ab-angle->ABCF A

Percentage Accurate: 79.8% → 79.7%

Time: 17.7s

Alternatives: 15

Speedup: N/A×

• 36.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, \pi \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right), 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 2000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot angle\_m, \left(\pi \cdot t\_0\right) \cdot \left(angle\_m \cdot \left(\pi \cdot \left(a \cdot t\_0\right)\right)\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, b \cdot b, a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0
         (fma
          PI
          (* PI (* (* angle_m angle_m) -2.8577960676726107e-8))
          0.005555555555555556)))
   (if (<= (/ angle_m 180.0) 2000000000.0)
     (fma
      (* a angle_m)
      (* (* PI t_0) (* angle_m (* PI (* a t_0))))
      (* (* b b) (fma 0.5 (cos (* PI (* angle_m 0.011111111111111112))) 0.5)))
     (fma
      1.0
      (* b b)
      (*
       a
       (*
        a
        (+ 0.5 (* -0.5 (cos (* 0.011111111111111112 (* angle_m PI)))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = fma(((double) M_PI), (((double) M_PI) * ((angle_m * angle_m) * -2.8577960676726107e-8)), 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 2000000000.0) {
		tmp = fma((a * angle_m), ((((double) M_PI) * t_0) * (angle_m * (((double) M_PI) * (a * t_0)))), ((b * b) * fma(0.5, cos((((double) M_PI) * (angle_m * 0.011111111111111112))), 0.5)));
	} else {
		tmp = fma(1.0, (b * b), (a * (a * (0.5 + (-0.5 * cos((0.011111111111111112 * (angle_m * ((double) M_PI)))))))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = fma(pi, Float64(pi * Float64(Float64(angle_m * angle_m) * -2.8577960676726107e-8)), 0.005555555555555556)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2000000000.0)
		tmp = fma(Float64(a * angle_m), Float64(Float64(pi * t_0) * Float64(angle_m * Float64(pi * Float64(a * t_0)))), Float64(Float64(b * b) * fma(0.5, cos(Float64(pi * Float64(angle_m * 0.011111111111111112))), 0.5)));
	else
		tmp = fma(1.0, Float64(b * b), Float64(a * Float64(a * Float64(0.5 + Float64(-0.5 * cos(Float64(0.011111111111111112 * Float64(angle_m * pi))))))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(Pi * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision]), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2000000000.0], N[(N[(a * angle$95$m), $MachinePrecision] * N[(N[(Pi * t$95$0), $MachinePrecision] * N[(angle$95$m * N[(Pi * N[(a * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(0.5 + N[(-0.5 * N[Cos[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2e9

   1. Initial program 91.4%

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

   3. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \left(a \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \color{blue}{\left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. associate-*l* N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a} + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. associate-*l* N/A

         \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \color{blue}{\frac{-1}{34992000} \cdot \left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \frac{-1}{34992000} \cdot \color{blue}{\left(a \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \frac{-1}{34992000} \cdot \left(a \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      11. *-commutative N/A

          \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \frac{-1}{34992000} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      12. associate-*r* N/A

          \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \color{blue}{\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot a}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. Simplified 89.7%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. Applied egg-rr 89.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot angle, \left(\pi \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right), 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(\pi \cdot \left(\mathsf{fma}\left(\pi, \pi \cdot \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right), 0.005555555555555556\right) \cdot a\right)\right)\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), 0.5\right)\right)} \]

   if 2e9 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 52.6%

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

   4. Applied egg-rr 53.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-PI.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 47.8

