raw-angle from scale-rotated-ellipse

Percentage Accurate: 17.0% → 58.6%

Time: 54.7s

Alternatives: 9

Speedup: 21.5×

• could not determine a ground truth

  1. a = 4.082402698287882e-55
  2. b = -2.9095929684695706e+166
  3. angle = -6.4782718527957345e+289
  4. x-scale = 4.632758810239966e-293
  5. y-scale = 9.222744580546651e-298

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: 17.0% 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: 58.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \frac{y-scale}{x-scale} \cdot -2\\ \mathbf{if}\;b\_m \leq 1.82 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1} \left(\left(\tan \left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot t\_0\right) \cdot -0.5\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{elif}\;b\_m \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\frac{\tan^{-1} \left(90 \cdot \frac{\frac{\frac{\left(b\_m \cdot b\_m\right) \cdot y-scale}{x-scale} \cdot -2}{\pi \cdot angle}}{\left(a + b\_m\right) \cdot \left(b\_m - a\right)}\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b\_m \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\pi}{\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot t\_0\right) \cdot -0.5\right) \cdot 180}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ y-scale x-scale) -2.0)))
   (if (<= b_m 1.82e+19)
     (*
      (atan
       (*
        (*
         (tan (* (* (sqrt PI) (sqrt PI)) (* 0.005555555555555556 angle)))
         t_0)
        -0.5))
      (* (/ 1.0 PI) 180.0))
     (if (<= b_m 3e+112)
       (*
        (/
         (atan
          (*
           90.0
           (/
            (/ (* (/ (* (* b_m b_m) y-scale) x-scale) -2.0) (* PI angle))
            (* (+ a b_m) (- b_m a)))))
         PI)
        180.0)
       (if (<= b_m 7.5e+153)
         (/
          1.0
          (/
           PI
           (*
            (atan (* (* (* (* PI angle) 0.005555555555555556) t_0) -0.5))
            180.0)))
         (*
          (/ (atan (* (* (/ y-scale (* (* PI x-scale) angle)) -2.0) 90.0)) PI)
          180.0))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale / x_45_scale) * -2.0;
	double tmp;
	if (b_m <= 1.82e+19) {
		tmp = atan(((tan(((sqrt(((double) M_PI)) * sqrt(((double) M_PI))) * (0.005555555555555556 * angle))) * t_0) * -0.5)) * ((1.0 / ((double) M_PI)) * 180.0);
	} else if (b_m <= 3e+112) {
		tmp = (atan((90.0 * ((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (((double) M_PI) * angle)) / ((a + b_m) * (b_m - a))))) / ((double) M_PI)) * 180.0;
	} else if (b_m <= 7.5e+153) {
		tmp = 1.0 / (((double) M_PI) / (atan(((((((double) M_PI) * angle) * 0.005555555555555556) * t_0) * -0.5)) * 180.0));
	} else {
		tmp = (atan((((y_45_scale / ((((double) M_PI) * x_45_scale) * angle)) * -2.0) * 90.0)) / ((double) M_PI)) * 180.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (y_45_scale / x_45_scale) * -2.0;
	double tmp;
	if (b_m <= 1.82e+19) {
		tmp = Math.atan(((Math.tan(((Math.sqrt(Math.PI) * Math.sqrt(Math.PI)) * (0.005555555555555556 * angle))) * t_0) * -0.5)) * ((1.0 / Math.PI) * 180.0);
	} else if (b_m <= 3e+112) {
		tmp = (Math.atan((90.0 * ((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (Math.PI * angle)) / ((a + b_m) * (b_m - a))))) / Math.PI) * 180.0;
	} else if (b_m <= 7.5e+153) {
		tmp = 1.0 / (Math.PI / (Math.atan(((((Math.PI * angle) * 0.005555555555555556) * t_0) * -0.5)) * 180.0));
	} else {
		tmp = (Math.atan((((y_45_scale / ((Math.PI * x_45_scale) * angle)) * -2.0) * 90.0)) / Math.PI) * 180.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (y_45_scale / x_45_scale) * -2.0
	tmp = 0
	if b_m <= 1.82e+19:
		tmp = math.atan(((math.tan(((math.sqrt(math.pi) * math.sqrt(math.pi)) * (0.005555555555555556 * angle))) * t_0) * -0.5)) * ((1.0 / math.pi) * 180.0)
	elif b_m <= 3e+112:
		tmp = (math.atan((90.0 * ((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (math.pi * angle)) / ((a + b_m) * (b_m - a))))) / math.pi) * 180.0
	elif b_m <= 7.5e+153:
		tmp = 1.0 / (math.pi / (math.atan(((((math.pi * angle) * 0.005555555555555556) * t_0) * -0.5)) * 180.0))
	else:
		tmp = (math.atan((((y_45_scale / ((math.pi * x_45_scale) * angle)) * -2.0) * 90.0)) / math.pi) * 180.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(y_45_scale / x_45_scale) * -2.0)
	tmp = 0.0
	if (b_m <= 1.82e+19)
		tmp = Float64(atan(Float64(Float64(tan(Float64(Float64(sqrt(pi) * sqrt(pi)) * Float64(0.005555555555555556 * angle))) * t_0) * -0.5)) * Float64(Float64(1.0 / pi) * 180.0));
	elseif (b_m <= 3e+112)
		tmp = Float64(Float64(atan(Float64(90.0 * Float64(Float64(Float64(Float64(Float64(Float64(b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / Float64(pi * angle)) / Float64(Float64(a + b_m) * Float64(b_m - a))))) / pi) * 180.0);
	elseif (b_m <= 7.5e+153)
		tmp = Float64(1.0 / Float64(pi / Float64(atan(Float64(Float64(Float64(Float64(pi * angle) * 0.005555555555555556) * t_0) * -0.5)) * 180.0)));
	else
		tmp = Float64(Float64(atan(Float64(Float64(Float64(y_45_scale / Float64(Float64(pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (y_45_scale / x_45_scale) * -2.0;
	tmp = 0.0;
	if (b_m <= 1.82e+19)
		tmp = atan(((tan(((sqrt(pi) * sqrt(pi)) * (0.005555555555555556 * angle))) * t_0) * -0.5)) * ((1.0 / pi) * 180.0);
	elseif (b_m <= 3e+112)
		tmp = (atan((90.0 * ((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (pi * angle)) / ((a + b_m) * (b_m - a))))) / pi) * 180.0;
	elseif (b_m <= 7.5e+153)
		tmp = 1.0 / (pi / (atan(((((pi * angle) * 0.005555555555555556) * t_0) * -0.5)) * 180.0));
	else
		tmp = (atan((((y_45_scale / ((pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision]}, If[LessEqual[b$95$m, 1.82e+19], N[(N[ArcTan[N[(N[(N[Tan[N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / Pi), $MachinePrecision] * 180.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3e+112], N[(N[(N[ArcTan[N[(90.0 * N[(N[(N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision] / N[(Pi * angle), $MachinePrecision]), $MachinePrecision] / N[(N[(a + b$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], If[LessEqual[b$95$m, 7.5e+153], N[(1.0 / N[(Pi / N[(N[ArcTan[N[(N[(N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * t$95$0), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[(N[(y$45$scale / N[(N[(Pi * x$45$scale), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * 90.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.82e19

