raw-angle from scale-rotated-ellipse

Percentage Accurate: 16.7% → 58.1%

Time: 36.6s

Alternatives: 15

Speedup: 22.2×

• could not determine a ground truth

  1. a = -6228579353074819.0
  2. b = -1.0997345978032144e+226
  3. angle = 2.2269887153864017e+161
  4. x-scale = -1.784091402410901e-263
  5. y-scale = -2.5386519198950378e+144

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: 16.7% 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: 58.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.9 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.2e-22)
   (/
    (*
     180.0
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
    PI)
   (if (<= b_m 1.9e+72)
     (*
      180.0
      (/
       (atan
        (/
         (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
         (* (* angle PI) (* (- b_m a) (+ b_m a)))))
       PI))
     (if (<= b_m 5.7e+119)
       (/
        1.0
        (/
         PI
         (*
          180.0
          (atan
           (*
            (/ y-scale x-scale)
            (tan (* PI (* 0.005555555555555556 angle))))))))
       (*
        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) {
	double tmp;
	if (b_m <= 2.2e-22) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 1.9e+72) {
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * ((double) M_PI)) * ((b_m - a) * (b_m + a))))) / ((double) M_PI));
	} else if (b_m <= 5.7e+119) {
		tmp = 1.0 / (((double) M_PI) / (180.0 * atan(((y_45_scale / x_45_scale) * tan((((double) M_PI) * (0.005555555555555556 * angle)))))));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 2.2e-22) {
		tmp = (180.0 * Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale))))) / Math.PI;
	} else if (b_m <= 1.9e+72) {
		tmp = 180.0 * (Math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * Math.PI) * ((b_m - a) * (b_m + a))))) / Math.PI);
	} else if (b_m <= 5.7e+119) {
		tmp = 1.0 / (Math.PI / (180.0 * Math.atan(((y_45_scale / x_45_scale) * Math.tan((Math.PI * (0.005555555555555556 * angle)))))));
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 2.2e-22:
		tmp = (180.0 * math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale))))) / math.pi
	elif b_m <= 1.9e+72:
		tmp = 180.0 * (math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * math.pi) * ((b_m - a) * (b_m + a))))) / math.pi)
	elif b_m <= 5.7e+119:
		tmp = 1.0 / (math.pi / (180.0 * math.atan(((y_45_scale / x_45_scale) * math.tan((math.pi * (0.005555555555555556 * angle)))))))
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 2.2e-22)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 1.9e+72)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / x_45_scale) / Float64(Float64(angle * pi) * Float64(Float64(b_m - a) * Float64(b_m + a))))) / pi));
	elseif (b_m <= 5.7e+119)
		tmp = Float64(1.0 / Float64(pi / Float64(180.0 * atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(pi * Float64(0.005555555555555556 * angle))))))));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * 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 (b_m <= 2.2e-22)
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale))))) / pi;
	elseif (b_m <= 1.9e+72)
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * pi) * ((b_m - a) * (b_m + a))))) / pi);
	elseif (b_m <= 5.7e+119)
		tmp = 1.0 / (pi / (180.0 * atan(((y_45_scale / x_45_scale) * tan((pi * (0.005555555555555556 * angle)))))));
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 2.2e-22], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 1.9e+72], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $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], If[LessEqual[b$95$m, 5.7e+119], N[(1.0 / N[(Pi / N[(180.0 * N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 4 regimes

2. if b < 2.2000000000000001e-22

   1. Initial program 17.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 29.3%

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

   6. Taylor expanded in b around 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 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.4

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.5

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

   14. Applied rewrites 55.5%

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

   if 2.2000000000000001e-22 < b < 1.90000000000000003e72

   1. Initial program 44.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 33.0%

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

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 40.6

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

   8. Applied rewrites 40.6%

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

   if 1.90000000000000003e72 < b < 5.7000000000000002e119

   1. Initial program 20.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 50.4%

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

   6. Taylor expanded in 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 50.3

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

   8. Applied rewrites 50.3%

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

   10. Applied rewrites 50.3%

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

   12. Applied rewrites 60.2%

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

   if 5.7000000000000002e119 < b

   1. Initial program 3.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 11.6%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 70.2

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

   8. Applied rewrites 70.2%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{1}{\frac{\pi}{180 \cdot \tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.5% accurate, 5.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ \mathbf{if}\;b\_m \leq 9.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.5 \cdot 10^{+115}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot {t\_1}^{2}}{x-scale \cdot \left(t\_1 \cdot \left(t\_2 \cdot \left(\left(b\_m + a\right) \cdot \left(a - b\_m\right)\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{t\_2 \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

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 (sin t_0)))
   (if (<= b_m 9.2e-51)
     (/
      (*
       180.0
       (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
      PI)
     (if (<= b_m 2.5e+115)
       (*
        180.0
        (/
         (atan
          (/
           (* (* y-scale (* b_m b_m)) (pow t_1 2.0))
           (* x-scale (* t_1 (* t_2 (* (+ b_m a) (- a b_m)))))))
         PI))
       (*
        180.0
        (/
         (atan
          (/
           (*
            y-scale
            (fma 0.5 (cos (* (* angle PI) 0.011111111111111112)) 0.5))
           (* 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 = sin(t_0);
	double tmp;
	if (b_m <= 9.2e-51) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 2.5e+115) {
		tmp = 180.0 * (atan((((y_45_scale * (b_m * b_m)) * pow(t_1, 2.0)) / (x_45_scale * (t_1 * (t_2 * ((b_m + a) * (a - b_m))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * fma(0.5, cos(((angle * ((double) M_PI)) * 0.011111111111111112)), 0.5)) / (t_2 * -x_45_scale))) / ((double) M_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 = sin(t_0)
	tmp = 0.0
	if (b_m <= 9.2e-51)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 2.5e+115)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * (t_1 ^ 2.0)) / Float64(x_45_scale * Float64(t_1 * Float64(t_2 * Float64(Float64(b_m + a) * Float64(a - b_m))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * fma(0.5, cos(Float64(Float64(angle * pi) * 0.011111111111111112)), 0.5)) / Float64(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[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[b$95$m, 9.2e-51], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 2.5e+115], N[(180.0 * N[(N[ArcTan[N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(t$95$1 * N[(t$95$2 * N[(N[(b$95$m + a), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.20000000000000007e-51

