ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.4%

Time: 12.6s

Alternatives: 12

Speedup: N/A×

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 1.8× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 5e-10)
   (+ (pow a 2.0) (pow (* 0.005555555555555556 (* angle_m (* PI b))) 2.0))
   (+
    (pow a 2.0)
    (*
     b
     (*
      b
      (-
       0.5
       (* 0.5 (cos (* 2.0 (* 0.005555555555555556 (* angle_m PI)))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-10) {
		tmp = pow(a, 2.0) + pow((0.005555555555555556 * (angle_m * (((double) M_PI) * b))), 2.0);
	} else {
		tmp = pow(a, 2.0) + (b * (b * (0.5 - (0.5 * cos((2.0 * (0.005555555555555556 * (angle_m * ((double) M_PI)))))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-10) {
		tmp = Math.pow(a, 2.0) + Math.pow((0.005555555555555556 * (angle_m * (Math.PI * b))), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (b * (b * (0.5 - (0.5 * Math.cos((2.0 * (0.005555555555555556 * (angle_m * Math.PI))))))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e-10:
		tmp = math.pow(a, 2.0) + math.pow((0.005555555555555556 * (angle_m * (math.pi * b))), 2.0)
	else:
		tmp = math.pow(a, 2.0) + (b * (b * (0.5 - (0.5 * math.cos((2.0 * (0.005555555555555556 * (angle_m * math.pi))))))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-10)
		tmp = Float64((a ^ 2.0) + (Float64(0.005555555555555556 * Float64(angle_m * Float64(pi * b))) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + Float64(b * Float64(b * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.005555555555555556 * Float64(angle_m * pi)))))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-10)
		tmp = (a ^ 2.0) + ((0.005555555555555556 * (angle_m * (pi * b))) ^ 2.0);
	else
		tmp = (a ^ 2.0) + (b * (b * (0.5 - (0.5 * cos((2.0 * (0.005555555555555556 * (angle_m * pi))))))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-10], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(b * N[(b * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000031e-10

   1. Initial program 86.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 87.0%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 87.1

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right) \cdot b\right)}^{2} \]

   8. Simplified 87.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}}^{2} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 87.1

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \left(\color{blue}{\left(\pi \cdot 0.005555555555555556\right)} \cdot angle\right) \cdot b\right)}^{2} \]

   10. Applied egg-rr 87.1%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)} \cdot b\right)}^{2} \]

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 84.5

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot b\right)\right)\right)}^{2} \]

   13. Simplified 84.5%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)}}^{2} \]

   if 5.00000000000000031e-10 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 58.8%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 59.0%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 59.4

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right) \cdot b\right)}^{2} \]

   8. Simplified 59.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}}^{2} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   10. Applied egg-rr 59.4%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot b\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {a}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle\_m \cdot \pi\right)\right) \cdot b\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow a 2.0) (pow (* (sin (* 0.005555555555555556 (* angle_m PI))) b) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow(a, 2.0) + pow((sin((0.005555555555555556 * (angle_m * ((double) M_PI)))) * b), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow(a, 2.0) + Math.pow((Math.sin((0.005555555555555556 * (angle_m * Math.PI))) * b), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow(a, 2.0) + math.pow((math.sin((0.005555555555555556 * (angle_m * math.pi))) * b), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((a ^ 2.0) + (Float64(sin(Float64(0.005555555555555556 * Float64(angle_m * pi))) * b) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a ^ 2.0) + ((sin((0.005555555555555556 * (angle_m * pi))) * b) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(N[Sin[N[(0.005555555555555556 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in angle around 0

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

5. Simplified 79.0%

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

6. Taylor expanded in b around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-sin.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 79.2

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right) \cdot b\right)}^{2} \]

8. Simplified 79.2%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}}^{2} \]

9. Final simplification 79.2%

   \[\leadsto {a}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {a}^{2} + {\left(b \cdot \sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow a 2.0) (pow (* b (sin (* angle_m (* 0.005555555555555556 PI)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow(a, 2.0) + pow((b * sin((angle_m * (0.005555555555555556 * ((double) M_PI))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow(a, 2.0) + Math.pow((b * Math.sin((angle_m * (0.005555555555555556 * Math.PI)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow(a, 2.0) + math.pow((b * math.sin((angle_m * (0.005555555555555556 * math.pi)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((a ^ 2.0) + (Float64(b * sin(Float64(angle_m * Float64(0.005555555555555556 * pi)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a ^ 2.0) + ((b * sin((angle_m * (0.005555555555555556 * pi)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in angle around 0

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

5. Simplified 79.0%

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

   1. lift-PI.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 79.1

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\color{blue}{\left(\pi \cdot 0.005555555555555556\right)} \cdot angle\right)\right)}^{2} \]

