ab-angle->ABCF A

Percentage Accurate: 79.9% → 79.8%

Time: 15.8s

Alternatives: 14

Speedup: 1.0×

• 36.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (angle / 180.0) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle / 180.0) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (angle / 180.0) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle / 180.0) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 79.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot angle\_m\right) \cdot \left(a \cdot angle\_m\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right), b \cdot b\right)}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 5000.0)
   (fma
    (* (* PI PI) 3.08641975308642e-5)
    (* (* a angle_m) (* a angle_m))
    (*
     b
     (*
      b
      (+ 0.5 (* 0.5 (cos (* 2.0 (* (* angle_m PI) 0.005555555555555556))))))))
   (/
    1.0
    (/
     1.0
     (fma
      a
      (*
       a
       (- 0.5 (* 0.5 (cos (* 2.0 (* PI (* angle_m 0.005555555555555556)))))))
      (* b b))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5000.0) {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5), ((a * angle_m) * (a * angle_m)), (b * (b * (0.5 + (0.5 * cos((2.0 * ((angle_m * ((double) M_PI)) * 0.005555555555555556))))))));
	} else {
		tmp = 1.0 / (1.0 / fma(a, (a * (0.5 - (0.5 * cos((2.0 * (((double) M_PI) * (angle_m * 0.005555555555555556))))))), (b * b)));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5000.0)
		tmp = fma(Float64(Float64(pi * pi) * 3.08641975308642e-5), Float64(Float64(a * angle_m) * Float64(a * angle_m)), Float64(b * Float64(b * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(angle_m * pi) * 0.005555555555555556))))))));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(a, Float64(a * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(angle_m * 0.005555555555555556))))))), Float64(b * b))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5000.0], N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(N[(a * angle$95$m), $MachinePrecision] * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(a * N[(a * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5e3

   1. Initial program 89.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. *-lowering-*.f64 85.4

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 85.4%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. Applied egg-rr 85.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot angle\right) \cdot \left(a \cdot angle\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \]

   if 5e3 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 59.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

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

   5. Simplified 59.4%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. clear-num N/A

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

      5. flip-+ N/A

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

   7. Applied egg-rr 59.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right), b \cdot b\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle\_m \cdot {\pi}^{0.8333333333333334}}} \cdot {\pi}^{0.16666666666666666}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (* (/ angle_m 180.0) PI))) 2.0)
  (pow
   (*
    b
    (cos
     (*
      (/ 1.0 (/ 180.0 (* angle_m (pow PI 0.8333333333333334))))
      (pow PI 0.16666666666666666))))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((angle_m / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((1.0 / (180.0 / (angle_m * pow(((double) M_PI), 0.8333333333333334)))) * pow(((double) M_PI), 0.16666666666666666)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin(((angle_m / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((1.0 / (180.0 / (angle_m * Math.pow(Math.PI, 0.8333333333333334)))) * Math.pow(Math.PI, 0.16666666666666666)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin(((angle_m / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((1.0 / (180.0 / (angle_m * math.pow(math.pi, 0.8333333333333334)))) * math.pow(math.pi, 0.16666666666666666)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(Float64(angle_m / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(1.0 / Float64(180.0 / Float64(angle_m * (pi ^ 0.8333333333333334)))) * (pi ^ 0.16666666666666666)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin(((angle_m / 180.0) * pi))) ^ 2.0) + ((b * cos(((1.0 / (180.0 / (angle_m * (pi ^ 0.8333333333333334)))) * (pi ^ 0.16666666666666666)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(1.0 / N[(180.0 / N[(angle$95$m * N[Power[Pi, 0.8333333333333334], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 82.3

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \color{blue}{\pi}}{180}\right)\right)}^{2} \]

4. Applied egg-rr 82.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. add-sqr-sqrt N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}^{2} \]

   3. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\frac{angle}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}^{2} \]

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}{\frac{180}{angle}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   7. un-div-inv N/A

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

   8. clear-num N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{angle}{180}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}^{2} \]