   1. Initial program 15.9%

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

   3. Taylor expanded in x-scale around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.0%

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

   10. Applied rewrites 54.0%

       \[\leadsto \color{blue}{\tan^{-1} \left(-0.5 \cdot \left(\left(-2 \cdot \frac{y-scale}{x-scale}\right) \cdot \tan \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.0%

       \[\leadsto \tan^{-1} \left(-0.5 \cdot \left(\left(-2 \cdot \frac{y-scale}{x-scale}\right) \cdot \tan \left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   if 1.82e19 < b < 2.99999999999999979e112

   1. Initial program 50.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 64.9%

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

   if 2.99999999999999979e112 < b < 7.50000000000000065e153

   1. Initial program 25.0%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 25.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 50.8%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 75.4%

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

   if 7.50000000000000065e153 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 66.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(-2 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot 90\right)}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.82 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1} \left(\left(\tan \left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\frac{\tan^{-1} \left(90 \cdot \frac{\frac{\frac{\left(b \cdot b\right) \cdot y-scale}{x-scale} \cdot -2}{\pi \cdot angle}}{\left(a + b\right) \cdot \left(b - a\right)}\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\pi}{\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot 180}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 60.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;b\_m \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;\left(\frac{1}{\pi} \cdot 180\right) \cdot \tan^{-1} \left(\left(\tan \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\left(\frac{\cos t\_0}{\sin t\_0 \cdot x-scale} \cdot y-scale\right) \cdot 2\right) \cdot -0.5\right)}{\pi} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= b_m 1.4e+19)
     (*
      (* (/ 1.0 PI) 180.0)
      (atan
       (*
        (*
         (tan (* (* 0.005555555555555556 angle) (cbrt (* (* PI PI) PI))))
         (* (/ y-scale x-scale) -2.0))
        -0.5)))
     (*
      (/
       (atan (* (* (* (/ (cos t_0) (* (sin t_0) x-scale)) y-scale) 2.0) -0.5))
       PI)
      180.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (b_m <= 1.4e+19) {
		tmp = ((1.0 / ((double) M_PI)) * 180.0) * atan(((tan(((0.005555555555555556 * angle) * cbrt(((((double) M_PI) * ((double) M_PI)) * ((double) M_PI))))) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	} else {
		tmp = (atan(((((cos(t_0) / (sin(t_0) * x_45_scale)) * y_45_scale) * 2.0) * -0.5)) / ((double) M_PI)) * 180.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (b_m <= 1.4e+19) {
		tmp = ((1.0 / Math.PI) * 180.0) * Math.atan(((Math.tan(((0.005555555555555556 * angle) * Math.cbrt(((Math.PI * Math.PI) * Math.PI)))) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	} else {
		tmp = (Math.atan(((((Math.cos(t_0) / (Math.sin(t_0) * x_45_scale)) * y_45_scale) * 2.0) * -0.5)) / Math.PI) * 180.0;
	}
	return tmp;
}

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (b_m <= 1.4e+19)
		tmp = Float64(Float64(Float64(1.0 / pi) * 180.0) * atan(Float64(Float64(tan(Float64(Float64(0.005555555555555556 * angle) * cbrt(Float64(Float64(pi * pi) * pi)))) * Float64(Float64(y_45_scale / x_45_scale) * -2.0)) * -0.5)));
	else
		tmp = Float64(Float64(atan(Float64(Float64(Float64(Float64(cos(t_0) / Float64(sin(t_0) * x_45_scale)) * y_45_scale) * 2.0) * -0.5)) / pi) * 180.0);
	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[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[b$95$m, 1.4e+19], N[(N[(N[(1.0 / Pi), $MachinePrecision] * 180.0), $MachinePrecision] * N[ArcTan[N[(N[(N[Tan[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * N[Power[N[(N[(Pi * Pi), $MachinePrecision] * Pi), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[(N[(N[(N[Cos[t$95$0], $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * 2.0), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e19