7. Applied egg-rr 79.1%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)}\right)}^{2} \]

8. Final simplification 79.1%

   \[\leadsto {a}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 4: 79.5% accurate, 1.8× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 5e-10)
   (+ (pow a 2.0) (pow (* 0.005555555555555556 (* angle_m (* PI b))) 2.0))
   (+
    (pow a 2.0)
    (*
     b
     (*
      b
      (-
       0.5
       (* 0.5 (cos (* 2.0 (* PI (* 0.005555555555555556 angle_m)))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-10) {
		tmp = pow(a, 2.0) + pow((0.005555555555555556 * (angle_m * (((double) M_PI) * b))), 2.0);
	} else {
		tmp = pow(a, 2.0) + (b * (b * (0.5 - (0.5 * cos((2.0 * (((double) M_PI) * (0.005555555555555556 * angle_m))))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 5e-10) {
		tmp = Math.pow(a, 2.0) + Math.pow((0.005555555555555556 * (angle_m * (Math.PI * b))), 2.0);
	} else {
		tmp = Math.pow(a, 2.0) + (b * (b * (0.5 - (0.5 * Math.cos((2.0 * (Math.PI * (0.005555555555555556 * angle_m))))))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if (angle_m / 180.0) <= 5e-10:
		tmp = math.pow(a, 2.0) + math.pow((0.005555555555555556 * (angle_m * (math.pi * b))), 2.0)
	else:
		tmp = math.pow(a, 2.0) + (b * (b * (0.5 - (0.5 * math.cos((2.0 * (math.pi * (0.005555555555555556 * angle_m))))))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-10)
		tmp = Float64((a ^ 2.0) + (Float64(0.005555555555555556 * Float64(angle_m * Float64(pi * b))) ^ 2.0));
	else
		tmp = Float64((a ^ 2.0) + Float64(b * Float64(b * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(0.005555555555555556 * angle_m)))))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if ((angle_m / 180.0) <= 5e-10)
		tmp = (a ^ 2.0) + ((0.005555555555555556 * (angle_m * (pi * b))) ^ 2.0);
	else
		tmp = (a ^ 2.0) + (b * (b * (0.5 - (0.5 * cos((2.0 * (pi * (0.005555555555555556 * angle_m))))))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-10], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(b * N[(b * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000031e-10

   1. Initial program 86.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 87.0%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 87.1

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right) \cdot b\right)}^{2} \]

   8. Simplified 87.1%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}}^{2} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 87.1

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \left(\color{blue}{\left(\pi \cdot 0.005555555555555556\right)} \cdot angle\right) \cdot b\right)}^{2} \]

   10. Applied egg-rr 87.1%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)} \cdot b\right)}^{2} \]

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 84.5

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot b\right)\right)\right)}^{2} \]

   13. Simplified 84.5%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)}}^{2} \]

   if 5.00000000000000031e-10 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 58.8%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 59.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. unpow1 N/A

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

      7. metadata-eval N/A

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

      8. pow2 N/A

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

      9. sqr-pow N/A

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

      10. lift-*.f64 N/A

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

      11. unpow-prod-down N/A

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

      12. pow2 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

   7. Applied egg-rr 59.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-10}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + b \cdot \left(b \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-57}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(angle\_m \cdot \left(\pi \cdot b\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.16e-57)
   (* a a)
   (+ (pow a 2.0) (pow (* 0.005555555555555556 (* angle_m (* PI b))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.16e-57) {
		tmp = a * a;
	} else {
		tmp = pow(a, 2.0) + pow((0.005555555555555556 * (angle_m * (((double) M_PI) * b))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.16e-57) {
		tmp = a * a;
	} else {
		tmp = Math.pow(a, 2.0) + Math.pow((0.005555555555555556 * (angle_m * (Math.PI * b))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 1.16e-57:
		tmp = a * a
	else:
		tmp = math.pow(a, 2.0) + math.pow((0.005555555555555556 * (angle_m * (math.pi * b))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.16e-57)
		tmp = Float64(a * a);
	else
		tmp = Float64((a ^ 2.0) + (Float64(0.005555555555555556 * Float64(angle_m * Float64(pi * b))) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.16e-57)
		tmp = a * a;
	else
		tmp = (a ^ 2.0) + ((0.005555555555555556 * (angle_m * (pi * b))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.16e-57], N[(a * a), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15999999999999996e-57

   1. Initial program 78.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 78.2%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 63.7

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 63.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]

   9. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 61.6

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

   11. Simplified 61.6%

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

   if 1.15999999999999996e-57 < b

   1. Initial program 81.4%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 81.3%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 81.3

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right) \cdot b\right)}^{2} \]

   8. Simplified 81.3%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot b\right)}}^{2} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 81.4

