)) * math.atan((-1.0 / ((0.005555555555555556 * (angle * math.pi)) * (x_45_scale / y_45_scale))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 2.4e-138)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	elseif (b_m <= 5.6e+146)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(-2.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / x_45_scale))) / Float64(Float64(angle * pi) * Float64(Float64(b_m + a) * Float64(b_m - a))))) / pi));
	else
		tmp = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(-1.0 / Float64(Float64(0.005555555555555556 * Float64(angle * pi)) * Float64(x_45_scale / y_45_scale)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 2.4e-138)
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	elseif (b_m <= 5.6e+146)
		tmp = 180.0 * (atan(((90.0 * (-2.0 * ((y_45_scale * (b_m * b_m)) / x_45_scale))) / ((angle * pi) * ((b_m + a) * (b_m - a))))) / pi);
	else
		tmp = (180.0 * (1.0 / pi)) * atan((-1.0 / ((0.005555555555555556 * (angle * pi)) * (x_45_scale / y_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 2.4e-138], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 5.6e+146], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(-2.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * Pi), $MachinePrecision] * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(-1.0 / N[(N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.3999999999999999e-138

   1. Initial program 14.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 23.6

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

   10. Applied rewrites 23.6%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 30.0

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 35.7%

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

   if 2.3999999999999999e-138 < b < 5.6000000000000002e146

   1. Initial program 36.1%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in x-scale around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \color{blue}{\left(-2 \cdot \frac{{b}^{2} \cdot y-scale}{x-scale}\right)}}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \color{blue}{\frac{{b}^{2} \cdot y-scale}{x-scale}}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{{b}^{2} \cdot y-scale}}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{x-scale}\right)}{\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 54.1

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

   8. Applied rewrites 54.1%

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

   if 5.6000000000000002e146 < b

   1. Initial program 0.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 0.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 65.2

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 65.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.4%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lower-PI.f64 63.6

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   13. Applied rewrites 63.6%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{-138}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.6 \cdot 10^{+146}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \left(-2 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{x-scale}\right)}{\left(angle \cdot \pi\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{y-scale}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 40.8% accurate, 17.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{-31}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b\_m \cdot -180\right) \cdot \frac{b\_m}{\left(b\_m - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a 1.85e-31)
   (* 180.0 (/ (atan (/ (* y-scale -180.0) (* angle (* PI x-scale)))) PI))
   (*
    180.0
    (/
     (atan
      (*
       (*
        (* b_m -180.0)
        (/
         b_m
         (* (- b_m a) (* (* x-scale x-scale) (* angle (* PI (+ b_m a)))))))
       (* y-scale x-scale)))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 1.85e-31) {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((double) M_PI) * x_45_scale)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (((double) M_PI) * (b_m + a))))))) * (y_45_scale * x_45_scale))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 1.85e-31) {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (Math.PI * x_45_scale)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (Math.PI * (b_m + a))))))) * (y_45_scale * x_45_scale))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 1.85e-31:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (math.pi * x_45_scale)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (math.pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 1.85e-31)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * Float64(pi * x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(b_m * -180.0) * Float64(b_m / Float64(Float64(b_m - a) * Float64(Float64(x_45_scale * x_45_scale) * Float64(angle * Float64(pi * Float64(b_m + a))))))) * Float64(y_45_scale * x_45_scale))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 1.85e-31)
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (pi * x_45_scale)))) / pi);
	else
		tmp = 180.0 * (atan((((b_m * -180.0) * (b_m / ((b_m - a) * ((x_45_scale * x_45_scale) * (angle * (pi * (b_m + a))))))) * (y_45_scale * x_45_scale))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 1.85e-31], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * -180.0), $MachinePrecision] / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(N[(b$95$m * -180.0), $MachinePrecision] * N[(b$95$m / N[(N[(b$95$m - a), $MachinePrecision] * N[(N[(x$45$scale * x$45$scale), $MachinePrecision] * N[(angle * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.8499999999999999e-31

   1. Initial program 19.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 34.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 47.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 47.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot y-scale}}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       5. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       7. lower-PI.f64 42.6

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)}\right)}{\pi} \]

   11. Applied rewrites 42.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}}{\pi} \]

   if 1.8499999999999999e-31 < a

   1. Initial program 11.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 14.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 14.6%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 23.2

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

   10. Applied rewrites 23.2%

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

       1. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{\left(-180 \cdot b\right) \cdot b}}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. lift-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. lift-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lift--.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \color{blue}{\left(b - a\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. lift-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\left(-180 \cdot b\right) \cdot b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. associate-/l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       11. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       12. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\color{blue}{\left(-180 \cdot b\right)} \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-/.f64 31.6