   1. Initial program 17.7%

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

   3. Taylor expanded in 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 29.8%

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

   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 54.8

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

   8. Applied rewrites 54.8%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 56.0

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

   11. Applied rewrites 56.0%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 56.0

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

   14. Applied rewrites 56.0%

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

   if 9.20000000000000007e-51 < b < 2.50000000000000004e115

   1. Initial program 31.6%

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

   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 44.4%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 45.5

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

   8. Applied rewrites 45.5%

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

   if 2.50000000000000004e115 < b

   1. Initial program 3.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 14.8%

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

   7. Applied rewrites 14.9%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 18.5%

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

   11. Taylor expanded in b around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-PI.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sin.f64 N/A

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

       14. lower-*.f64 N/A

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

   13. Applied rewrites 75.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+115}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot {\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \sin t\_0\\ \mathbf{if}\;b\_m \leq 9.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.5 \cdot 10^{+115}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \left(b\_m \cdot b\_m\right)}{x-scale \cdot \left(\cos t\_0 \cdot \left(t\_1 \cdot \left(\left(b\_m + a\right) \cdot \left(a - b\_m\right)\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{t\_1 \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

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 (sin t_0)))
   (if (<= b_m 9.2e-51)
     (/
      (*
       180.0
       (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
      PI)
     (if (<= b_m 2.5e+115)
       (*
        180.0
        (/
         (atan
          (/
           (* y-scale (* b_m b_m))
           (* x-scale (* (cos t_0) (* t_1 (* (+ b_m a) (- a b_m)))))))
         PI))
       (*
        180.0
        (/
         (atan
          (/
           (*
            y-scale
            (fma 0.5 (cos (* (* angle PI) 0.011111111111111112)) 0.5))
           (* t_1 (- 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 = sin(t_0);
	double tmp;
	if (b_m <= 9.2e-51) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 2.5e+115) {
		tmp = 180.0 * (atan(((y_45_scale * (b_m * b_m)) / (x_45_scale * (cos(t_0) * (t_1 * ((b_m + a) * (a - b_m))))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * fma(0.5, cos(((angle * ((double) M_PI)) * 0.011111111111111112)), 0.5)) / (t_1 * -x_45_scale))) / ((double) M_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 = sin(t_0)
	tmp = 0.0
	if (b_m <= 9.2e-51)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 2.5e+115)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * Float64(b_m * b_m)) / Float64(x_45_scale * Float64(cos(t_0) * Float64(t_1 * Float64(Float64(b_m + a) * Float64(a - b_m))))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * fma(0.5, cos(Float64(Float64(angle * pi) * 0.011111111111111112)), 0.5)) / Float64(t_1 * 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[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[b$95$m, 9.2e-51], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 2.5e+115], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * N[(N[Cos[t$95$0], $MachinePrecision] * N[(t$95$1 * N[(N[(b$95$m + a), $MachinePrecision] * N[(a - b$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 9.20000000000000007e-51

   1. Initial program 17.7%

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

   3. Taylor expanded in 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 29.8%

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

   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 54.8

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

   8. Applied rewrites 54.8%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 56.0

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

   11. Applied rewrites 56.0%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 56.0

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

   14. Applied rewrites 56.0%

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

   if 9.20000000000000007e-51 < b < 2.50000000000000004e115

   1. Initial program 31.6%

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

   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 44.4%

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

   6. Taylor expanded in angle around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 45.4

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

   8. Applied rewrites 45.4%

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

   if 2.50000000000000004e115 < b

   1. Initial program 3.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 14.8%

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

   7. Applied rewrites 14.9%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 18.5%

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

   11. Taylor expanded in b around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-PI.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sin.f64 N/A