   10. clear-num N/A

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

   11. un-div-inv N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{180}{angle}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   12. *-lft-identity N/A

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

   13. associate-*l/ N/A

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

   14. clear-num N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\frac{angle}{180}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   15. *-lowering-*.f64 N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\frac{angle}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   16. div-inv N/A

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

   17. metadata-eval N/A

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

   18. *-lowering-*.f64 N/A

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

   19. sqrt-lowering-sqrt.f64 N/A

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

   20. PI-lowering-PI.f64 N/A

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

   21. sqrt-lowering-sqrt.f64 N/A

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

   22. PI-lowering-PI.f64 82.2

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\color{blue}{\pi}}\right)\right)}^{2} \]

6. Applied egg-rr 82.2%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{\pi}\right) \cdot \sqrt{\pi}\right)}\right)}^{2} \]
7. Step-by-step derivation

   1. metadata-eval N/A

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

   2. div-inv N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\left(\color{blue}{\frac{angle}{180}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   3. *-commutative N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \frac{angle}{180}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   4. clear-num N/A

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

   5. un-div-inv N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)}}{\frac{180}{angle}}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} \]

   6. associate-*l/ N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\frac{180}{angle}}\right)}\right)}^{2} \]

   7. remove-double-neg N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{180}{angle}\right)\right)\right)}}\right)\right)}^{2} \]

   8. neg-mul-1 N/A

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

   9. times-frac N/A

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

   10. *-lowering-*.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. sqrt-lowering-sqrt.f64 N/A

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

   13. PI-lowering-PI.f64 N/A

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

   14. /-lowering-/.f64 N/A

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

   15. sqrt-lowering-sqrt.f64 N/A

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

   16. PI-lowering-PI.f64 N/A

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

   17. distribute-neg-frac N/A

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

   18. /-lowering-/.f64 N/A

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

   19. metadata-eval 82.3

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

8. Applied egg-rr 82.3%

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

   1. associate-/r/ N/A

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

   2. associate-*l/ N/A

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

   3. frac-times N/A

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

   4. associate-*l* N/A

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

   5. add-sqr-sqrt N/A

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

   6. add-cbrt-cube N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\color{blue}{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \cdot angle}{-1 \cdot -180}\right)\right)}^{2} \]

   7. add-sqr-sqrt N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}} \cdot angle}{-1 \cdot -180}\right)\right)}^{2} \]

   8. associate-*r* N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\sqrt[3]{\color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}}} \cdot angle}{-1 \cdot -180}\right)\right)}^{2} \]

   9. cbrt-unprod N/A

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\color{blue}{\left(\sqrt[3]{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)} \cdot angle}{-1 \cdot -180}\right)\right)}^{2} \]

   10. unpow1/3 N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\left(\color{blue}{{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot angle}{-1 \cdot -180}\right)\right)}^{2} \]

   11. *-commutative N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\color{blue}{angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}}{-1 \cdot -180}\right)\right)}^{2} \]

   12. metadata-eval N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}{\color{blue}{180}}\right)\right)}^{2} \]

   13. clear-num N/A

       \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot \left({\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}^{\frac{1}{3}} \cdot \sqrt[3]{\sqrt{\mathsf{PI}\left(\right)}}\right)}}\right)}\right)}^{2} \]

10. Applied egg-rr 82.3%

    \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{1}{\frac{180}{angle \cdot {\pi}^{0.8333333333333334}}} \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} \]
11. Add Preprocessing

Alternative 3: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\pi \cdot \frac{1}{\frac{180}{angle\_m}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle\_m \cdot \pi}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (* PI (/ 1.0 (/ 180.0 angle_m))))) 2.0)
  (pow (* b (cos (/ (* angle_m PI) 180.0))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin((((double) M_PI) * (1.0 / (180.0 / angle_m))))), 2.0) + pow((b * cos(((angle_m * ((double) M_PI)) / 180.0))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin((Math.PI * (1.0 / (180.0 / angle_m))))), 2.0) + Math.pow((b * Math.cos(((angle_m * Math.PI) / 180.0))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin((math.pi * (1.0 / (180.0 / angle_m))))), 2.0) + math.pow((b * math.cos(((angle_m * math.pi) / 180.0))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(pi * Float64(1.0 / Float64(180.0 / angle_m))))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle_m * pi) / 180.0))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin((pi * (1.0 / (180.0 / angle_m))))) ^ 2.0) + ((b * cos(((angle_m * pi) / 180.0))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(Pi * N[(1.0 / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 82.3