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

       14. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\color{blue}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       15. lift-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(\left(-180 \cdot b\right) \cdot \frac{b}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)}}\right) \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. Applied rewrites 39.8%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.85 \cdot 10^{-31}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(\left(b \cdot -180\right) \cdot \frac{b}{\left(b - a\right) \cdot \left(\left(x-scale \cdot x-scale\right) \cdot \left(angle \cdot \left(\pi \cdot \left(b + a\right)\right)\right)\right)}\right) \cdot \left(y-scale \cdot x-scale\right)\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.0% accurate, 17.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(y-scale \cdot x-scale\right) \cdot \frac{-180}{\pi \cdot \left(angle \cdot \left(x-scale \cdot x-scale\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right)}{\left(angle \cdot x-scale\right) \cdot \left(\left(b\_m - a\right) \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{y-scale}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.65e-130)
   (*
    180.0
    (/
     (atan
      (* (* y-scale x-scale) (/ -180.0 (* PI (* angle (* x-scale x-scale))))))
     PI))
   (if (<= b_m 4.2e+109)
     (*
      180.0
      (/
       (atan
        (/
         (* -180.0 (* y-scale (* b_m b_m)))
         (* (* angle x-scale) (* (- b_m a) (* PI (+ b_m a))))))
       PI))
     (*
      (* 180.0 (/ 1.0 PI))
      (atan
       (/
        -1.0
        (* (* 0.005555555555555556 (* angle PI)) (/ x-scale y-scale))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.65e-130) {
		tmp = 180.0 * (atan(((y_45_scale * x_45_scale) * (-180.0 / (((double) M_PI) * (angle * (x_45_scale * x_45_scale)))))) / ((double) M_PI));
	} else if (b_m <= 4.2e+109) {
		tmp = 180.0 * (atan(((-180.0 * (y_45_scale * (b_m * b_m))) / ((angle * x_45_scale) * ((b_m - a) * (((double) M_PI) * (b_m + a)))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * (1.0 / ((double) M_PI))) * atan((-1.0 / ((0.005555555555555556 * (angle * ((double) M_PI))) * (x_45_scale / y_45_scale))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.65e-130) {
		tmp = 180.0 * (Math.atan(((y_45_scale * x_45_scale) * (-180.0 / (Math.PI * (angle * (x_45_scale * x_45_scale)))))) / Math.PI);
	} else if (b_m <= 4.2e+109) {
		tmp = 180.0 * (Math.atan(((-180.0 * (y_45_scale * (b_m * b_m))) / ((angle * x_45_scale) * ((b_m - a) * (Math.PI * (b_m + a)))))) / Math.PI);
	} else {
		tmp = (180.0 * (1.0 / Math.PI)) * Math.atan((-1.0 / ((0.005555555555555556 * (angle * Math.PI)) * (x_45_scale / y_45_scale))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 1.65e-130:
		tmp = 180.0 * (math.atan(((y_45_scale * x_45_scale) * (-180.0 / (math.pi * (angle * (x_45_scale * x_45_scale)))))) / math.pi)
	elif b_m <= 4.2e+109:
		tmp = 180.0 * (math.atan(((-180.0 * (y_45_scale * (b_m * b_m))) / ((angle * x_45_scale) * ((b_m - a) * (math.pi * (b_m + a)))))) / math.pi)
	else:
		tmp = (180.0 * (1.0 / math.pi)) * math.atan((-1.0 / ((0.005555555555555556 * (angle * math.pi)) * (x_45_scale / y_45_scale))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 1.65e-130)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * x_45_scale) * Float64(-180.0 / Float64(pi * Float64(angle * Float64(x_45_scale * x_45_scale)))))) / pi));
	elseif (b_m <= 4.2e+109)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(-180.0 * Float64(y_45_scale * Float64(b_m * b_m))) / Float64(Float64(angle * x_45_scale) * Float64(Float64(b_m - a) * Float64(pi * Float64(b_m + a)))))) / pi));
	else
		tmp = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(-1.0 / Float64(Float64(0.005555555555555556 * Float64(angle * pi)) * Float64(x_45_scale / y_45_scale)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 1.65e-130)
		tmp = 180.0 * (atan(((y_45_scale * x_45_scale) * (-180.0 / (pi * (angle * (x_45_scale * x_45_scale)))))) / pi);
	elseif (b_m <= 4.2e+109)
		tmp = 180.0 * (atan(((-180.0 * (y_45_scale * (b_m * b_m))) / ((angle * x_45_scale) * ((b_m - a) * (pi * (b_m + a)))))) / pi);
	else
		tmp = (180.0 * (1.0 / pi)) * atan((-1.0 / ((0.005555555555555556 * (angle * pi)) * (x_45_scale / y_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 1.65e-130], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(-180.0 / N[(Pi * N[(angle * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 4.2e+109], N[(180.0 * N[(N[ArcTan[N[(N[(-180.0 * N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(angle * x$45$scale), $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(Pi * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(-1.0 / N[(N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.6499999999999999e-130