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

       14. lower-*.f64 N/A

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

   13. Applied rewrites 75.6%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+115}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \left(b \cdot b\right)}{x-scale \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b + a\right) \cdot \left(a - b\right)\right)\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.6% accurate, 8.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;b\_m \leq 1.62 \cdot 10^{-25}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.85 \cdot 10^{+115}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -0.5}{x-scale} \cdot \frac{\left(b\_m \cdot b\_m\right) \cdot 2}{t\_0 \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{t\_0 \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (sin (* 0.005555555555555556 (* angle PI)))))
   (if (<= b_m 1.62e-25)
     (/
      (*
       180.0
       (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
      PI)
     (if (<= b_m 1.85e+115)
       (*
        180.0
        (/
         (atan
          (*
           (/ (* y-scale -0.5) x-scale)
           (/ (* (* b_m b_m) 2.0) (* t_0 (* (- b_m a) (+ b_m a))))))
         PI))
       (*
        180.0
        (/
         (atan
          (/
           (*
            y-scale
            (fma 0.5 (cos (* (* angle PI) 0.011111111111111112)) 0.5))
           (* t_0 (- 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 = sin((0.005555555555555556 * (angle * ((double) M_PI))));
	double tmp;
	if (b_m <= 1.62e-25) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 1.85e+115) {
		tmp = 180.0 * (atan((((y_45_scale * -0.5) / x_45_scale) * (((b_m * b_m) * 2.0) / (t_0 * ((b_m - a) * (b_m + a)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * fma(0.5, cos(((angle * ((double) M_PI)) * 0.011111111111111112)), 0.5)) / (t_0 * -x_45_scale))) / ((double) M_PI));
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = sin(Float64(0.005555555555555556 * Float64(angle * pi)))
	tmp = 0.0
	if (b_m <= 1.62e-25)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 1.85e+115)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(y_45_scale * -0.5) / x_45_scale) * Float64(Float64(Float64(b_m * b_m) * 2.0) / Float64(t_0 * Float64(Float64(b_m - a) * Float64(b_m + a)))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * fma(0.5, cos(Float64(Float64(angle * pi) * 0.011111111111111112)), 0.5)) / Float64(t_0 * 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[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 1.62e-25], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 1.85e+115], N[(180.0 * N[(N[ArcTan[N[(N[(N[(y$45$scale * -0.5), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t$95$0 * N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale * N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * (-x$45$scale)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.62e-25

   1. Initial program 17.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 29.4%

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

   6. Taylor expanded in 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 54.6

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

   8. Applied rewrites 54.6%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.7

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

   11. Applied rewrites 55.7%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.7

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

   14. Applied rewrites 55.7%

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

   if 1.62e-25 < b < 1.85000000000000003e115

   1. Initial program 36.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 47.1%

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

   7. Applied rewrites 47.7%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 47.2%

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

   11. Taylor expanded in angle around 0

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

       1. unpow2 N/A

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

       2. lower-*.f64 44.4

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

   13. Applied rewrites 44.4%

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

   if 1.85000000000000003e115 < b

   1. Initial program 3.7%

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

   3. Taylor expanded in 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 17.9%

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

   7. Applied rewrites 17.9%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 17.9%

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

   11. Taylor expanded in b around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 N/A

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

       11. lower-PI.f64 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-sin.f64 N/A

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

       14. lower-*.f64 N/A

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

   13. Applied rewrites 72.9%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(-\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}}{\pi} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.62 \cdot 10^{-25}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{+115}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -0.5}{x-scale} \cdot \frac{\left(b \cdot b\right) \cdot 2}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.62 \cdot 10^{-25}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.55 \cdot 10^{+136}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -0.5}{x-scale} \cdot \frac{\left(b\_m \cdot b\_m\right) \cdot 2}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.62e-25)
   (/
    (*
     180.0
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
    PI)
   (if (<= b_m 1.55e+136)
     (*
      180.0
      (/
       (atan
        (*
         (/ (* y-scale -0.5) x-scale)
         (/
          (* (* b_m b_m) 2.0)
          (*
           (sin (* 0.005555555555555556 (* angle PI)))
           (* (- b_m a) (+ b_m a))))))
       PI))
     (* 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) {
	double tmp;
	if (b_m <= 1.62e-25) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 1.55e+136) {
		tmp = 180.0 * (atan((((y_45_scale * -0.5) / x_45_scale) * (((b_m * b_m) * 2.0) / (sin((0.005555555555555556 * (angle * ((double) M_PI)))) * ((b_m - a) * (b_m + a)))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 1.62e-25) {
		tmp = (180.0 * Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale))))) / Math.PI;
	} else if (b_m <= 1.55e+136) {
		tmp = 180.0 * (Math.atan((((y_45_scale * -0.5) / x_45_scale) * (((b_m * b_m) * 2.0) / (Math.sin((0.005555555555555556 * (angle * Math.PI))) * ((b_m - a) * (b_m + a)))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 1.62e-25:
		tmp = (180.0 * math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale))))) / math.pi
	elif b_m <= 1.55e+136:
		tmp = 180.0 * (math.atan((((y_45_scale * -0.5) / x_45_scale) * (((b_m * b_m) * 2.0) / (math.sin((0.005555555555555556 * (angle * math.pi))) * ((b_m - a) * (b_m + a)))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 1.62e-25)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 1.55e+136)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(y_45_scale * -0.5) / x_45_scale) * Float64(Float64(Float64(b_m * b_m) * 2.0) / Float64(sin(Float64(0.005555555555555556 * 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 * -180.0) / Float64(angle * 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 (b_m <= 1.62e-25)
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale))))) / pi;
	elseif (b_m <= 1.55e+136)
		tmp = 180.0 * (atan((((y_45_scale * -0.5) / x_45_scale) * (((b_m * b_m) * 2.0) / (sin((0.005555555555555556 * (angle * pi))) * ((b_m - a) * (b_m + a)))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 1.62e-25], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 1.55e+136], N[(180.0 * N[(N[ArcTan[N[(N[(N[(y$45$scale * -0.5), $MachinePrecision] / x$45$scale), $MachinePrecision] * N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * 2.0), $MachinePrecision] / N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 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. Split input into 3 regimes

2. if b < 1.62e-25

   1. Initial program 17.5%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 29.4%

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

   6. Taylor expanded in 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 54.6

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

   8. Applied rewrites 54.6%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.7

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

   11. Applied rewrites 55.7%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.7

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

   14. Applied rewrites 55.7%

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

   if 1.62e-25 < b < 1.54999999999999992e136

   1. Initial program 32.7%

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

   3. Taylor expanded in 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 47.7%

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

   7. Applied rewrites 48.2%

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

   8. Taylor expanded in angle around 0

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

   10. Applied rewrites 47.8%

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

   11. Taylor expanded in angle around 0

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

       1. unpow2 N/A

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

       2. lower-*.f64 45.8

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

   13. Applied rewrites 45.8%

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

   if 1.54999999999999992e136 < 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 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 9.1%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 73.7