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \color{blue}{\pi}}{180}\right)\right)}^{2} \]

4. Applied egg-rr 82.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 82.3

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

6. Applied egg-rr 82.3%

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

7. Final simplification 82.3%

   \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \frac{1}{\frac{180}{angle}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 4: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \cos \left(\frac{angle\_m \cdot \pi}{180}\right)\right)}^{2} + {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (cos (/ (* angle_m PI) 180.0))) 2.0)
  (pow (* a (sin (/ PI (/ 180.0 angle_m)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * cos(((angle_m * ((double) M_PI)) / 180.0))), 2.0) + pow((a * sin((((double) M_PI) / (180.0 / angle_m)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((b * Math.cos(((angle_m * Math.PI) / 180.0))), 2.0) + Math.pow((a * Math.sin((Math.PI / (180.0 / angle_m)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((b * math.cos(((angle_m * math.pi) / 180.0))), 2.0) + math.pow((a * math.sin((math.pi / (180.0 / angle_m)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * cos(Float64(Float64(angle_m * pi) / 180.0))) ^ 2.0) + (Float64(a * sin(Float64(pi / Float64(180.0 / angle_m)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * cos(((angle_m * pi) / 180.0))) ^ 2.0) + ((a * sin((pi / (180.0 / angle_m)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 82.3

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \color{blue}{\pi}}{180}\right)\right)}^{2} \]

4. Applied egg-rr 82.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

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

   5. PI-lowering-PI.f64 N/A

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

   6. /-lowering-/.f64 82.3

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\color{blue}{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} \]

6. Applied egg-rr 82.3%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} \]

7. Final simplification 82.3%

   \[\leadsto {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 5: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle\_m \cdot \pi}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (sin (* (/ angle_m 180.0) PI))) 2.0)
  (pow (* b (cos (/ (* angle_m PI) 180.0))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((angle_m / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle_m * ((double) M_PI)) / 180.0))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin(((angle_m / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle_m * Math.PI) / 180.0))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin(((angle_m / 180.0) * math.pi))), 2.0) + math.pow((b * math.cos(((angle_m * math.pi) / 180.0))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(Float64(angle_m / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle_m * pi) / 180.0))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin(((angle_m / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle_m * pi) / 180.0))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 82.3

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \color{blue}{\pi}}{180}\right)\right)}^{2} \]

4. Applied egg-rr 82.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \frac{angle\_m}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (/ angle_m 180.0) PI)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (angle_m / 180.0) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (angle_m / 180.0) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(angle_m / 180.0) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = (angle_m / 180.0) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 82.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\pi \cdot \frac{1}{\frac{180}{angle\_m}}\right)\right)}^{2} + {b}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (* PI (/ 1.0 (/ 180.0 angle_m))))) 2.0) (pow b 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin((((double) M_PI) * (1.0 / (180.0 / angle_m))))), 2.0) + pow(b, 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin((Math.PI * (1.0 / (180.0 / angle_m))))), 2.0) + Math.pow(b, 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin((math.pi * (1.0 / (180.0 / angle_m))))), 2.0) + math.pow(b, 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(pi * Float64(1.0 / Float64(180.0 / angle_m))))) ^ 2.0) + (b ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin((pi * (1.0 / (180.0 / angle_m))))) ^ 2.0) + (b ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(Pi * N[(1.0 / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. PI-lowering-PI.f64 82.3

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \color{blue}{\pi}}{180}\right)\right)}^{2} \]