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \left(\color{blue}{\left(\pi \cdot 0.005555555555555556\right)} \cdot angle\right) \cdot b\right)}^{2} \]

   10. Applied egg-rr 81.4%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(\sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot angle\right)} \cdot b\right)}^{2} \]

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 78.9

          \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\color{blue}{\pi} \cdot b\right)\right)\right)}^{2} \]

   13. Simplified 78.9%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + {\color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)}}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-57}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 9.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot b\right) \cdot \left(angle\_m \cdot -0.005555555555555556\right)\\ \mathbf{if}\;b \leq 1.16 \cdot 10^{-57}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI b) (* angle_m -0.005555555555555556))))
   (if (<= b 1.16e-57) (* a a) (fma t_0 t_0 (* a a)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * b) * (angle_m * -0.005555555555555556);
	double tmp;
	if (b <= 1.16e-57) {
		tmp = a * a;
	} else {
		tmp = fma(t_0, t_0, (a * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(pi * b) * Float64(angle_m * -0.005555555555555556))
	tmp = 0.0
	if (b <= 1.16e-57)
		tmp = Float64(a * a);
	else
		tmp = fma(t_0, t_0, Float64(a * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * b), $MachinePrecision] * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 1.16e-57], N[(a * a), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15999999999999996e-57

   1. Initial program 78.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 78.2%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 63.7

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 63.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]

   9. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 61.6

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

   11. Simplified 61.6%

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

   if 1.15999999999999996e-57 < b

   1. Initial program 81.4%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 81.3%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 63.0

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 63.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   10. Applied egg-rr 78.8%

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

Alternative 7: 71.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \left(angle\_m \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\pi \cdot \pi\right), 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(angle\_m \cdot \pi\right)\right), b \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.4e+154)
   (fma
    (* (* angle_m (* angle_m (* b b))) (* PI PI))
    3.08641975308642e-5
    (* a a))
   (fma
    (* b (* (* angle_m PI) (* angle_m PI)))
    (* b 3.08641975308642e-5)
    (* a a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.4e+154) {
		tmp = fma(((angle_m * (angle_m * (b * b))) * (((double) M_PI) * ((double) M_PI))), 3.08641975308642e-5, (a * a));
	} else {
		tmp = fma((b * ((angle_m * ((double) M_PI)) * (angle_m * ((double) M_PI)))), (b * 3.08641975308642e-5), (a * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.4e+154)
		tmp = fma(Float64(Float64(angle_m * Float64(angle_m * Float64(b * b))) * Float64(pi * pi)), 3.08641975308642e-5, Float64(a * a));
	else
		tmp = fma(Float64(b * Float64(Float64(angle_m * pi) * Float64(angle_m * pi))), Float64(b * 3.08641975308642e-5), Float64(a * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.4e+154], N[(N[(N[(angle$95$m * N[(angle$95$m * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5 + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * 3.08641975308642e-5), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e154

   1. Initial program 76.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 76.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 63.4

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 63.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   10. Applied egg-rr 69.3%

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

   if 1.4e154 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 64.8

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 64.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   10. Applied egg-rr 84.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\pi \cdot \pi\right), 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right), b \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \left(angle\_m \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\pi \cdot \pi\right), 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle\_m \cdot angle\_m\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.4e+154)
   (fma
    (* (* angle_m (* angle_m (* b b))) (* PI PI))
    3.08641975308642e-5
    (* a a))
   (fma
    b
    (* (* b (* (* PI PI) 3.08641975308642e-5)) (* angle_m angle_m))
    (* a a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.4e+154) {
		tmp = fma(((angle_m * (angle_m * (b * b))) * (((double) M_PI) * ((double) M_PI))), 3.08641975308642e-5, (a * a));
	} else {
		tmp = fma(b, ((b * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5)) * (angle_m * angle_m)), (a * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.4e+154)
		tmp = fma(Float64(Float64(angle_m * Float64(angle_m * Float64(b * b))) * Float64(pi * pi)), 3.08641975308642e-5, Float64(a * a));
	else
		tmp = fma(b, Float64(Float64(b * Float64(Float64(pi * pi) * 3.08641975308642e-5)) * Float64(angle_m * angle_m)), Float64(a * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.4e+154], N[(N[(N[(angle$95$m * N[(angle$95$m * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5 + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.4e154

   1. Initial program 76.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 76.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 63.4

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 63.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   10. Applied egg-rr 69.3%