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

   8. Applied rewrites 73.7%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.62 \cdot 10^{-25}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{+136}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -0.5}{x-scale} \cdot \frac{\left(b \cdot b\right) \cdot 2}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.9 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 5.7 \cdot 10^{+119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.2e-22)
   (/
    (*
     180.0
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
    PI)
   (if (<= b_m 1.9e+72)
     (*
      180.0
      (/
       (atan
        (/
         (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
         (* (* angle PI) (* (- b_m a) (+ b_m a)))))
       PI))
     (if (<= b_m 5.7e+119)
       (*
        180.0
        (/
         (atan
          (* (/ y-scale x-scale) (tan (* PI (* 0.005555555555555556 angle)))))
         PI))
       (*
        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) {
	double tmp;
	if (b_m <= 2.2e-22) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 1.9e+72) {
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * ((double) M_PI)) * ((b_m - a) * (b_m + a))))) / ((double) M_PI));
	} else if (b_m <= 5.7e+119) {
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((((double) M_PI) * (0.005555555555555556 * angle))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 2.2e-22) {
		tmp = (180.0 * Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale))))) / Math.PI;
	} else if (b_m <= 1.9e+72) {
		tmp = 180.0 * (Math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * Math.PI) * ((b_m - a) * (b_m + a))))) / Math.PI);
	} else if (b_m <= 5.7e+119) {
		tmp = 180.0 * (Math.atan(((y_45_scale / x_45_scale) * Math.tan((Math.PI * (0.005555555555555556 * angle))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 2.2e-22:
		tmp = (180.0 * math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale))))) / math.pi
	elif b_m <= 1.9e+72:
		tmp = 180.0 * (math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * math.pi) * ((b_m - a) * (b_m + a))))) / math.pi)
	elif b_m <= 5.7e+119:
		tmp = 180.0 * (math.atan(((y_45_scale / x_45_scale) * math.tan((math.pi * (0.005555555555555556 * angle))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 2.2e-22)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 1.9e+72)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / x_45_scale) / Float64(Float64(angle * pi) * Float64(Float64(b_m - a) * Float64(b_m + a))))) / pi));
	elseif (b_m <= 5.7e+119)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale / x_45_scale) * tan(Float64(pi * Float64(0.005555555555555556 * angle))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * 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 (b_m <= 2.2e-22)
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale))))) / pi;
	elseif (b_m <= 1.9e+72)
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * pi) * ((b_m - a) * (b_m + a))))) / pi);
	elseif (b_m <= 5.7e+119)
		tmp = 180.0 * (atan(((y_45_scale / x_45_scale) * tan((pi * (0.005555555555555556 * angle))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 2.2e-22], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 1.9e+72], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $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], If[LessEqual[b$95$m, 5.7e+119], N[(180.0 * N[(N[ArcTan[N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 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. Split input into 4 regimes

2. if b < 2.2000000000000001e-22

   1. Initial program 17.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 29.3%

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

   6. Taylor expanded in b around 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 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.4

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.5

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

   14. Applied rewrites 55.5%

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

   if 2.2000000000000001e-22 < b < 1.90000000000000003e72

   1. Initial program 44.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 33.0%

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

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 40.6

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

   8. Applied rewrites 40.6%

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

   if 1.90000000000000003e72 < b < 5.7000000000000002e119

   1. Initial program 20.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 50.4%

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

   6. Taylor expanded in 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 50.3

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

   8. Applied rewrites 50.3%

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

   10. Applied rewrites 50.3%

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

   12. Applied rewrites 60.2%

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

   if 5.7000000000000002e119 < b

   1. Initial program 3.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 11.6%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 70.2

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

   8. Applied rewrites 70.2%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 5.7 \cdot 10^{+119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale}{x-scale} \cdot \tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.9 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2 \cdot 10^{+119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.2e-22)
   (/
    (*
     180.0
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
    PI)
   (if (<= b_m 1.9e+72)
     (*
      180.0
      (/
       (atan
        (/
         (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
         (* (* angle PI) (* (- b_m a) (+ b_m a)))))
       PI))
     (if (<= b_m 2e+119)
       (*
        180.0
        (/
         (atan
          (* y-scale (/ (tan (* PI (* 0.005555555555555556 angle))) x-scale)))
         PI))
       (*
        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) {
	double tmp;
	if (b_m <= 2.2e-22) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 1.9e+72) {
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * ((double) M_PI)) * ((b_m - a) * (b_m + a))))) / ((double) M_PI));
	} else if (b_m <= 2e+119) {
		tmp = 180.0 * (atan((y_45_scale * (tan((((double) M_PI) * (0.005555555555555556 * angle))) / x_45_scale))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 2.2e-22) {
		tmp = (180.0 * Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale))))) / Math.PI;
	} else if (b_m <= 1.9e+72) {
		tmp = 180.0 * (Math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * Math.PI) * ((b_m - a) * (b_m + a))))) / Math.PI);
	} else if (b_m <= 2e+119) {
		tmp = 180.0 * (Math.atan((y_45_scale * (Math.tan((Math.PI * (0.005555555555555556 * angle))) / x_45_scale))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 2.2e-22:
		tmp = (180.0 * math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale))))) / math.pi
	elif b_m <= 1.9e+72:
		tmp = 180.0 * (math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * math.pi) * ((b_m - a) * (b_m + a))))) / math.pi)
	elif b_m <= 2e+119:
		tmp = 180.0 * (math.atan((y_45_scale * (math.tan((math.pi * (0.005555555555555556 * angle))) / x_45_scale))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 2.2e-22)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 1.9e+72)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / x_45_scale) / Float64(Float64(angle * pi) * Float64(Float64(b_m - a) * Float64(b_m + a))))) / pi));
	elseif (b_m <= 2e+119)
		tmp = Float64(180.0 * Float64(atan(Float64(y_45_scale * Float64(tan(Float64(pi * Float64(0.005555555555555556 * angle))) / x_45_scale))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * 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 (b_m <= 2.2e-22)
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale))))) / pi;
	elseif (b_m <= 1.9e+72)
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * pi) * ((b_m - a) * (b_m + a))))) / pi);
	elseif (b_m <= 2e+119)
		tmp = 180.0 * (atan((y_45_scale * (tan((pi * (0.005555555555555556 * angle))) / x_45_scale))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 2.2e-22], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 1.9e+72], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $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], If[LessEqual[b$95$m, 2e+119], N[(180.0 * N[(N[ArcTan[N[(y$45$scale * N[(N[Tan[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 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. Split input into 4 regimes