4. Applied egg-rr 82.3%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. /-lowering-/.f64 82.3

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

6. Applied egg-rr 82.3%

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

7. Taylor expanded in angle around 0

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

9. Simplified 82.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle}} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]

10. Final simplification 82.0%

    \[\leadsto {\left(a \cdot \sin \left(\pi \cdot \frac{1}{\frac{180}{angle}}\right)\right)}^{2} + {b}^{2} \]
11. Add Preprocessing

Alternative 8: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle\_m}}\right)\right)}^{2} + {b}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (/ PI (/ 180.0 angle_m)))) 2.0) (pow b 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin((((double) M_PI) / (180.0 / angle_m)))), 2.0) + pow(b, 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin((Math.PI / (180.0 / angle_m)))), 2.0) + Math.pow(b, 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin((math.pi / (180.0 / angle_m)))), 2.0) + math.pow(b, 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(pi / Float64(180.0 / angle_m)))) ^ 2.0) + (b ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin((pi / (180.0 / angle_m)))) ^ 2.0) + (b ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 82.2

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

4. Applied egg-rr 82.2%

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

5. Taylor expanded in angle around 0

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

7. Simplified 81.9%

   \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
8. Step-by-step derivation

   1. associate-/r* N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. PI-lowering-PI.f64 N/A

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

   5. /-lowering-/.f64 82.0

      \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\color{blue}{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

9. Applied egg-rr 82.0%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\frac{\pi}{\frac{180}{angle}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

10. Final simplification 82.0%

    \[\leadsto {\left(a \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\right)}^{2} + {b}^{2} \]
11. Add Preprocessing

Alternative 9: 79.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\right)}^{2} + b \cdot b \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* a (sin (* (/ angle_m 180.0) PI))) 2.0) (* b b)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * sin(((angle_m / 180.0) * ((double) M_PI)))), 2.0) + (b * b);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.sin(((angle_m / 180.0) * Math.PI))), 2.0) + (b * b);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.sin(((angle_m / 180.0) * math.pi))), 2.0) + (b * b)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * sin(Float64(Float64(angle_m / 180.0) * pi))) ^ 2.0) + Float64(b * b))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin(((angle_m / 180.0) * pi))) ^ 2.0) + (b * b);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

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

5. Simplified 82.0%

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. *-rgt-identity 82.0

      \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + b \cdot \color{blue}{b} \]

7. Applied egg-rr 82.0%

   \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + \color{blue}{b \cdot b} \]
8. Add Preprocessing

Alternative 10: 66.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 2.95 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right), a \cdot a, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot angle\_m\right) \cdot \left(angle\_m \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 2.95e+122)
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (* PI (* angle_m 0.005555555555555556))))))
    (* a a)
    (* b b))
   (* (* a angle_m) (* angle_m (* a (* PI (* PI 3.08641975308642e-5)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 2.95e+122) {
		tmp = fma((0.5 - (0.5 * cos((2.0 * (((double) M_PI) * (angle_m * 0.005555555555555556)))))), (a * a), (b * b));
	} else {
		tmp = (a * angle_m) * (angle_m * (a * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 2.95e+122)
		tmp = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(angle_m * 0.005555555555555556)))))), Float64(a * a), Float64(b * b));
	else
		tmp = Float64(Float64(a * angle_m) * Float64(angle_m * Float64(a * Float64(pi * Float64(pi * 3.08641975308642e-5)))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 2.95e+122], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(a * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.95000000000000016e122

   1. Initial program 79.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

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

   5. Simplified 79.6%

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

      1. *-commutative N/A

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

      2. unpow-prod-down N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2}, {a}^{2}, {\left(b \cdot 1\right)}^{2}\right)} \]

   7. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right), a \cdot a, b \cdot b\right)} \]

   if 2.95000000000000016e122 < a

   1. Initial program 96.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. *-lowering-*.f64 96.9

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 96.9%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      13. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \]

      16. PI-lowering-PI.f64 56.2

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \]