   1. Initial program 14.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 23.4

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

   10. Applied rewrites 23.4%

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

   11. Taylor expanded in b around inf

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{-180}{angle \cdot \left({x-scale}^{2} \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{-180}{angle \cdot \left({x-scale}^{2} \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\left({x-scale}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot {x-scale}^{2}\right)} \cdot angle} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left({x-scale}^{2} \cdot angle\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot {x-scale}^{2}\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(angle \cdot {x-scale}^{2}\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot {x-scale}^{2}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot {x-scale}^{2}\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\mathsf{PI}\left(\right) \cdot \left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 34.5

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\pi \cdot \left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   13. Applied rewrites 34.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{-180}{\pi \cdot \left(angle \cdot \left(x-scale \cdot x-scale\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   if 1.6499999999999999e-130 < b < 4.2000000000000003e109

   1. Initial program 37.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   5. Applied rewrites 29.0%

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

   7. Applied rewrites 29.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{x-scale \cdot \left(\left(y-scale \cdot 2\right) \cdot \mathsf{fma}\left(a, \frac{a}{y-scale \cdot y-scale}, \frac{b \cdot b}{x-scale \cdot \left(-x-scale\right)}\right)\right)}{\left(b + a\right) \cdot \left(b - a\right)} \cdot \frac{90}{angle \cdot \pi}\right)}}{\pi} \]

   8. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. difference-of-squares N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left({b}^{2} \cdot y-scale\right)}{angle \cdot \left(\left(x-scale \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(b \cdot b - a \cdot a\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left({b}^{2} \cdot y-scale\right)}{angle \cdot \left(\left(x-scale \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{{b}^{2}} - a \cdot a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left({b}^{2} \cdot y-scale\right)}{angle \cdot \left(\left(x-scale \cdot \mathsf{PI}\left(\right)\right) \cdot \left({b}^{2} - \color{blue}{{a}^{2}}\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      7. associate-*r* N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left({b}^{2} \cdot y-scale\right)}{angle \cdot \color{blue}{\left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot \left({b}^{2} \cdot y-scale\right)}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot \left({b}^{2} \cdot y-scale\right)}}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      10. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \color{blue}{\left(y-scale \cdot {b}^{2}\right)}}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \color{blue}{\left(y-scale \cdot {b}^{2}\right)}}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      12. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(y-scale \cdot \color{blue}{\left(b \cdot b\right)}\right)}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(y-scale \cdot \color{blue}{\left(b \cdot b\right)}\right)}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      14. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(angle \cdot x-scale\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      15. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{\left(angle \cdot x-scale\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{b \cdot b} - {a}^{2}\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      16. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{\left(angle \cdot x-scale\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b - \color{blue}{a \cdot a}\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

   10. Applied rewrites 58.8%

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

   if 4.2000000000000003e109 < b

   1. Initial program 7.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 7.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 56.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 56.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lower-PI.f64 55.7