2. if b < 2.2000000000000001e-22

   1. Initial program 17.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 29.3%

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

   6. Taylor expanded in b around 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 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.4

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.5

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

   14. Applied rewrites 55.5%

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

   if 2.2000000000000001e-22 < b < 1.90000000000000003e72

   1. Initial program 44.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 33.0%

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

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 40.6

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

   8. Applied rewrites 40.6%

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

   if 1.90000000000000003e72 < b < 1.99999999999999989e119

   1. Initial program 20.3%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 50.4%

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

   6. Taylor expanded in 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 50.3

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

   8. Applied rewrites 50.3%

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

      1. lift-PI.f64 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 60.2

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

   10. Applied rewrites 60.2%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sin.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-PI.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. lift-cos.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-/l* N/A

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

       11. *-commutative N/A

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

       12. lower-*.f64 N/A

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

   12. Applied rewrites 60.2%

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

   if 1.99999999999999989e119 < b

   1. Initial program 3.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 11.6%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 70.2

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

   8. Applied rewrites 70.2%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+119}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(y-scale \cdot \frac{\tan \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}{x-scale}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.3% accurate, 16.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.35 \cdot 10^{+96}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(b\_m - a\right) \cdot \left(b\_m + a\right)}}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.2e-22)
   (/
    (*
     180.0
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
    PI)
   (if (<= b_m 1.35e+96)
     (*
      180.0
      (/
       (atan
        (/
         (/
          (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
          (* (- b_m a) (+ b_m a)))
         (* angle PI)))
       PI))
     (* 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) {
	double tmp;
	if (b_m <= 2.2e-22) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 1.35e+96) {
		tmp = 180.0 * (atan((((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((b_m - a) * (b_m + a))) / (angle * ((double) M_PI)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 2.2e-22) {
		tmp = (180.0 * Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale))))) / Math.PI;
	} else if (b_m <= 1.35e+96) {
		tmp = 180.0 * (Math.atan((((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((b_m - a) * (b_m + a))) / (angle * Math.PI))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 2.2e-22:
		tmp = (180.0 * math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale))))) / math.pi
	elif b_m <= 1.35e+96:
		tmp = 180.0 * (math.atan((((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((b_m - a) * (b_m + a))) / (angle * math.pi))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 2.2e-22)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 1.35e+96)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / x_45_scale) / Float64(Float64(b_m - a) * Float64(b_m + a))) / Float64(angle * pi))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * 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 (b_m <= 2.2e-22)
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale))))) / pi;
	elseif (b_m <= 1.35e+96)
		tmp = 180.0 * (atan((((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((b_m - a) * (b_m + a))) / (angle * pi))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 2.2e-22], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 1.35e+96], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $MachinePrecision] / N[(N[(b$95$m - a), $MachinePrecision] * N[(b$95$m + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 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. Split input into 3 regimes

2. if b < 2.2000000000000001e-22

   1. Initial program 17.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 29.3%

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

   6. Taylor expanded in b around 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 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.4

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.5

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

   14. Applied rewrites 55.5%

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

   if 2.2000000000000001e-22 < b < 1.35000000000000011e96

   1. Initial program 37.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 28.0%

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

   6. Taylor expanded in x-scale around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 32.9

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

   8. Applied rewrites 32.9%

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

   10. Applied rewrites 38.4%

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

   11. Taylor expanded in x-scale around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 40.0

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

   13. Applied rewrites 40.0%

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

   if 1.35000000000000011e96 < b

   1. Initial program 9.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 14.6%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 64.8