   1. Initial program 15.9%

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

   3. Taylor expanded in x-scale around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.0%

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

   10. Applied rewrites 54.0%

       \[\leadsto \color{blue}{\tan^{-1} \left(-0.5 \cdot \left(\left(-2 \cdot \frac{y-scale}{x-scale}\right) \cdot \tan \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.0%

       \[\leadsto \tan^{-1} \left(-0.5 \cdot \left(\left(-2 \cdot \frac{y-scale}{x-scale}\right) \cdot \tan \left(\sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   if 1.4e19 < b

   1. Initial program 20.2%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 22.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 58.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(2 \cdot \left(y-scale \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot -0.5\right)}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;\left(\frac{1}{\pi} \cdot 180\right) \cdot \tan^{-1} \left(\left(\tan \left(\left(0.005555555555555556 \cdot angle\right) \cdot \sqrt[3]{\left(\pi \cdot \pi\right) \cdot \pi}\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\left(\frac{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot x-scale} \cdot y-scale\right) \cdot 2\right) \cdot -0.5\right)}{\pi} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;b\_m \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1} \left(\left(\tan \left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\left(\frac{\cos t\_0}{\sin t\_0 \cdot x-scale} \cdot y-scale\right) \cdot 2\right) \cdot -0.5\right)}{\pi} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= b_m 1.4e+19)
     (*
      (atan
       (*
        (*
         (tan (* (* (sqrt PI) (sqrt PI)) (* 0.005555555555555556 angle)))
         (* (/ y-scale x-scale) -2.0))
        -0.5))
      (* (/ 1.0 PI) 180.0))
     (*
      (/
       (atan (* (* (* (/ (cos t_0) (* (sin t_0) x-scale)) y-scale) 2.0) -0.5))
       PI)
      180.0))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (b_m <= 1.4e+19) {
		tmp = atan(((tan(((sqrt(((double) M_PI)) * sqrt(((double) M_PI))) * (0.005555555555555556 * angle))) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5)) * ((1.0 / ((double) M_PI)) * 180.0);
	} else {
		tmp = (atan(((((cos(t_0) / (sin(t_0) * x_45_scale)) * y_45_scale) * 2.0) * -0.5)) / ((double) M_PI)) * 180.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (b_m <= 1.4e+19) {
		tmp = Math.atan(((Math.tan(((Math.sqrt(Math.PI) * Math.sqrt(Math.PI)) * (0.005555555555555556 * angle))) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5)) * ((1.0 / Math.PI) * 180.0);
	} else {
		tmp = (Math.atan(((((Math.cos(t_0) / (Math.sin(t_0) * x_45_scale)) * y_45_scale) * 2.0) * -0.5)) / Math.PI) * 180.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if b_m <= 1.4e+19:
		tmp = math.atan(((math.tan(((math.sqrt(math.pi) * math.sqrt(math.pi)) * (0.005555555555555556 * angle))) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5)) * ((1.0 / math.pi) * 180.0)
	else:
		tmp = (math.atan(((((math.cos(t_0) / (math.sin(t_0) * x_45_scale)) * y_45_scale) * 2.0) * -0.5)) / math.pi) * 180.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (b_m <= 1.4e+19)
		tmp = Float64(atan(Float64(Float64(tan(Float64(Float64(sqrt(pi) * sqrt(pi)) * Float64(0.005555555555555556 * angle))) * Float64(Float64(y_45_scale / x_45_scale) * -2.0)) * -0.5)) * Float64(Float64(1.0 / pi) * 180.0));
	else
		tmp = Float64(Float64(atan(Float64(Float64(Float64(Float64(cos(t_0) / Float64(sin(t_0) * x_45_scale)) * y_45_scale) * 2.0) * -0.5)) / pi) * 180.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (b_m <= 1.4e+19)
		tmp = atan(((tan(((sqrt(pi) * sqrt(pi)) * (0.005555555555555556 * angle))) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5)) * ((1.0 / pi) * 180.0);
	else
		tmp = (atan(((((cos(t_0) / (sin(t_0) * x_45_scale)) * y_45_scale) * 2.0) * -0.5)) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[b$95$m, 1.4e+19], N[(N[ArcTan[N[(N[(N[Tan[N[(N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / Pi), $MachinePrecision] * 180.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[(N[(N[(N[Cos[t$95$0], $MachinePrecision] / N[(N[Sin[t$95$0], $MachinePrecision] * x$45$scale), $MachinePrecision]), $MachinePrecision] * y$45$scale), $MachinePrecision] * 2.0), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e19

   1. Initial program 15.9%

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

   3. Taylor expanded in x-scale around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.0%

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

   10. Applied rewrites 54.0%

       \[\leadsto \color{blue}{\tan^{-1} \left(-0.5 \cdot \left(\left(-2 \cdot \frac{y-scale}{x-scale}\right) \cdot \tan \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)} \]
   11. Step-by-step derivation

   12. Applied rewrites 56.0%

       \[\leadsto \tan^{-1} \left(-0.5 \cdot \left(\left(-2 \cdot \frac{y-scale}{x-scale}\right) \cdot \tan \left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \left(\frac{1}{\pi} \cdot 180\right) \]