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   13. Applied rewrites 55.7%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(y-scale \cdot x-scale\right) \cdot \frac{-180}{\pi \cdot \left(angle \cdot \left(x-scale \cdot x-scale\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{\left(angle \cdot x-scale\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{y-scale}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.1% accurate, 17.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(y-scale \cdot x-scale\right) \cdot \frac{-180}{\pi \cdot \left(angle \cdot \left(x-scale \cdot x-scale\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.04 \cdot 10^{+108}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(\left(b\_m + a\right) \cdot \left(b\_m - a\right)\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{y-scale}}\right)\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.65e-130)
   (*
    180.0
    (/
     (atan
      (* (* y-scale x-scale) (/ -180.0 (* PI (* angle (* x-scale x-scale))))))
     PI))
   (if (<= b_m 1.04e+108)
     (*
      180.0
      (/
       (atan
        (*
         -180.0
         (/
          (* y-scale (* b_m b_m))
          (* angle (* x-scale (* PI (* (+ b_m a) (- b_m a))))))))
       PI))
     (*
      (* 180.0 (/ 1.0 PI))
      (atan
       (/
        -1.0
        (* (* 0.005555555555555556 (* angle PI)) (/ x-scale y-scale))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.65e-130) {
		tmp = 180.0 * (atan(((y_45_scale * x_45_scale) * (-180.0 / (((double) M_PI) * (angle * (x_45_scale * x_45_scale)))))) / ((double) M_PI));
	} else if (b_m <= 1.04e+108) {
		tmp = 180.0 * (atan((-180.0 * ((y_45_scale * (b_m * b_m)) / (angle * (x_45_scale * (((double) M_PI) * ((b_m + a) * (b_m - a)))))))) / ((double) M_PI));
	} else {
		tmp = (180.0 * (1.0 / ((double) M_PI))) * atan((-1.0 / ((0.005555555555555556 * (angle * ((double) M_PI))) * (x_45_scale / y_45_scale))));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.65e-130) {
		tmp = 180.0 * (Math.atan(((y_45_scale * x_45_scale) * (-180.0 / (Math.PI * (angle * (x_45_scale * x_45_scale)))))) / Math.PI);
	} else if (b_m <= 1.04e+108) {
		tmp = 180.0 * (Math.atan((-180.0 * ((y_45_scale * (b_m * b_m)) / (angle * (x_45_scale * (Math.PI * ((b_m + a) * (b_m - a)))))))) / Math.PI);
	} else {
		tmp = (180.0 * (1.0 / Math.PI)) * Math.atan((-1.0 / ((0.005555555555555556 * (angle * Math.PI)) * (x_45_scale / y_45_scale))));
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if b_m <= 1.65e-130:
		tmp = 180.0 * (math.atan(((y_45_scale * x_45_scale) * (-180.0 / (math.pi * (angle * (x_45_scale * x_45_scale)))))) / math.pi)
	elif b_m <= 1.04e+108:
		tmp = 180.0 * (math.atan((-180.0 * ((y_45_scale * (b_m * b_m)) / (angle * (x_45_scale * (math.pi * ((b_m + a) * (b_m - a)))))))) / math.pi)
	else:
		tmp = (180.0 * (1.0 / math.pi)) * math.atan((-1.0 / ((0.005555555555555556 * (angle * math.pi)) * (x_45_scale / y_45_scale))))
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (b_m <= 1.65e-130)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * x_45_scale) * Float64(-180.0 / Float64(pi * Float64(angle * Float64(x_45_scale * x_45_scale)))))) / pi));
	elseif (b_m <= 1.04e+108)
		tmp = Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(Float64(y_45_scale * Float64(b_m * b_m)) / Float64(angle * Float64(x_45_scale * Float64(pi * Float64(Float64(b_m + a) * Float64(b_m - a)))))))) / pi));
	else
		tmp = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(-1.0 / Float64(Float64(0.005555555555555556 * Float64(angle * pi)) * Float64(x_45_scale / y_45_scale)))));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (b_m <= 1.65e-130)
		tmp = 180.0 * (atan(((y_45_scale * x_45_scale) * (-180.0 / (pi * (angle * (x_45_scale * x_45_scale)))))) / pi);
	elseif (b_m <= 1.04e+108)
		tmp = 180.0 * (atan((-180.0 * ((y_45_scale * (b_m * b_m)) / (angle * (x_45_scale * (pi * ((b_m + a) * (b_m - a)))))))) / pi);
	else
		tmp = (180.0 * (1.0 / pi)) * atan((-1.0 / ((0.005555555555555556 * (angle * pi)) * (x_45_scale / y_45_scale))));
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 1.65e-130], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(-180.0 / N[(Pi * N[(angle * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 1.04e+108], N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(angle * N[(x$45$scale * N[(Pi * N[(N[(b$95$m + a), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(-1.0 / N[(N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.6499999999999999e-130

   1. Initial program 14.7%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 17.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 18.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 23.4

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

   10. Applied rewrites 23.4%

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

   11. Taylor expanded in b around inf

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{-180}{angle \cdot \left({x-scale}^{2} \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{-180}{angle \cdot \left({x-scale}^{2} \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\left({x-scale}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot {x-scale}^{2}\right)} \cdot angle} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left({x-scale}^{2} \cdot angle\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot {x-scale}^{2}\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(angle \cdot {x-scale}^{2}\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot {x-scale}^{2}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot {x-scale}^{2}\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\mathsf{PI}\left(\right) \cdot \left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 34.5

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\pi \cdot \left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   13. Applied rewrites 34.5%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{-180}{\pi \cdot \left(angle \cdot \left(x-scale \cdot x-scale\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   if 1.6499999999999999e-130 < b < 1.04e108

   1. Initial program 37.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 57.4%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in angle around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\color{blue}{\left(b \cdot b\right)} \cdot y-scale}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\left(b \cdot b\right) \cdot y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\left(b \cdot b\right) \cdot y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\left(b \cdot b\right) \cdot y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\left(b \cdot b\right) \cdot y-scale}{angle \cdot \left(x-scale \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\left(b \cdot b\right) \cdot y-scale}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{\left(b \cdot b\right) \cdot y-scale}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(a + b\right)} \cdot \left(b - a\right)\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

      12. lower--.f64 56.6

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

   8. Applied rewrites 56.6%

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

   if 1.04e108 < b

   1. Initial program 7.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 7.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 56.9

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 56.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.1%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lower-PI.f64 55.7