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

   8. Applied rewrites 64.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+96}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(b - a\right) \cdot \left(b + a\right)}}{angle \cdot \pi}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.1% accurate, 16.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -2}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.2e-22)
   (/
    (*
     180.0
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
    PI)
   (if (<= b_m 4.1e+99)
     (*
      180.0
      (/
       (atan
        (/
         (* 90.0 (/ (* (* y-scale (* b_m b_m)) -2.0) x-scale))
         (* (* angle PI) (* (- b_m a) (+ b_m a)))))
       PI))
     (* 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) {
	double tmp;
	if (b_m <= 2.2e-22) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 4.1e+99) {
		tmp = 180.0 * (atan(((90.0 * (((y_45_scale * (b_m * b_m)) * -2.0) / x_45_scale)) / ((angle * ((double) M_PI)) * ((b_m - a) * (b_m + a))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 2.2e-22) {
		tmp = (180.0 * Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale))))) / Math.PI;
	} else if (b_m <= 4.1e+99) {
		tmp = 180.0 * (Math.atan(((90.0 * (((y_45_scale * (b_m * b_m)) * -2.0) / x_45_scale)) / ((angle * Math.PI) * ((b_m - a) * (b_m + a))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 2.2e-22:
		tmp = (180.0 * math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale))))) / math.pi
	elif b_m <= 4.1e+99:
		tmp = 180.0 * (math.atan(((90.0 * (((y_45_scale * (b_m * b_m)) * -2.0) / x_45_scale)) / ((angle * math.pi) * ((b_m - a) * (b_m + a))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 2.2e-22)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 4.1e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(90.0 * Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -2.0) / 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 * -180.0) / Float64(angle * 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 (b_m <= 2.2e-22)
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale))))) / pi;
	elseif (b_m <= 4.1e+99)
		tmp = 180.0 * (atan(((90.0 * (((y_45_scale * (b_m * b_m)) * -2.0) / x_45_scale)) / ((angle * pi) * ((b_m - a) * (b_m + a))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 2.2e-22], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 4.1e+99], N[(180.0 * N[(N[ArcTan[N[(N[(90.0 * N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] / x$45$scale), $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 * -180.0), $MachinePrecision] / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.2000000000000001e-22

   1. Initial program 17.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 29.3%

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

   6. Taylor expanded in b around 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 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.4

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.5

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

   14. Applied rewrites 55.5%

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

   if 2.2000000000000001e-22 < b < 4.09999999999999979e99

   1. Initial program 37.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 28.0%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 39.9

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

   8. Applied rewrites 39.9%

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

   if 4.09999999999999979e99 < b

   1. Initial program 9.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 14.6%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 64.8

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

   8. Applied rewrites 64.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{90 \cdot \frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -2}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.1% accurate, 17.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b\_m - a\right) \cdot \left(b\_m + a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.2e-22)
   (/
    (*
     180.0
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
    PI)
   (if (<= b_m 4.1e+99)
     (*
      180.0
      (/
       (atan
        (/
         (/ (* (* y-scale (* b_m b_m)) -180.0) x-scale)
         (* (* angle PI) (* (- b_m a) (+ b_m a)))))
       PI))
     (* 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) {
	double tmp;
	if (b_m <= 2.2e-22) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 4.1e+99) {
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * ((double) M_PI)) * ((b_m - a) * (b_m + a))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 2.2e-22) {
		tmp = (180.0 * Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale))))) / Math.PI;
	} else if (b_m <= 4.1e+99) {
		tmp = 180.0 * (Math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * Math.PI) * ((b_m - a) * (b_m + a))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 2.2e-22:
		tmp = (180.0 * math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale))))) / math.pi
	elif b_m <= 4.1e+99:
		tmp = 180.0 * (math.atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * math.pi) * ((b_m - a) * (b_m + a))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 2.2e-22)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 4.1e+99)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / 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 * -180.0) / Float64(angle * 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 (b_m <= 2.2e-22)
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale))))) / pi;
	elseif (b_m <= 4.1e+99)
		tmp = 180.0 * (atan(((((y_45_scale * (b_m * b_m)) * -180.0) / x_45_scale) / ((angle * pi) * ((b_m - a) * (b_m + a))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 2.2e-22], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 4.1e+99], N[(180.0 * N[(N[ArcTan[N[(N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $MachinePrecision] / x$45$scale), $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 * -180.0), $MachinePrecision] / N[(angle * N[(Pi * x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 2.2000000000000001e-22

   1. Initial program 17.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 29.3%

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

   6. Taylor expanded in b around 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 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.4

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.5

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

   14. Applied rewrites 55.5%

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

   if 2.2000000000000001e-22 < b < 4.09999999999999979e99

   1. Initial program 37.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 28.0%

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

   6. Taylor expanded in x-scale around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 39.9

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

   8. Applied rewrites 39.9%

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

   if 4.09999999999999979e99 < b

   1. Initial program 9.2%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 14.6%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 64.8