   if 1.4e19 < b

   1. Initial program 20.2%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 22.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 58.8%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(2 \cdot \left(y-scale \cdot \frac{\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}{x-scale \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot -0.5\right)}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1} \left(\left(\tan \left(\left(\sqrt{\pi} \cdot \sqrt{\pi}\right) \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\left(\frac{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot x-scale} \cdot y-scale\right) \cdot 2\right) \cdot -0.5\right)}{\pi} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.7% accurate, 16.0× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right)\\ \mathbf{if}\;b\_m \leq 8 \cdot 10^{+18}:\\ \;\;\;\;t\_0 \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{elif}\;b\_m \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\frac{\tan^{-1} \left(90 \cdot \frac{\frac{\frac{\left(b\_m \cdot b\_m\right) \cdot y-scale}{x-scale} \cdot -2}{\pi \cdot angle}}{\left(a + b\_m\right) \cdot \left(b\_m - a\right)}\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b\_m \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\pi}{t\_0 \cdot 180}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (atan
          (*
           (*
            (* (* PI angle) 0.005555555555555556)
            (* (/ y-scale x-scale) -2.0))
           -0.5))))
   (if (<= b_m 8e+18)
     (* t_0 (* (/ 1.0 PI) 180.0))
     (if (<= b_m 3e+112)
       (*
        (/
         (atan
          (*
           90.0
           (/
            (/ (* (/ (* (* b_m b_m) y-scale) x-scale) -2.0) (* PI angle))
            (* (+ a b_m) (- b_m a)))))
         PI)
        180.0)
       (if (<= b_m 7.5e+153)
         (/ 1.0 (/ PI (* t_0 180.0)))
         (*
          (/ (atan (* (* (/ y-scale (* (* PI x-scale) angle)) -2.0) 90.0)) PI)
          180.0))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = atan(((((((double) M_PI) * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	double tmp;
	if (b_m <= 8e+18) {
		tmp = t_0 * ((1.0 / ((double) M_PI)) * 180.0);
	} else if (b_m <= 3e+112) {
		tmp = (atan((90.0 * ((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (((double) M_PI) * angle)) / ((a + b_m) * (b_m - a))))) / ((double) M_PI)) * 180.0;
	} else if (b_m <= 7.5e+153) {
		tmp = 1.0 / (((double) M_PI) / (t_0 * 180.0));
	} else {
		tmp = (atan((((y_45_scale / ((((double) M_PI) * x_45_scale) * angle)) * -2.0) * 90.0)) / ((double) M_PI)) * 180.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.atan(((((Math.PI * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	double tmp;
	if (b_m <= 8e+18) {
		tmp = t_0 * ((1.0 / Math.PI) * 180.0);
	} else if (b_m <= 3e+112) {
		tmp = (Math.atan((90.0 * ((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (Math.PI * angle)) / ((a + b_m) * (b_m - a))))) / Math.PI) * 180.0;
	} else if (b_m <= 7.5e+153) {
		tmp = 1.0 / (Math.PI / (t_0 * 180.0));
	} else {
		tmp = (Math.atan((((y_45_scale / ((Math.PI * x_45_scale) * angle)) * -2.0) * 90.0)) / Math.PI) * 180.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.atan(((((math.pi * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5))
	tmp = 0
	if b_m <= 8e+18:
		tmp = t_0 * ((1.0 / math.pi) * 180.0)
	elif b_m <= 3e+112:
		tmp = (math.atan((90.0 * ((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (math.pi * angle)) / ((a + b_m) * (b_m - a))))) / math.pi) * 180.0
	elif b_m <= 7.5e+153:
		tmp = 1.0 / (math.pi / (t_0 * 180.0))
	else:
		tmp = (math.atan((((y_45_scale / ((math.pi * x_45_scale) * angle)) * -2.0) * 90.0)) / math.pi) * 180.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = atan(Float64(Float64(Float64(Float64(pi * angle) * 0.005555555555555556) * Float64(Float64(y_45_scale / x_45_scale) * -2.0)) * -0.5))
	tmp = 0.0
	if (b_m <= 8e+18)
		tmp = Float64(t_0 * Float64(Float64(1.0 / pi) * 180.0));
	elseif (b_m <= 3e+112)
		tmp = Float64(Float64(atan(Float64(90.0 * Float64(Float64(Float64(Float64(Float64(Float64(b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / Float64(pi * angle)) / Float64(Float64(a + b_m) * Float64(b_m - a))))) / pi) * 180.0);
	elseif (b_m <= 7.5e+153)
		tmp = Float64(1.0 / Float64(pi / Float64(t_0 * 180.0)));
	else
		tmp = Float64(Float64(atan(Float64(Float64(Float64(y_45_scale / Float64(Float64(pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = atan(((((pi * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	tmp = 0.0;
	if (b_m <= 8e+18)
		tmp = t_0 * ((1.0 / pi) * 180.0);
	elseif (b_m <= 3e+112)
		tmp = (atan((90.0 * ((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (pi * angle)) / ((a + b_m) * (b_m - a))))) / pi) * 180.0;
	elseif (b_m <= 7.5e+153)
		tmp = 1.0 / (pi / (t_0 * 180.0));
	else
		tmp = (atan((((y_45_scale / ((pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[ArcTan[N[(N[(N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 8e+18], N[(t$95$0 * N[(N[(1.0 / Pi), $MachinePrecision] * 180.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3e+112], N[(N[(N[ArcTan[N[(90.0 * N[(N[(N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision] / N[(Pi * angle), $MachinePrecision]), $MachinePrecision] / N[(N[(a + b$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], If[LessEqual[b$95$m, 7.5e+153], N[(1.0 / N[(Pi / N[(t$95$0 * 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[(N[(y$45$scale / N[(N[(Pi * x$45$scale), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * 90.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 8e18

   1. Initial program 15.9%

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

   3. Taylor expanded in x-scale around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.0%

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

   10. Applied rewrites 54.0%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 55.3%

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

   if 8e18 < b < 2.99999999999999979e112

   1. Initial program 50.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 50.2%

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 64.9%

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

   if 2.99999999999999979e112 < b < 7.50000000000000065e153

   1. Initial program 25.0%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 25.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 50.8%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 75.4%