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   13. Applied rewrites 55.7%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.65 \cdot 10^{-130}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(y-scale \cdot x-scale\right) \cdot \frac{-180}{\pi \cdot \left(angle \cdot \left(x-scale \cdot x-scale\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.04 \cdot 10^{+108}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale \cdot \left(b \cdot b\right)}{angle \cdot \left(x-scale \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \frac{x-scale}{y-scale}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 39.0% accurate, 18.8× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 8.1 \cdot 10^{+89}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot x-scale\right)\right)}{y-scale}}\right)\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right)}{a \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot x-scale\right)\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a 8.1e+89)
   (*
    (* 180.0 (/ 1.0 PI))
    (atan
     (/ -1.0 (/ (* 0.005555555555555556 (* angle (* PI x-scale))) y-scale))))
   (*
    180.0
    (/
     (atan
      (/
       (* 180.0 (* y-scale (* b_m b_m)))
       (* a (* angle (* a (* PI x-scale))))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 8.1e+89) {
		tmp = (180.0 * (1.0 / ((double) M_PI))) * atan((-1.0 / ((0.005555555555555556 * (angle * (((double) M_PI) * x_45_scale))) / y_45_scale)));
	} else {
		tmp = 180.0 * (atan(((180.0 * (y_45_scale * (b_m * b_m))) / (a * (angle * (a * (((double) M_PI) * x_45_scale)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 8.1e+89) {
		tmp = (180.0 * (1.0 / Math.PI)) * Math.atan((-1.0 / ((0.005555555555555556 * (angle * (Math.PI * x_45_scale))) / y_45_scale)));
	} else {
		tmp = 180.0 * (Math.atan(((180.0 * (y_45_scale * (b_m * b_m))) / (a * (angle * (a * (Math.PI * x_45_scale)))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 8.1e+89:
		tmp = (180.0 * (1.0 / math.pi)) * math.atan((-1.0 / ((0.005555555555555556 * (angle * (math.pi * x_45_scale))) / y_45_scale)))
	else:
		tmp = 180.0 * (math.atan(((180.0 * (y_45_scale * (b_m * b_m))) / (a * (angle * (a * (math.pi * x_45_scale)))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 8.1e+89)
		tmp = Float64(Float64(180.0 * Float64(1.0 / pi)) * atan(Float64(-1.0 / Float64(Float64(0.005555555555555556 * Float64(angle * Float64(pi * x_45_scale))) / y_45_scale))));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(180.0 * Float64(y_45_scale * Float64(b_m * b_m))) / Float64(a * Float64(angle * Float64(a * Float64(pi * x_45_scale)))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 8.1e+89)
		tmp = (180.0 * (1.0 / pi)) * atan((-1.0 / ((0.005555555555555556 * (angle * (pi * x_45_scale))) / y_45_scale)));
	else
		tmp = 180.0 * (atan(((180.0 * (y_45_scale * (b_m * b_m))) / (a * (angle * (a * (pi * x_45_scale)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 8.1e+89], N[(N[(180.0 * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision] * N[ArcTan[N[(-1.0 / N[(N[(0.005555555555555556 * N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(180.0 * N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(angle * N[(a * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.1e89

   1. Initial program 20.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 36.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 47.3

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 47.3%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
   9. Step-by-step derivation

   10. Applied rewrites 48.3%

       \[\leadsto \color{blue}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{\frac{1}{180} \cdot \frac{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}{y-scale}}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{\frac{\frac{1}{180} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}{y-scale}}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       2. lower-/.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{\frac{\frac{1}{180} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}{y-scale}}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{\color{blue}{\frac{1}{180} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}}{y-scale}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}}{y-scale}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       5. *-commutative N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}\right)}{y-scale}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}\right)}{y-scale}}\right) \cdot \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot 180\right) \]

       7. lower-PI.f64 42.7

          \[\leadsto \tan^{-1} \left(\frac{-1}{\frac{0.005555555555555556 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)\right)}{y-scale}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   13. Applied rewrites 42.7%

       \[\leadsto \tan^{-1} \left(\frac{-1}{\color{blue}{\frac{0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot x-scale\right)\right)}{y-scale}}}\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   if 8.1e89 < a

   1. Initial program 5.0%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 7.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 4.9%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 17.5

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

   10. Applied rewrites 17.5%

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

   11. Taylor expanded in b around 0

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

       1. associate-*r/ N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{180 \cdot \left({b}^{2} \cdot y-scale\right)}{{a}^{2} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{180 \cdot \left({b}^{2} \cdot y-scale\right)}{{a}^{2} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{180 \cdot \left({b}^{2} \cdot y-scale\right)}}{{a}^{2} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \color{blue}{\left(y-scale \cdot {b}^{2}\right)}}{{a}^{2} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       5. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \color{blue}{\left(y-scale \cdot {b}^{2}\right)}}{{a}^{2} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       6. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \color{blue}{\left(b \cdot b\right)}\right)}{{a}^{2} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \color{blue}{\left(b \cdot b\right)}\right)}{{a}^{2} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       8. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{\color{blue}{\left(a \cdot a\right)} \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       9. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{\color{blue}{a \cdot \left(a \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{\color{blue}{a \cdot \left(a \cdot \left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       11. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{a \cdot \color{blue}{\left(\left(angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)\right) \cdot a\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       12. associate-*l* N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{a \cdot \color{blue}{\left(angle \cdot \left(\left(x-scale \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       13. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{a \cdot \color{blue}{\left(angle \cdot \left(\left(x-scale \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       14. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{a \cdot \left(angle \cdot \color{blue}{\left(\left(x-scale \cdot \mathsf{PI}\left(\right)\right) \cdot a\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