   8. Simplified 56.2%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \color{blue}{{a}^{2}} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right)\right)} \cdot {a}^{2} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      7. associate-*l* N/A

         \[\leadsto angle \cdot \left(\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \cdot {a}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto angle \cdot \left(\color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)} \cdot {a}^{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto angle \cdot \left(\color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)} \cdot {a}^{2}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right) \cdot {a}^{2}\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      14. pow2 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

      15. *-lowering-*.f64 65.4

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   10. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{angle \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(a \cdot a\right)\right) \cdot angle} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot a\right)} \cdot angle \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot \left(a \cdot angle\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot \left(a \cdot angle\right)} \]

       5. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)} \cdot a\right) \cdot \left(a \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right)\right)} \cdot \left(a \cdot angle\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right)\right)} \cdot \left(a \cdot angle\right) \]

       8. *-commutative N/A

          \[\leadsto \left(angle \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)}\right) \cdot \left(a \cdot angle\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)}\right) \cdot \left(a \cdot angle\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right)\right) \cdot \left(a \cdot angle\right) \]

       11. PI-lowering-PI.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot \left(a \cdot angle\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right)\right) \cdot \left(a \cdot angle\right) \]

       13. PI-lowering-PI.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot \left(a \cdot angle\right) \]

       14. *-commutative N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot \color{blue}{\left(angle \cdot a\right)} \]

       15. *-lowering-*.f64 78.5

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right) \cdot \color{blue}{\left(angle \cdot a\right)} \]

   12. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right) \cdot \left(angle \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.95 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right), a \cdot a, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot angle\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.1 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), a \cdot a, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot angle\_m\right) \cdot \left(angle\_m \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.1e+122)
   (fma
    (- 0.5 (* 0.5 (cos (* 2.0 (* angle_m (* PI 0.005555555555555556))))))
    (* a a)
    (* b b))
   (* (* a angle_m) (* angle_m (* a (* PI (* PI 3.08641975308642e-5)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.1e+122) {
		tmp = fma((0.5 - (0.5 * cos((2.0 * (angle_m * (((double) M_PI) * 0.005555555555555556)))))), (a * a), (b * b));
	} else {
		tmp = (a * angle_m) * (angle_m * (a * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3.1e+122)
		tmp = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(angle_m * Float64(pi * 0.005555555555555556)))))), Float64(a * a), Float64(b * b));
	else
		tmp = Float64(Float64(a * angle_m) * Float64(angle_m * Float64(a * Float64(pi * Float64(pi * 3.08641975308642e-5)))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3.1e+122], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(a * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.09999999999999999e122

   1. Initial program 79.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ N/A

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

      2. clear-num N/A

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

      3. /-lowering-/.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. PI-lowering-PI.f64 79.8

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

   4. Applied egg-rr 79.8%

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

   5. Taylor expanded in angle around 0

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

   7. Simplified 79.5%

      \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-/r/ N/A

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

      4. clear-num N/A

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

      5. unpow-prod-down N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({\sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}^{2}, {a}^{2}, {\left(b \cdot 1\right)}^{2}\right)} \]

   9. Applied egg-rr 68.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), a \cdot a, b \cdot b\right)} \]

   if 3.09999999999999999e122 < a

   1. Initial program 96.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. *-lowering-*.f64 96.9

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 96.9%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      13. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \]

      16. PI-lowering-PI.f64 56.2

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \]

   8. Simplified 56.2%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \color{blue}{{a}^{2}} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right)\right)} \cdot {a}^{2} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      7. associate-*l* N/A

         \[\leadsto angle \cdot \left(\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \cdot {a}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto angle \cdot \left(\color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)} \cdot {a}^{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto angle \cdot \left(\color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)} \cdot {a}^{2}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right) \cdot {a}^{2}\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      14. pow2 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

      15. *-lowering-*.f64 65.4

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   10. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{angle \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(a \cdot a\right)\right) \cdot angle} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot a\right)} \cdot angle \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot \left(a \cdot angle\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot \left(a \cdot angle\right)} \]

       5. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)} \cdot a\right) \cdot \left(a \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right)\right)} \cdot \left(a \cdot angle\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right)\right)} \cdot \left(a \cdot angle\right) \]