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

   if 7.50000000000000065e153 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 66.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(-2 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot 90\right)}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\frac{\tan^{-1} \left(90 \cdot \frac{\frac{\frac{\left(b \cdot b\right) \cdot y-scale}{x-scale} \cdot -2}{\pi \cdot angle}}{\left(a + b\right) \cdot \left(b - a\right)}\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\pi}{\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot 180}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.6% accurate, 16.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right)\\ \mathbf{if}\;b\_m \leq 1550000000:\\ \;\;\;\;t\_0 \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{elif}\;b\_m \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\frac{\left(b\_m \cdot b\_m\right) \cdot y-scale}{x-scale} \cdot -2}{\left(\left(a + b\_m\right) \cdot \left(b\_m - a\right)\right) \cdot \left(\pi \cdot angle\right)} \cdot 90\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b\_m \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\pi}{t\_0 \cdot 180}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (atan
          (*
           (*
            (* (* PI angle) 0.005555555555555556)
            (* (/ y-scale x-scale) -2.0))
           -0.5))))
   (if (<= b_m 1550000000.0)
     (* t_0 (* (/ 1.0 PI) 180.0))
     (if (<= b_m 3e+112)
       (*
        (/
         (atan
          (*
           (/
            (* (/ (* (* b_m b_m) y-scale) x-scale) -2.0)
            (* (* (+ a b_m) (- b_m a)) (* PI angle)))
           90.0))
         PI)
        180.0)
       (if (<= b_m 7.5e+153)
         (/ 1.0 (/ PI (* t_0 180.0)))
         (*
          (/ (atan (* (* (/ y-scale (* (* PI x-scale) angle)) -2.0) 90.0)) PI)
          180.0))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = atan(((((((double) M_PI) * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	double tmp;
	if (b_m <= 1550000000.0) {
		tmp = t_0 * ((1.0 / ((double) M_PI)) * 180.0);
	} else if (b_m <= 3e+112) {
		tmp = (atan(((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (((a + b_m) * (b_m - a)) * (((double) M_PI) * angle))) * 90.0)) / ((double) M_PI)) * 180.0;
	} else if (b_m <= 7.5e+153) {
		tmp = 1.0 / (((double) M_PI) / (t_0 * 180.0));
	} else {
		tmp = (atan((((y_45_scale / ((((double) M_PI) * x_45_scale) * angle)) * -2.0) * 90.0)) / ((double) M_PI)) * 180.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.atan(((((Math.PI * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	double tmp;
	if (b_m <= 1550000000.0) {
		tmp = t_0 * ((1.0 / Math.PI) * 180.0);
	} else if (b_m <= 3e+112) {
		tmp = (Math.atan(((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (((a + b_m) * (b_m - a)) * (Math.PI * angle))) * 90.0)) / Math.PI) * 180.0;
	} else if (b_m <= 7.5e+153) {
		tmp = 1.0 / (Math.PI / (t_0 * 180.0));
	} else {
		tmp = (Math.atan((((y_45_scale / ((Math.PI * x_45_scale) * angle)) * -2.0) * 90.0)) / Math.PI) * 180.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.atan(((((math.pi * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5))
	tmp = 0
	if b_m <= 1550000000.0:
		tmp = t_0 * ((1.0 / math.pi) * 180.0)
	elif b_m <= 3e+112:
		tmp = (math.atan(((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (((a + b_m) * (b_m - a)) * (math.pi * angle))) * 90.0)) / math.pi) * 180.0
	elif b_m <= 7.5e+153:
		tmp = 1.0 / (math.pi / (t_0 * 180.0))
	else:
		tmp = (math.atan((((y_45_scale / ((math.pi * x_45_scale) * angle)) * -2.0) * 90.0)) / math.pi) * 180.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = atan(Float64(Float64(Float64(Float64(pi * angle) * 0.005555555555555556) * Float64(Float64(y_45_scale / x_45_scale) * -2.0)) * -0.5))
	tmp = 0.0
	if (b_m <= 1550000000.0)
		tmp = Float64(t_0 * Float64(Float64(1.0 / pi) * 180.0));
	elseif (b_m <= 3e+112)
		tmp = Float64(Float64(atan(Float64(Float64(Float64(Float64(Float64(Float64(b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / Float64(Float64(Float64(a + b_m) * Float64(b_m - a)) * Float64(pi * angle))) * 90.0)) / pi) * 180.0);
	elseif (b_m <= 7.5e+153)
		tmp = Float64(1.0 / Float64(pi / Float64(t_0 * 180.0)));
	else
		tmp = Float64(Float64(atan(Float64(Float64(Float64(y_45_scale / Float64(Float64(pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = atan(((((pi * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	tmp = 0.0;
	if (b_m <= 1550000000.0)
		tmp = t_0 * ((1.0 / pi) * 180.0);
	elseif (b_m <= 3e+112)
		tmp = (atan(((((((b_m * b_m) * y_45_scale) / x_45_scale) * -2.0) / (((a + b_m) * (b_m - a)) * (pi * angle))) * 90.0)) / pi) * 180.0;
	elseif (b_m <= 7.5e+153)
		tmp = 1.0 / (pi / (t_0 * 180.0));
	else
		tmp = (atan((((y_45_scale / ((pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[ArcTan[N[(N[(N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 1550000000.0], N[(t$95$0 * N[(N[(1.0 / Pi), $MachinePrecision] * 180.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 3e+112], N[(N[(N[ArcTan[N[(N[(N[(N[(N[(N[(b$95$m * b$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision] / N[(N[(N[(a + b$95$m), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 90.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], If[LessEqual[b$95$m, 7.5e+153], N[(1.0 / N[(Pi / N[(t$95$0 * 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[(N[(y$45$scale / N[(N[(Pi * x$45$scale), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * 90.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 1.55e9