       15. *-commutative N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{a \cdot \left(angle \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)} \cdot a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       16. lower-*.f64 N/A

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{a \cdot \left(angle \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)} \cdot a\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       17. lower-PI.f64 23.5

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

   13. Applied rewrites 23.5%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.1 \cdot 10^{+89}:\\ \;\;\;\;\left(180 \cdot \frac{1}{\pi}\right) \cdot \tan^{-1} \left(\frac{-1}{\frac{0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot x-scale\right)\right)}{y-scale}}\right)\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{180 \cdot \left(y-scale \cdot \left(b \cdot b\right)\right)}{a \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot x-scale\right)\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 37.7% accurate, 20.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{-123}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(y-scale \cdot x-scale\right) \cdot \frac{-180}{\pi \cdot \left(angle \cdot \left(x-scale \cdot x-scale\right)\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= a 6e-123)
   (* 180.0 (/ (atan (/ (* y-scale -180.0) (* angle (* PI x-scale)))) PI))
   (*
    180.0
    (/
     (atan
      (* (* y-scale x-scale) (/ -180.0 (* PI (* angle (* x-scale x-scale))))))
     PI))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 6e-123) {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((double) M_PI) * x_45_scale)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * x_45_scale) * (-180.0 / (((double) M_PI) * (angle * (x_45_scale * x_45_scale)))))) / ((double) M_PI));
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (a <= 6e-123) {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (Math.PI * x_45_scale)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * x_45_scale) * (-180.0 / (Math.PI * (angle * (x_45_scale * x_45_scale)))))) / Math.PI);
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	tmp = 0
	if a <= 6e-123:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (math.pi * x_45_scale)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * x_45_scale) * (-180.0 / (math.pi * (angle * (x_45_scale * x_45_scale)))))) / math.pi)
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0
	if (a <= 6e-123)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * Float64(pi * x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * x_45_scale) * Float64(-180.0 / Float64(pi * Float64(angle * Float64(x_45_scale * x_45_scale)))))) / pi));
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
	if (a <= 6e-123)
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (pi * x_45_scale)))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * x_45_scale) * (-180.0 / (pi * (angle * (x_45_scale * x_45_scale)))))) / pi);
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[a, 6e-123], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * -180.0), $MachinePrecision] / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * x$45$scale), $MachinePrecision] * N[(-180.0 / N[(Pi * N[(angle * N[(x$45$scale * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.99999999999999968e-123

   1. Initial program 19.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   3. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 36.2%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

   6. Taylor expanded in b around inf

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      5. lower-cos.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      8. lower-PI.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      9. lower-*.f64 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-sin.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-PI.f64 49.1

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

   8. Applied rewrites 49.1%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

       2. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot y-scale}}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       5. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

       7. lower-PI.f64 44.0

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)}\right)}{\pi} \]

   11. Applied rewrites 44.0%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}}{\pi} \]

   if 5.99999999999999968e-123 < a

   1. Initial program 14.6%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   2. Add Preprocessing

   4. Applied rewrites 15.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale} - \left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} + \mathsf{hypot}\left(\frac{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(a \cdot a, 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)}{y-scale \cdot y-scale}, \frac{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}{x-scale \cdot y-scale}\right)\right)}{\left(\left(b + a\right) \cdot \left(b - a\right)\right) \cdot \sin \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}}{\pi} \]

   5. Taylor expanded in y-scale around inf

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

   7. Applied rewrites 16.0%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\left(-\frac{2 \cdot \left(\frac{\left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale} + \frac{\left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)}{x-scale \cdot x-scale}\right)}{\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   8. Taylor expanded in angle around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      11. associate-*r* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a + b\right)\right)} \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      14. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      15. +-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      16. lower-+.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot \left(b \cdot b\right)}{\left(angle \cdot \left(x-scale \cdot x-scale\right)\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(b - a\right)\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

      17. lower--.f64 22.0

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

   10. Applied rewrites 22.0%

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

   11. Taylor expanded in b around inf

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{-180}{angle \cdot \left({x-scale}^{2} \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]
   12. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{-180}{angle \cdot \left({x-scale}^{2} \cdot \mathsf{PI}\left(\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       2. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\left({x-scale}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot angle}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       3. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot {x-scale}^{2}\right)} \cdot angle} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       4. associate-*l* N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left({x-scale}^{2} \cdot angle\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       5. *-commutative N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot {x-scale}^{2}\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(angle \cdot {x-scale}^{2}\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       7. lower-PI.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(angle \cdot {x-scale}^{2}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(angle \cdot {x-scale}^{2}\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       9. unpow2 N/A