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

   if 1.4e154 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 64.8

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 64.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   10. Applied egg-rr 84.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right)\right) \cdot \left(\pi \cdot \pi\right), 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot angle\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{if}\;b \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot \left(angle\_m \cdot \left(b \cdot b\right)\right), t\_0, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(b \cdot t\_0\right) \cdot \left(angle\_m \cdot angle\_m\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI PI) 3.08641975308642e-5)))
   (if (<= b 1.3e+154)
     (fma (* angle_m (* angle_m (* b b))) t_0 (* a a))
     (fma b (* (* b t_0) (* angle_m angle_m)) (* a a)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5;
	double tmp;
	if (b <= 1.3e+154) {
		tmp = fma((angle_m * (angle_m * (b * b))), t_0, (a * a));
	} else {
		tmp = fma(b, ((b * t_0) * (angle_m * angle_m)), (a * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(pi * pi) * 3.08641975308642e-5)
	tmp = 0.0
	if (b <= 1.3e+154)
		tmp = fma(Float64(angle_m * Float64(angle_m * Float64(b * b))), t_0, Float64(a * a));
	else
		tmp = fma(b, Float64(Float64(b * t_0) * Float64(angle_m * angle_m)), Float64(a * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]}, If[LessEqual[b, 1.3e+154], N[(N[(angle$95$m * N[(angle$95$m * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(b * N[(N[(b * t$95$0), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.29999999999999994e154

   1. Initial program 76.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 76.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 63.4

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 63.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-*r* N/A

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

   10. Applied egg-rr 69.3%

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

   if 1.29999999999999994e154 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 64.8

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 64.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   10. Applied egg-rr 84.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.3 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(b \cdot b\right)\right), \left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot angle\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.1% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-57}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle\_m \cdot angle\_m\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.16e-57)
   (* a a)
   (fma
    b
    (* (* b (* (* PI PI) 3.08641975308642e-5)) (* angle_m angle_m))
    (* a a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.16e-57) {
		tmp = a * a;
	} else {
		tmp = fma(b, ((b * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5)) * (angle_m * angle_m)), (a * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.16e-57)
		tmp = Float64(a * a);
	else
		tmp = fma(b, Float64(Float64(b * Float64(Float64(pi * pi) * 3.08641975308642e-5)) * Float64(angle_m * angle_m)), Float64(a * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.16e-57], N[(a * a), $MachinePrecision], N[(b * N[(N[(b * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.15999999999999996e-57

   1. Initial program 78.0%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 78.2%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 63.7

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 63.7%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]

   9. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 61.6

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

   11. Simplified 61.6%

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

   if 1.15999999999999996e-57 < b

   1. Initial program 81.4%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 81.3%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 63.0

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 63.0%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. +-commutative N/A

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

   10. Applied egg-rr 70.5%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-57}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(angle \cdot angle\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{+162}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 7.5e+162)
   (* a a)
   (* (* angle_m angle_m) (* PI (* PI (* (* b b) 3.08641975308642e-5))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 7.5e+162) {
		tmp = a * a;
	} else {
		tmp = (angle_m * angle_m) * (((double) M_PI) * (((double) M_PI) * ((b * b) * 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 (b <= 7.5e+162) {
		tmp = a * a;
	} else {
		tmp = (angle_m * angle_m) * (Math.PI * (Math.PI * ((b * b) * 3.08641975308642e-5)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 7.5e+162:
		tmp = a * a
	else:
		tmp = (angle_m * angle_m) * (math.pi * (math.pi * ((b * b) * 3.08641975308642e-5)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 7.5e+162)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(pi * Float64(Float64(b * b) * 3.08641975308642e-5))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 7.5e+162)
		tmp = a * a;
	else
		tmp = (angle_m * angle_m) * (pi * (pi * ((b * b) * 3.08641975308642e-5)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 7.5e+162], N[(a * a), $MachinePrecision], N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.50000000000000033e162

   1. Initial program 76.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 76.8%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 63.4

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 63.4%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]

   9. Taylor expanded in a around inf

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

       1. unpow2 N/A

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

       2. lower-*.f64 58.8

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

   11. Simplified 58.8%

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

   if 7.50000000000000033e162 < b

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 99.7%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-PI.f64 N/A

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

      17. lower-PI.f64 64.8

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

   8. Simplified 64.8%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]

   9. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. unpow2 N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. lower-PI.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lower-PI.f64 N/A

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

       16. lower-*.f64 N/A

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

       17. unpow2 N/A

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

       18. lower-*.f64 64.8

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

   11. Simplified 64.8%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7.5 \cdot 10^{+162}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.2% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

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 = a * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in angle around 0

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

5. Simplified 79.0%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-PI.f64 N/A

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

   17. lower-PI.f64 63.5

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right) \]

8. Simplified 63.5%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(angle \cdot angle\right) \cdot \left(\left(b \cdot b\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)} \]

9. Taylor expanded in a around inf

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

    1. unpow2 N/A

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

    2. lower-*.f64 56.4

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

11. Simplified 56.4%

    \[\leadsto \color{blue}{a \cdot a} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))