   1. Initial program 16.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.0%

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

   10. Applied rewrites 54.0%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 55.4%

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

   if 1.55e9 < b < 2.99999999999999979e112

   1. Initial program 45.1%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 57.9%

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

   if 2.99999999999999979e112 < b < 7.50000000000000065e153

   1. Initial program 25.0%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 25.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 50.8%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 75.4%

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

   if 7.50000000000000065e153 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 66.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(-2 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot 90\right)}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1550000000:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+112}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{\frac{\left(b \cdot b\right) \cdot y-scale}{x-scale} \cdot -2}{\left(\left(a + b\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot angle\right)} \cdot 90\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\pi}{\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot 180}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} t_0 := \tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right)\\ \mathbf{if}\;b\_m \leq 8 \cdot 10^{+18}:\\ \;\;\;\;t\_0 \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{elif}\;b\_m \leq 2.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{360 \cdot \left(\left(b\_m \cdot b\_m\right) \cdot y-scale\right)}{\left(\left(\left(a + b\_m\right) \cdot \pi\right) \cdot \left(b\_m - a\right)\right) \cdot \left(x-scale \cdot angle\right)} \cdot -0.5\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b\_m \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\pi}{t\_0 \cdot 180}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (let* ((t_0
         (atan
          (*
           (*
            (* (* PI angle) 0.005555555555555556)
            (* (/ y-scale x-scale) -2.0))
           -0.5))))
   (if (<= b_m 8e+18)
     (* t_0 (* (/ 1.0 PI) 180.0))
     (if (<= b_m 2.9e+112)
       (*
        (/
         (atan
          (*
           (/
            (* 360.0 (* (* b_m b_m) y-scale))
            (* (* (* (+ a b_m) PI) (- b_m a)) (* x-scale angle)))
           -0.5))
         PI)
        180.0)
       (if (<= b_m 7.5e+153)
         (/ 1.0 (/ PI (* t_0 180.0)))
         (*
          (/ (atan (* (* (/ y-scale (* (* PI x-scale) angle)) -2.0) 90.0)) PI)
          180.0))))))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = atan(((((((double) M_PI) * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	double tmp;
	if (b_m <= 8e+18) {
		tmp = t_0 * ((1.0 / ((double) M_PI)) * 180.0);
	} else if (b_m <= 2.9e+112) {
		tmp = (atan((((360.0 * ((b_m * b_m) * y_45_scale)) / ((((a + b_m) * ((double) M_PI)) * (b_m - a)) * (x_45_scale * angle))) * -0.5)) / ((double) M_PI)) * 180.0;
	} else if (b_m <= 7.5e+153) {
		tmp = 1.0 / (((double) M_PI) / (t_0 * 180.0));
	} else {
		tmp = (atan((((y_45_scale / ((((double) M_PI) * x_45_scale) * angle)) * -2.0) * 90.0)) / ((double) M_PI)) * 180.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = Math.atan(((((Math.PI * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	double tmp;
	if (b_m <= 8e+18) {
		tmp = t_0 * ((1.0 / Math.PI) * 180.0);
	} else if (b_m <= 2.9e+112) {
		tmp = (Math.atan((((360.0 * ((b_m * b_m) * y_45_scale)) / ((((a + b_m) * Math.PI) * (b_m - a)) * (x_45_scale * angle))) * -0.5)) / Math.PI) * 180.0;
	} else if (b_m <= 7.5e+153) {
		tmp = 1.0 / (Math.PI / (t_0 * 180.0));
	} else {
		tmp = (Math.atan((((y_45_scale / ((Math.PI * x_45_scale) * angle)) * -2.0) * 90.0)) / Math.PI) * 180.0;
	}
	return tmp;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	t_0 = math.atan(((((math.pi * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5))
	tmp = 0
	if b_m <= 8e+18:
		tmp = t_0 * ((1.0 / math.pi) * 180.0)
	elif b_m <= 2.9e+112:
		tmp = (math.atan((((360.0 * ((b_m * b_m) * y_45_scale)) / ((((a + b_m) * math.pi) * (b_m - a)) * (x_45_scale * angle))) * -0.5)) / math.pi) * 180.0
	elif b_m <= 7.5e+153:
		tmp = 1.0 / (math.pi / (t_0 * 180.0))
	else:
		tmp = (math.atan((((y_45_scale / ((math.pi * x_45_scale) * angle)) * -2.0) * 90.0)) / math.pi) * 180.0
	return tmp

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = atan(Float64(Float64(Float64(Float64(pi * angle) * 0.005555555555555556) * Float64(Float64(y_45_scale / x_45_scale) * -2.0)) * -0.5))
	tmp = 0.0
	if (b_m <= 8e+18)
		tmp = Float64(t_0 * Float64(Float64(1.0 / pi) * 180.0));
	elseif (b_m <= 2.9e+112)
		tmp = Float64(Float64(atan(Float64(Float64(Float64(360.0 * Float64(Float64(b_m * b_m) * y_45_scale)) / Float64(Float64(Float64(Float64(a + b_m) * pi) * Float64(b_m - a)) * Float64(x_45_scale * angle))) * -0.5)) / pi) * 180.0);
	elseif (b_m <= 7.5e+153)
		tmp = Float64(1.0 / Float64(pi / Float64(t_0 * 180.0)));
	else
		tmp = Float64(Float64(atan(Float64(Float64(Float64(y_45_scale / Float64(Float64(pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0);
	end
	return tmp
end