          \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\mathsf{PI}\left(\right) \cdot \left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\mathsf{PI}\left(\right)} \]

       10. lower-*.f64 31.7

           \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180}{\pi \cdot \left(angle \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}\right)} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]

   13. Applied rewrites 31.7%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\color{blue}{\frac{-180}{\pi \cdot \left(angle \cdot \left(x-scale \cdot x-scale\right)\right)}} \cdot \left(x-scale \cdot y-scale\right)\right)}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 6 \cdot 10^{-123}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\left(y-scale \cdot x-scale\right) \cdot \frac{-180}{\pi \cdot \left(angle \cdot \left(x-scale \cdot x-scale\right)\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 38.1% accurate, 21.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \frac{180}{\frac{\pi}{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (/ 180.0 (/ PI (atan (* -180.0 (/ y-scale (* angle (* PI x-scale))))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 / (((double) M_PI) / atan((-180.0 * (y_45_scale / (angle * (((double) M_PI) * x_45_scale))))));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 / (Math.PI / Math.atan((-180.0 * (y_45_scale / (angle * (Math.PI * x_45_scale))))));
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 / (math.pi / math.atan((-180.0 * (y_45_scale / (angle * (math.pi * x_45_scale))))))

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 / Float64(pi / atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(pi * x_45_scale)))))))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 / (pi / atan((-180.0 * (y_45_scale / (angle * (pi * x_45_scale))))));
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 / N[(Pi / N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
2. Add Preprocessing

3. Taylor expanded in x-scale around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

5. Applied rewrites 33.0%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

6. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   5. lower-cos.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   8. lower-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

   10. lower-sin.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   13. lower-PI.f64 42.8

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

8. Applied rewrites 42.8%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
9. Step-by-step derivation

10. Applied rewrites 43.8%

    \[\leadsto \color{blue}{\frac{180}{\frac{\pi}{\tan^{-1} \left(\frac{-1}{\frac{x-scale}{y-scale} \cdot \tan \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}} \]

11. Taylor expanded in angle around 0

    \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}} \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}} \]

    2. lower-/.f64 N/A

       \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}} \]

    3. lower-*.f64 N/A

       \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}} \]

    4. *-commutative N/A

       \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}} \]

    5. lower-*.f64 N/A

       \[\leadsto \frac{180}{\frac{\mathsf{PI}\left(\right)}{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}} \]

    6. lower-PI.f64 38.2

       \[\leadsto \frac{180}{\frac{\pi}{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)}\right)}} \]

13. Applied rewrites 38.2%

    \[\leadsto \frac{180}{\frac{\pi}{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}}} \]
14. Add Preprocessing

Alternative 25: 38.0% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (/ (* y-scale -180.0) (* angle (* PI x-scale)))) PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((double) M_PI) * x_45_scale)))) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (Math.PI * x_45_scale)))) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (math.pi * x_45_scale)))) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * Float64(pi * x_45_scale)))) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (pi * x_45_scale)))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * -180.0), $MachinePrecision] / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
2. Add Preprocessing

3. Taylor expanded in x-scale around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

5. Applied rewrites 33.0%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-0.5 \cdot \left(y-scale \cdot \left(2 \cdot \mathsf{fma}\left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b, {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot \left(a \cdot a\right)\right)\right)\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(b - a\right)\right)\right)\right)}\right)}}{\pi} \]

6. Taylor expanded in b around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-1 \cdot \frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-neg.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}}{\mathsf{PI}\left(\right)} \]

   3. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\color{blue}{\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{\color{blue}{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   5. lower-cos.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \color{blue}{\cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   8. lower-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{\color{blue}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

   10. lower-sin.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}\right)\right)}{\mathsf{PI}\left(\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\mathsf{neg}\left(\frac{y-scale \cdot \cos \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}{x-scale \cdot \sin \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)}\right)\right)}{\mathsf{PI}\left(\right)} \]

   13. lower-PI.f64 42.8

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right)}\right)}{\pi} \]

8. Applied rewrites 42.8%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]

9. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

    2. lower-/.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

    3. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{\color{blue}{-180 \cdot y-scale}}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}{\mathsf{PI}\left(\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

    5. *-commutative N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

    6. lower-*.f64 N/A

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x-scale\right)}}\right)}{\mathsf{PI}\left(\right)} \]

    7. lower-PI.f64 38.0

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{-180 \cdot y-scale}{angle \cdot \left(\color{blue}{\pi} \cdot x-scale\right)}\right)}{\pi} \]

11. Applied rewrites 38.0%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}}{\pi} \]

12. Final simplification 38.0%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi} \]
13. Add Preprocessing

Alternative 26: 38.0% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* -180.0 (/ y-scale (* angle (* PI x-scale))))) PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-180.0 * (y_45_scale / (angle * (((double) M_PI) * x_45_scale))))) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-180.0 * (y_45_scale / (angle * (Math.PI * x_45_scale))))) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-180.0 * (y_45_scale / (angle * (math.pi * x_45_scale))))) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(y_45_scale / Float64(angle * Float64(pi * x_45_scale))))) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-180.0 * (y_45_scale / (angle * (pi * x_45_scale))))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(y$45$scale / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