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

   8. Applied rewrites 64.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{x-scale}}{\left(angle \cdot \pi\right) \cdot \left(\left(b - a\right) \cdot \left(b + a\right)\right)}\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.2% accurate, 17.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 1.9 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(y-scale \cdot \left(b\_m \cdot b\_m\right)\right) \cdot -180}{\left(angle \cdot x-scale\right) \cdot \left(\left(b\_m - a\right) \cdot \left(\pi \cdot \left(b\_m + a\right)\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b\_m \leq 2.15 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(y-scale \cdot \left(angle \cdot \frac{\pi}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 2.2e-22)
   (/
    (*
     180.0
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
    PI)
   (if (<= b_m 1.9e+72)
     (*
      180.0
      (/
       (atan
        (/
         (* (* y-scale (* b_m b_m)) -180.0)
         (* (* angle x-scale) (* (- b_m a) (* PI (+ b_m a))))))
       PI))
     (if (<= b_m 2.15e+103)
       (*
        180.0
        (/
         (atan (* 0.005555555555555556 (* y-scale (* angle (/ PI x-scale)))))
         PI))
       (*
        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) {
	double tmp;
	if (b_m <= 2.2e-22) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else if (b_m <= 1.9e+72) {
		tmp = 180.0 * (atan((((y_45_scale * (b_m * b_m)) * -180.0) / ((angle * x_45_scale) * ((b_m - a) * (((double) M_PI) * (b_m + a)))))) / ((double) M_PI));
	} else if (b_m <= 2.15e+103) {
		tmp = 180.0 * (atan((0.005555555555555556 * (y_45_scale * (angle * (((double) M_PI) / x_45_scale))))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 2.2e-22) {
		tmp = (180.0 * Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale))))) / Math.PI;
	} else if (b_m <= 1.9e+72) {
		tmp = 180.0 * (Math.atan((((y_45_scale * (b_m * b_m)) * -180.0) / ((angle * x_45_scale) * ((b_m - a) * (Math.PI * (b_m + a)))))) / Math.PI);
	} else if (b_m <= 2.15e+103) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * (y_45_scale * (angle * (Math.PI / x_45_scale))))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 2.2e-22:
		tmp = (180.0 * math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale))))) / math.pi
	elif b_m <= 1.9e+72:
		tmp = 180.0 * (math.atan((((y_45_scale * (b_m * b_m)) * -180.0) / ((angle * x_45_scale) * ((b_m - a) * (math.pi * (b_m + a)))))) / math.pi)
	elif b_m <= 2.15e+103:
		tmp = 180.0 * (math.atan((0.005555555555555556 * (y_45_scale * (angle * (math.pi / x_45_scale))))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 2.2e-22)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	elseif (b_m <= 1.9e+72)
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(Float64(y_45_scale * Float64(b_m * b_m)) * -180.0) / Float64(Float64(angle * x_45_scale) * Float64(Float64(b_m - a) * Float64(pi * Float64(b_m + a)))))) / pi));
	elseif (b_m <= 2.15e+103)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(y_45_scale * Float64(angle * Float64(pi / x_45_scale))))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * 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 (b_m <= 2.2e-22)
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale))))) / pi;
	elseif (b_m <= 1.9e+72)
		tmp = 180.0 * (atan((((y_45_scale * (b_m * b_m)) * -180.0) / ((angle * x_45_scale) * ((b_m - a) * (pi * (b_m + a)))))) / pi);
	elseif (b_m <= 2.15e+103)
		tmp = 180.0 * (atan((0.005555555555555556 * (y_45_scale * (angle * (pi / x_45_scale))))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 2.2e-22], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], If[LessEqual[b$95$m, 1.9e+72], N[(180.0 * N[(N[ArcTan[N[(N[(N[(y$45$scale * N[(b$95$m * b$95$m), $MachinePrecision]), $MachinePrecision] * -180.0), $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], If[LessEqual[b$95$m, 2.15e+103], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(y$45$scale * N[(angle * N[(Pi / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 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. Split input into 4 regimes

2. if b < 2.2000000000000001e-22

   1. Initial program 17.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 29.3%

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

   6. Taylor expanded in b around 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 54.3

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

   8. Applied rewrites 54.3%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.4

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

   11. Applied rewrites 55.4%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.5

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

   14. Applied rewrites 55.5%

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

   if 2.2000000000000001e-22 < b < 1.90000000000000003e72

   1. Initial program 44.4%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 48.0%

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

   6. Taylor expanded in 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. 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. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 40.6

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

   8. Applied rewrites 40.6%

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

   if 1.90000000000000003e72 < b < 2.14999999999999984e103

   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 40.7%

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

   6. Taylor expanded in b around 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 61.6

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

   8. Applied rewrites 61.6%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 61.6

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

   11. Applied rewrites 61.6%

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

       1. lift-PI.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 61.6

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-/.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-/l* N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-/.f64 61.6

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

   13. Applied rewrites 61.6%

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

   if 2.14999999999999984e103 < b

   1. Initial program 9.8%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 13.2%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 65.7

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

   8. Applied rewrites 65.7%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{elif}\;b \leq 1.9 \cdot 10^{+72}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{\left(y-scale \cdot \left(b \cdot b\right)\right) \cdot -180}{\left(angle \cdot x-scale\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot \left(b + a\right)\right)\right)}\right)}{\pi}\\ \mathbf{elif}\;b \leq 2.15 \cdot 10^{+103}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(y-scale \cdot \left(angle \cdot \frac{\pi}{x-scale}\right)\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.9% accurate, 21.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 0.0073:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 0.0073)
   (/
    (*
     180.0
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale)))))
    PI)
   (* 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) {
	double tmp;
	if (b_m <= 0.0073) {
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale))))) / ((double) M_PI);
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 0.0073) {
		tmp = (180.0 * Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale))))) / Math.PI;
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 0.0073:
		tmp = (180.0 * math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale))))) / math.pi
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 0.0073)
		tmp = Float64(Float64(180.0 * atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale))))) / pi);
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * 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 (b_m <= 0.0073)
		tmp = (180.0 * atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale))))) / pi;
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 0.0073], N[(N[(180.0 * N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 0.00730000000000000007

   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 29.7%

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

   6. Taylor expanded in b around 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 54.1

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

   8. Applied rewrites 54.1%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.2

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

   11. Applied rewrites 55.2%

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

   12. Taylor expanded in angle around 0

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

       1. associate-*r/ N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. associate-/l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower-PI.f64 55.2

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

   14. Applied rewrites 55.2%

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

   if 0.00730000000000000007 < b

   1. Initial program 17.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 17.9%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 52.8