MATLAB

b_m = abs(b);
function tmp_2 = code(a, b_m, angle, x_45_scale, y_45_scale)
	t_0 = atan(((((pi * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5));
	tmp = 0.0;
	if (b_m <= 8e+18)
		tmp = t_0 * ((1.0 / pi) * 180.0);
	elseif (b_m <= 2.9e+112)
		tmp = (atan((((360.0 * ((b_m * b_m) * y_45_scale)) / ((((a + b_m) * pi) * (b_m - a)) * (x_45_scale * angle))) * -0.5)) / pi) * 180.0;
	elseif (b_m <= 7.5e+153)
		tmp = 1.0 / (pi / (t_0 * 180.0));
	else
		tmp = (atan((((y_45_scale / ((pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[ArcTan[N[(N[(N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b$95$m, 8e+18], N[(t$95$0 * N[(N[(1.0 / Pi), $MachinePrecision] * 180.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$m, 2.9e+112], N[(N[(N[ArcTan[N[(N[(N[(360.0 * N[(N[(b$95$m * b$95$m), $MachinePrecision] * y$45$scale), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(a + b$95$m), $MachinePrecision] * Pi), $MachinePrecision] * N[(b$95$m - a), $MachinePrecision]), $MachinePrecision] * N[(x$45$scale * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision], If[LessEqual[b$95$m, 7.5e+153], N[(1.0 / N[(Pi / N[(t$95$0 * 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[(N[(y$45$scale / N[(N[(Pi * x$45$scale), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * 90.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if b < 8e18

   1. Initial program 15.9%

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

   3. Taylor expanded in x-scale around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.0%

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

   10. Applied rewrites 54.0%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 55.3%

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

   if 8e18 < b < 2.9000000000000002e112

   1. Initial program 50.5%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 20.6%

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

   9. Taylor expanded in angle around 0

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(360 \cdot \frac{{b}^{2} \cdot y-scale}{angle \cdot \left(x-scale \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)}\right) \cdot \frac{-1}{2}\right)}{\mathsf{PI}\left(\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 58.1%

       \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\frac{360 \cdot \left(\left(b \cdot b\right) \cdot y-scale\right)}{\left(angle \cdot x-scale\right) \cdot \left(\left(\pi \cdot \left(b + a\right)\right) \cdot \left(b - a\right)\right)} \cdot -0.5\right)}{\pi} \]

   if 2.9000000000000002e112 < b < 7.50000000000000065e153

   1. Initial program 25.0%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 25.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 50.8%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 75.4%

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

   if 7.50000000000000065e153 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 0.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 66.5%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(-2 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot 90\right)}{\pi} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{+112}:\\ \;\;\;\;\frac{\tan^{-1} \left(\frac{360 \cdot \left(\left(b \cdot b\right) \cdot y-scale\right)}{\left(\left(\left(a + b\right) \cdot \pi\right) \cdot \left(b - a\right)\right) \cdot \left(x-scale \cdot angle\right)} \cdot -0.5\right)}{\pi} \cdot 180\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\pi}{\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot 180}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.6% accurate, 19.4× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \begin{array}{l} \mathbf{if}\;b\_m \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (if (<= b_m 1.4e+19)
   (*
    (atan
     (*
      (* (* (* PI angle) 0.005555555555555556) (* (/ y-scale x-scale) -2.0))
      -0.5))
    (* (/ 1.0 PI) 180.0))
   (*
    (/ (atan (* (* (/ y-scale (* (* PI x-scale) angle)) -2.0) 90.0)) PI)
    180.0)))

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 <= 1.4e+19) {
		tmp = atan(((((((double) M_PI) * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5)) * ((1.0 / ((double) M_PI)) * 180.0);
	} else {
		tmp = (atan((((y_45_scale / ((((double) M_PI) * x_45_scale) * angle)) * -2.0) * 90.0)) / ((double) M_PI)) * 180.0;
	}
	return tmp;
}

Java

b_m = Math.abs(b);
public static double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	double tmp;
	if (b_m <= 1.4e+19) {
		tmp = Math.atan(((((Math.PI * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5)) * ((1.0 / Math.PI) * 180.0);
	} else {
		tmp = (Math.atan((((y_45_scale / ((Math.PI * x_45_scale) * angle)) * -2.0) * 90.0)) / Math.PI) * 180.0;
	}
	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 <= 1.4e+19:
		tmp = math.atan(((((math.pi * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5)) * ((1.0 / math.pi) * 180.0)
	else:
		tmp = (math.atan((((y_45_scale / ((math.pi * x_45_scale) * angle)) * -2.0) * 90.0)) / math.pi) * 180.0
	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 <= 1.4e+19)
		tmp = Float64(atan(Float64(Float64(Float64(Float64(pi * angle) * 0.005555555555555556) * Float64(Float64(y_45_scale / x_45_scale) * -2.0)) * -0.5)) * Float64(Float64(1.0 / pi) * 180.0));
	else
		tmp = Float64(Float64(atan(Float64(Float64(Float64(y_45_scale / Float64(Float64(pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0);
	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 <= 1.4e+19)
		tmp = atan(((((pi * angle) * 0.005555555555555556) * ((y_45_scale / x_45_scale) * -2.0)) * -0.5)) * ((1.0 / pi) * 180.0);
	else
		tmp = (atan((((y_45_scale / ((pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0;
	end
	tmp_2 = tmp;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := If[LessEqual[b$95$m, 1.4e+19], N[(N[ArcTan[N[(N[(N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * N[(N[(y$45$scale / x$45$scale), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(1.0 / Pi), $MachinePrecision] * 180.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[(N[(N[(y$45$scale / N[(N[(Pi * x$45$scale), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * 90.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e19

   1. Initial program 15.9%

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

   3. Taylor expanded in x-scale around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 30.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.0%