       8. *-commutative N/A

          \[\leadsto \left(angle \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)}\right) \cdot \left(a \cdot angle\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)}\right) \cdot \left(a \cdot angle\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right)\right) \cdot \left(a \cdot angle\right) \]

       11. PI-lowering-PI.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot \left(a \cdot angle\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right)\right) \cdot \left(a \cdot angle\right) \]

       13. PI-lowering-PI.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot \left(a \cdot angle\right) \]

       14. *-commutative N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot \color{blue}{\left(angle \cdot a\right)} \]

       15. *-lowering-*.f64 78.5

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right) \cdot \color{blue}{\left(angle \cdot a\right)} \]

   12. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right) \cdot \left(angle \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.1 \cdot 10^{+122}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right), a \cdot a, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot angle\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.1 \cdot 10^{+122}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot angle\_m\right) \cdot \left(angle\_m \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.1e+122)
   (* b b)
   (* (* a angle_m) (* angle_m (* a (* PI (* PI 3.08641975308642e-5)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.1e+122) {
		tmp = b * b;
	} else {
		tmp = (a * angle_m) * (angle_m * (a * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.1e+122) {
		tmp = b * b;
	} else {
		tmp = (a * angle_m) * (angle_m * (a * (Math.PI * (Math.PI * 3.08641975308642e-5))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 3.1e+122:
		tmp = b * b
	else:
		tmp = (a * angle_m) * (angle_m * (a * (math.pi * (math.pi * 3.08641975308642e-5))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3.1e+122)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(a * angle_m) * Float64(angle_m * Float64(a * Float64(pi * Float64(pi * 3.08641975308642e-5)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 3.1e+122)
		tmp = b * b;
	else
		tmp = (a * angle_m) * (angle_m * (a * (pi * (pi * 3.08641975308642e-5))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3.1e+122], N[(b * b), $MachinePrecision], N[(N[(a * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(a * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.09999999999999999e122

   1. Initial program 79.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. *-lowering-*.f64 63.4

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Simplified 63.4%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 3.09999999999999999e122 < a

   1. Initial program 96.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. *-lowering-*.f64 96.9

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 96.9%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      13. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \]

      16. PI-lowering-PI.f64 56.2

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \]

   8. Simplified 56.2%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \color{blue}{{a}^{2}} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right)\right)} \cdot {a}^{2} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      7. associate-*l* N/A

         \[\leadsto angle \cdot \left(\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \cdot {a}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto angle \cdot \left(\color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)} \cdot {a}^{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto angle \cdot \left(\color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)} \cdot {a}^{2}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right) \cdot {a}^{2}\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      14. pow2 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

      15. *-lowering-*.f64 65.4

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   10. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{angle \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(a \cdot a\right)\right) \cdot angle} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot a\right)} \cdot angle \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot \left(a \cdot angle\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot \left(a \cdot angle\right)} \]

       5. associate-*l* N/A

          \[\leadsto \left(\color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)} \cdot a\right) \cdot \left(a \cdot angle\right) \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right)\right)} \cdot \left(a \cdot angle\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right)\right)} \cdot \left(a \cdot angle\right) \]

       8. *-commutative N/A

          \[\leadsto \left(angle \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)}\right) \cdot \left(a \cdot angle\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)}\right) \cdot \left(a \cdot angle\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right)\right) \cdot \left(a \cdot angle\right) \]

       11. PI-lowering-PI.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot \left(a \cdot angle\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right)\right) \cdot \left(a \cdot angle\right) \]

       13. PI-lowering-PI.f64 N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot \left(a \cdot angle\right) \]

       14. *-commutative N/A

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot \color{blue}{\left(angle \cdot a\right)} \]

       15. *-lowering-*.f64 78.5

           \[\leadsto \left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right) \cdot \color{blue}{\left(angle \cdot a\right)} \]