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

   8. Applied rewrites 52.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0073:\\ \;\;\;\;\frac{180 \cdot \tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.9% accurate, 21.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 0.0073:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 0.0073)
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale))))
     PI))
   (* 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) {
	double tmp;
	if (b_m <= 0.0073) {
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (((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 (b_m <= 0.0073) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan(((y_45_scale * -180.0) / (angle * (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 b_m <= 0.0073:
		tmp = 180.0 * (math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan(((y_45_scale * -180.0) / (angle * (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 (b_m <= 0.0073)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(Float64(y_45_scale * -180.0) / Float64(angle * 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 (b_m <= 0.0073)
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale)))) / pi);
	else
		tmp = 180.0 * (atan(((y_45_scale * -180.0) / (angle * (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[b$95$m, 0.0073], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 0.00730000000000000007

   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 29.7%

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

   6. Taylor expanded in b around 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 54.1

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

   8. Applied rewrites 54.1%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 55.2

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

   11. Applied rewrites 55.2%

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

   if 0.00730000000000000007 < b

   1. Initial program 17.9%

      \[180 \cdot \frac{\tan^{-1} \left(\frac{\left(\frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} - \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) - \sqrt{{\left(\frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale} - \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale}\right)}^{2} + {\left(\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}\right)}^{2}}}{\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale}}\right)}{\pi} \]
   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 17.9%

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

   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. 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. lower-*.f64 N/A

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

      6. lower-PI.f64 52.8

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

   8. Applied rewrites 52.8%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{-180 \cdot y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right)}}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.0073:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(\frac{y-scale \cdot -180}{angle \cdot \left(\pi \cdot x-scale\right)}\right)}{\pi}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.8% accurate, 21.3× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.9 \cdot 10^{+220}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;180 \cdot \frac{\tan^{-1} \left(x-scale \cdot \frac{-180}{y-scale \cdot \left(angle \cdot \pi\right)}\right)}{\pi}\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.9e+220)
   (*
    180.0
    (/
     (atan (* 0.005555555555555556 (* angle (/ (* y-scale PI) x-scale))))
     PI))
   (* 180.0 (/ (atan (* x-scale (/ -180.0 (* y-scale (* angle PI))))) 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 (b_m <= 1.9e+220) {
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * ((y_45_scale * ((double) M_PI)) / x_45_scale)))) / ((double) M_PI));
	} else {
		tmp = 180.0 * (atan((x_45_scale * (-180.0 / (y_45_scale * (angle * ((double) M_PI)))))) / ((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 (b_m <= 1.9e+220) {
		tmp = 180.0 * (Math.atan((0.005555555555555556 * (angle * ((y_45_scale * Math.PI) / x_45_scale)))) / Math.PI);
	} else {
		tmp = 180.0 * (Math.atan((x_45_scale * (-180.0 / (y_45_scale * (angle * Math.PI))))) / 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 b_m <= 1.9e+220:
		tmp = 180.0 * (math.atan((0.005555555555555556 * (angle * ((y_45_scale * math.pi) / x_45_scale)))) / math.pi)
	else:
		tmp = 180.0 * (math.atan((x_45_scale * (-180.0 / (y_45_scale * (angle * math.pi))))) / 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 (b_m <= 1.9e+220)
		tmp = Float64(180.0 * Float64(atan(Float64(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * pi) / x_45_scale)))) / pi));
	else
		tmp = Float64(180.0 * Float64(atan(Float64(x_45_scale * Float64(-180.0 / Float64(y_45_scale * Float64(angle * pi))))) / 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 (b_m <= 1.9e+220)
		tmp = 180.0 * (atan((0.005555555555555556 * (angle * ((y_45_scale * pi) / x_45_scale)))) / pi);
	else
		tmp = 180.0 * (atan((x_45_scale * (-180.0 / (y_45_scale * (angle * pi))))) / 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[b$95$m, 1.9e+220], N[(180.0 * N[(N[ArcTan[N[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision], N[(180.0 * N[(N[ArcTan[N[(x$45$scale * N[(-180.0 / N[(y$45$scale * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.89999999999999992e220

   1. Initial program 18.6%

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

   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 31.2%

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

   6. Taylor expanded in b around 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 50.4

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

   8. Applied rewrites 50.4%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. associate-/l* N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-PI.f64 51.2

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

   11. Applied rewrites 51.2%

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

   if 1.89999999999999992e220 < 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 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 0.0%

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

   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. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 31.2

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

   8. Applied rewrites 31.2%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 31.2

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 31.2

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

   10. Applied rewrites 31.2%

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

Alternative 15: 45.4% accurate, 22.2× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ 180 \cdot \frac{\tan^{-1} \left(0.005555555555555556 \cdot \left(angle \cdot \frac{y-scale \cdot \pi}{x-scale}\right)\right)}{\pi} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  180.0
  (/ (atan (* 0.005555555555555556 (* angle (/ (* y-scale 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((0.005555555555555556 * (angle * ((y_45_scale * ((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((0.005555555555555556 * (angle * ((y_45_scale * 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((0.005555555555555556 * (angle * ((y_45_scale * 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(0.005555555555555556 * Float64(angle * Float64(Float64(y_45_scale * 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((0.005555555555555556 * (angle * ((y_45_scale * 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[(0.005555555555555556 * N[(angle * N[(N[(y$45$scale * Pi), $MachinePrecision] / 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 29.9%

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

6. Taylor expanded in b around 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 48.8

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

8. Applied rewrites 48.8%

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

9. Taylor expanded in angle around 0

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

    1. lower-*.f64 N/A

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

    2. associate-/l* N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lower-PI.f64 49.5

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

11. Applied rewrites 49.5%

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

Reproduce

herbie shell --seed 2024214 
(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)))