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

   10. Applied rewrites 54.0%

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

   11. Taylor expanded in angle around 0

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

   13. Applied rewrites 55.3%

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

   if 1.4e19 < b

   1. Initial program 20.2%

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

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 18.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 52.7%

      \[\leadsto 180 \cdot \frac{\tan^{-1} \left(\left(-2 \cdot \frac{y-scale}{angle \cdot \left(x-scale \cdot \pi\right)}\right) \cdot 90\right)}{\pi} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1} \left(\left(\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\frac{y-scale}{x-scale} \cdot -2\right)\right) \cdot -0.5\right) \cdot \left(\frac{1}{\pi} \cdot 180\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.4% accurate, 21.5× speedup

Math

\[\begin{array}{l} b_m = \left|b\right| \\ \frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180 \end{array} \]

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (*
  (/ (atan (* (* (/ y-scale (* (* PI x-scale) angle)) -2.0) 90.0)) PI)
  180.0))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return (atan((((y_45_scale / ((((double) M_PI) * x_45_scale) * angle)) * -2.0) * 90.0)) / ((double) M_PI)) * 180.0;
}

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 (Math.atan((((y_45_scale / ((Math.PI * x_45_scale) * angle)) * -2.0) * 90.0)) / Math.PI) * 180.0;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return (math.atan((((y_45_scale / ((math.pi * x_45_scale) * angle)) * -2.0) * 90.0)) / math.pi) * 180.0

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(Float64(atan(Float64(Float64(Float64(y_45_scale / Float64(Float64(pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0)
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = (atan((((y_45_scale / ((pi * x_45_scale) * angle)) * -2.0) * 90.0)) / pi) * 180.0;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[ArcTan[N[(N[(N[(y$45$scale / N[(N[(Pi * x$45$scale), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * 90.0), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]

Derivation

1. Initial program 16.7%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 11.9%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 31.9%

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

9. Final simplification 31.9%

   \[\leadsto \frac{\tan^{-1} \left(\left(\frac{y-scale}{\left(\pi \cdot x-scale\right) \cdot angle} \cdot -2\right) \cdot 90\right)}{\pi} \cdot 180 \]
10. Add Preprocessing

Alternative 9: 14.7% accurate, 22.2× speedup

Math

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

FPCore

b_m = (fabs.f64 b)
(FPCore (a b_m angle x-scale y-scale)
 :precision binary64
 (* (/ (atan (* -180.0 (/ x-scale (* (* PI y-scale) angle)))) PI) 180.0))

C

b_m = fabs(b);
double code(double a, double b_m, double angle, double x_45_scale, double y_45_scale) {
	return (atan((-180.0 * (x_45_scale / ((((double) M_PI) * y_45_scale) * angle)))) / ((double) M_PI)) * 180.0;
}

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 (Math.atan((-180.0 * (x_45_scale / ((Math.PI * y_45_scale) * angle)))) / Math.PI) * 180.0;
}

Python

b_m = math.fabs(b)
def code(a, b_m, angle, x_45_scale, y_45_scale):
	return (math.atan((-180.0 * (x_45_scale / ((math.pi * y_45_scale) * angle)))) / math.pi) * 180.0

Julia

b_m = abs(b)
function code(a, b_m, angle, x_45_scale, y_45_scale)
	return Float64(Float64(atan(Float64(-180.0 * Float64(x_45_scale / Float64(Float64(pi * y_45_scale) * angle)))) / pi) * 180.0)
end

MATLAB

b_m = abs(b);
function tmp = code(a, b_m, angle, x_45_scale, y_45_scale)
	tmp = (atan((-180.0 * (x_45_scale / ((pi * y_45_scale) * angle)))) / pi) * 180.0;
end

Wolfram

b_m = N[Abs[b], $MachinePrecision]
code[a_, b$95$m_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[ArcTan[N[(-180.0 * N[(x$45$scale / N[(N[(Pi * y$45$scale), $MachinePrecision] * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] * 180.0), $MachinePrecision]

Derivation

1. Initial program 16.7%

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

3. Taylor expanded in a around inf

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

5. Applied rewrites 18.5%

   \[\leadsto 180 \cdot \frac{\tan^{-1} \color{blue}{\left(\frac{\left(x-scale \cdot y-scale\right) \cdot \left(\left(\frac{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale} - \sqrt{\mathsf{fma}\left({\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2} \cdot \frac{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}, 4, {\left(\frac{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale \cdot x-scale} - \frac{{\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{y-scale \cdot y-scale}\right)}^{2}\right)}\right) - \frac{{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}}{x-scale \cdot x-scale}\right)}{\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)} \cdot -0.5\right)}}{\pi} \]

7. Applied rewrites 11.4%

   \[\leadsto \color{blue}{\frac{180 \cdot \tan^{-1} \left(\left(-0.5 \cdot \left(x-scale \cdot y-scale\right)\right) \cdot \frac{\left(\frac{\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right)}{y-scale \cdot y-scale} - \sqrt{\mathsf{fma}\left(4 \cdot \frac{0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5}{\left(\left(y-scale \cdot y-scale\right) \cdot x-scale\right) \cdot x-scale}, \mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right), {\left(\frac{0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5}{x-scale \cdot x-scale} - \frac{\mathsf{fma}\left(\cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right), 0.5, 0.5\right)}{y-scale \cdot y-scale}\right)}^{2}\right)}\right) - \frac{0.5 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right) \cdot 0.5}{x-scale \cdot x-scale}}{\left(\sin 0 + \sin \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot 0.5}\right)}{\pi}} \]

8. Taylor expanded in angle around 0

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

10. Applied rewrites 13.6%

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

12. Applied rewrites 13.6%

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

13. Final simplification 13.6%

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

Reproduce

herbie shell --seed 2024240 
(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)))