   12. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right) \cdot \left(angle \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.1 \cdot 10^{+122}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot angle\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+122}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(a \cdot \left(angle\_m \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3e+122)
   (* b b)
   (* angle_m (* a (* angle_m (* a (* PI (* PI 3.08641975308642e-5))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3e+122) {
		tmp = b * b;
	} else {
		tmp = angle_m * (a * (angle_m * (a * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3e+122) {
		tmp = b * b;
	} else {
		tmp = angle_m * (a * (angle_m * (a * (Math.PI * (Math.PI * 3.08641975308642e-5)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 3e+122:
		tmp = b * b
	else:
		tmp = angle_m * (a * (angle_m * (a * (math.pi * (math.pi * 3.08641975308642e-5)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3e+122)
		tmp = Float64(b * b);
	else
		tmp = Float64(angle_m * Float64(a * Float64(angle_m * Float64(a * Float64(pi * Float64(pi * 3.08641975308642e-5))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 3e+122)
		tmp = b * b;
	else
		tmp = angle_m * (a * (angle_m * (a * (pi * (pi * 3.08641975308642e-5)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3e+122], N[(b * b), $MachinePrecision], N[(angle$95$m * N[(a * N[(angle$95$m * N[(a * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.99999999999999986e122

   1. Initial program 79.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. *-lowering-*.f64 63.4

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Simplified 63.4%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 2.99999999999999986e122 < a

   1. Initial program 96.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. PI-lowering-PI.f64 N/A

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

      10. *-lowering-*.f64 96.9

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 96.9%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      8. associate-*l* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)} \]

      10. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right) \]

      13. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right)\right) \]

      15. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \]

      16. PI-lowering-PI.f64 56.2

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \]

   8. Simplified 56.2%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \color{blue}{{a}^{2}} \cdot \left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right)\right)} \cdot {a}^{2} \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right)} \]

      7. associate-*l* N/A

         \[\leadsto angle \cdot \left(\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right) \cdot {a}^{2}\right) \]

      8. associate-*r* N/A

         \[\leadsto angle \cdot \left(\color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)} \cdot {a}^{2}\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto angle \cdot \left(\color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)} \cdot {a}^{2}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right) \cdot {a}^{2}\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right)\right) \cdot {a}^{2}\right) \]

      14. pow2 N/A

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

      15. *-lowering-*.f64 65.4

          \[\leadsto angle \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) \]

   10. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{angle \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(a \cdot a\right)\right)} \]
   11. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot a\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right) \cdot a\right)} \]

       3. associate-*l* N/A

          \[\leadsto angle \cdot \left(\left(\color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)} \cdot a\right) \cdot a\right) \]

       4. associate-*l* N/A

          \[\leadsto angle \cdot \left(\color{blue}{\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right)\right)} \cdot a\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\color{blue}{\left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right) \cdot a\right)\right)} \cdot a\right) \]

       6. *-commutative N/A

          \[\leadsto angle \cdot \left(\left(angle \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)}\right) \cdot a\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(angle \cdot \color{blue}{\left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)}\right) \cdot a\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto angle \cdot \left(\left(angle \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)}\right)\right) \cdot a\right) \]

       9. PI-lowering-PI.f64 N/A

          \[\leadsto angle \cdot \left(\left(angle \cdot \left(a \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)\right)\right)\right) \cdot a\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto angle \cdot \left(\left(angle \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}\right)\right)\right) \cdot a\right) \]

       11. PI-lowering-PI.f64 70.9

           \[\leadsto angle \cdot \left(\left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\color{blue}{\pi} \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right) \cdot a\right) \]

   12. Applied egg-rr 70.9%

       \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right) \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3 \cdot 10^{+122}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.1% accurate, 74.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ b \cdot b \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* b b))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return b * b;
}

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = b * b
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return b * b;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return b * b

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(b * b)
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = b * b;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 82.3%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{b}^{2}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{b \cdot b} \]

   2. *-lowering-*.f64 62.0

      \[\leadsto \color{blue}{b \cdot b} \]

5. Simplified 62.0%

   \[\leadsto \color{blue}{b \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))