5. Applied rewrites 15.0%

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

6. Taylor expanded in a around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{y-scale}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{\color{blue}{angle \cdot \left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \color{blue}{\left(x-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   5. lower-PI.f64 38.0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \color{blue}{\pi}\right)}\right)}{\pi} \]

8. Applied rewrites 38.0%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}}{\pi} \]

9. Final simplification 38.0%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{y-scale}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi} \]
10. Add Preprocessing

Alternative 27: 14.8% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\pi \cdot \left(y-scale \cdot angle\right)}\right)}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* -180.0 (/ x-scale (* PI (* y-scale angle))))) PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-180.0 * (x_45_scale / (((double) M_PI) * (y_45_scale * angle))))) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-180.0 * (x_45_scale / (Math.PI * (y_45_scale * angle))))) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-180.0 * (x_45_scale / (math.pi * (y_45_scale * angle))))) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(x_45_scale / Float64(pi * Float64(y_45_scale * angle))))) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-180.0 * (x_45_scale / (pi * (y_45_scale * angle))))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(x$45$scale / N[(Pi * N[(y$45$scale * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

5. Applied rewrites 15.0%

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

6. Taylor expanded in a around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\color{blue}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \color{blue}{\left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   5. lower-PI.f64 15.4

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \color{blue}{\pi}\right)}\right)}{\pi} \]

8. Applied rewrites 15.4%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \pi\right)}\right)}}{\pi} \]
9. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right)}{\mathsf{PI}\left(\right)} \]

   2. associate-*r* N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\color{blue}{\left(angle \cdot y-scale\right) \cdot \mathsf{PI}\left(\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\color{blue}{\left(angle \cdot y-scale\right) \cdot \mathsf{PI}\left(\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   4. *-commutative N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\color{blue}{\left(y-scale \cdot angle\right)} \cdot \mathsf{PI}\left(\right)}\right)}{\mathsf{PI}\left(\right)} \]

   5. lower-*.f64 15.4

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\color{blue}{\left(y-scale \cdot angle\right)} \cdot \pi}\right)}{\pi} \]

10. Applied rewrites 15.4%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\color{blue}{\left(y-scale \cdot angle\right) \cdot \pi}}\right)}{\pi} \]

11. Final simplification 15.4%

    \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\pi \cdot \left(y-scale \cdot angle\right)}\right)}{\pi} \]
12. Add Preprocessing

Alternative 28: 14.7% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \pi\right)}\right)}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* 180.0 (/ (atan (* -180.0 (/ x-scale (* angle (* y-scale PI))))) PI)))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (atan((-180.0 * (x_45_scale / (angle * (y_45_scale * ((double) M_PI)))))) / ((double) M_PI));
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return 180.0 * (Math.atan((-180.0 * (x_45_scale / (angle * (y_45_scale * Math.PI))))) / Math.PI);
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return 180.0 * (math.atan((-180.0 * (x_45_scale / (angle * (y_45_scale * math.pi))))) / math.pi)

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(180.0 * Float64(atan(Float64(-180.0 * Float64(x_45_scale / Float64(angle * Float64(y_45_scale * pi))))) / pi))
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = 180.0 * (atan((-180.0 * (x_45_scale / (angle * (y_45_scale * pi))))) / pi);
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(180.0 * N[(N[ArcTan[N[(-180.0 * N[(x$45$scale / N[(angle * N[(y$45$scale * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

   \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(90 \cdot \frac{x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{90 \cdot \left(x-scale \cdot \left(y-scale \cdot \left(2 \cdot \frac{{a}^{2}}{{y-scale}^{2}} - 2 \cdot \frac{{b}^{2}}{{x-scale}^{2}}\right)\right)\right)}{angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

5. Applied rewrites 15.0%

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

6. Taylor expanded in a around inf

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}\right)}}{\mathsf{PI}\left(\right)} \]

   2. lower-/.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \color{blue}{\frac{x-scale}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{\color{blue}{angle \cdot \left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \color{blue}{\left(y-scale \cdot \mathsf{PI}\left(\right)\right)}}\right)}{\mathsf{PI}\left(\right)} \]

   5. lower-PI.f64 15.4

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \color{blue}{\pi}\right)}\right)}{\pi} \]

8. Applied rewrites 15.4%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-180 \cdot \frac{x-scale}{angle \cdot \left(y-scale \cdot \pi\right)}\right)}}{\pi} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (a b angle x-scale y-scale)
  :name "raw-angle from scale-rotated-ellipse"
  :precision binary64
  (* 180.0 (/ (atan (/ (- (- (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale) (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale)) (sqrt (+ (pow (- (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale)) 2.0) (pow (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) 2.0)))) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale))) PI)))