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

   6. Applied egg-rr 47.8%

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

   8. Applied egg-rr 52.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right), b \cdot b, a \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right)\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, b \cdot b, a \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 53.0%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1}, b \cdot b, a \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot angle, \left(\pi \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right), 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot \mathsf{fma}\left(\pi, \pi \cdot \left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}\right), 0.005555555555555556\right)\right)\right)\right), \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, b \cdot b, a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := a \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \pi\right)\\ t_1 := {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}\\ \mathsf{fma}\left(t\_0, t\_0, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{\sqrt{t\_1 \cdot t\_1}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle\_m}}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* a (sin (* (* angle_m 0.005555555555555556) PI))))
        (t_1 (pow (cbrt (sqrt PI)) 3.0)))
   (fma
    t_0
    t_0
    (*
     (* b b)
     (+
      0.5
      (*
       0.5
       (cos
        (*
         2.0
         (* (/ (sqrt (* t_1 t_1)) 180.0) (/ (sqrt PI) (/ 1.0 angle_m)))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = a * sin(((angle_m * 0.005555555555555556) * ((double) M_PI)));
	double t_1 = pow(cbrt(sqrt(((double) M_PI))), 3.0);
	return fma(t_0, t_0, ((b * b) * (0.5 + (0.5 * cos((2.0 * ((sqrt((t_1 * t_1)) / 180.0) * (sqrt(((double) M_PI)) / (1.0 / angle_m)))))))));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(a * sin(Float64(Float64(angle_m * 0.005555555555555556) * pi)))
	t_1 = cbrt(sqrt(pi)) ^ 3.0
	return fma(t_0, t_0, Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(sqrt(Float64(t_1 * t_1)) / 180.0) * Float64(sqrt(pi) / Float64(1.0 / angle_m)))))))))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(a * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision] / 180.0), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 83.5%

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

4. Applied egg-rr 83.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. div-inv N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 83.7

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

6. Applied egg-rr 83.7%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{\sqrt{\pi}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle}}\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. add-cube-cbrt N/A

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

   2. pow3 N/A

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

   3. lift-PI.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. cbrt-prod N/A

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

   8. unpow-prod-down N/A

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

   9. lower-*.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-cbrt.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-cbrt.f64 83.7

       \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{\sqrt{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\color{blue}{\left(\sqrt[3]{\sqrt{\pi}}\right)}}^{3}}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle}}\right)\right)\right)\right) \]

8. Applied egg-rr 83.7%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{\sqrt{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3} \cdot {\left(\sqrt[3]{\sqrt{\pi}}\right)}^{3}}}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle}}\right)\right)\right)\right) \]
9. Add Preprocessing

Alternative 3: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := a \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \pi\right)\\ \mathsf{fma}\left(t\_0, t\_0, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{\sqrt{\pi}}{180} \cdot \frac{1}{\frac{1}{angle\_m \cdot \sqrt{\pi}}}\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* a (sin (* (* angle_m 0.005555555555555556) PI)))))
   (fma
    t_0
    t_0
    (*
     (* b b)
     (+
      0.5
      (*
       0.5
       (cos
        (*
         2.0
         (* (/ (sqrt PI) 180.0) (/ 1.0 (/ 1.0 (* angle_m (sqrt PI)))))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = a * sin(((angle_m * 0.005555555555555556) * ((double) M_PI)));
	return fma(t_0, t_0, ((b * b) * (0.5 + (0.5 * cos((2.0 * ((sqrt(((double) M_PI)) / 180.0) * (1.0 / (1.0 / (angle_m * sqrt(((double) M_PI))))))))))));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(a * sin(Float64(Float64(angle_m * 0.005555555555555556) * pi)))
	return fma(t_0, t_0, Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(sqrt(pi) / 180.0) * Float64(1.0 / Float64(1.0 / Float64(angle_m * sqrt(pi)))))))))))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(a * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[Sqrt[Pi], $MachinePrecision] / 180.0), $MachinePrecision] * N[(1.0 / N[(1.0 / N[(angle$95$m * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 83.5%

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

4. Applied egg-rr 83.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. div-inv N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 83.7

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

6. Applied egg-rr 83.7%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{\sqrt{\pi}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle}}\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. clear-num N/A

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

   7. lift-/.f64 N/A

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

   8. lower-/.f64 83.7

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/r/ N/A

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

   12. /-rgt-identity N/A

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

   13. lower-*.f64 83.7

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

8. Applied egg-rr 83.7%

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

9. Final simplification 83.7%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{\sqrt{\pi}}{180} \cdot \frac{1}{\frac{1}{angle \cdot \sqrt{\pi}}}\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 4: 79.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := a \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \pi\right)\\ \mathsf{fma}\left(t\_0, t\_0, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{\sqrt{\pi}}{180} \cdot \left(angle\_m \cdot \sqrt{\pi}\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* a (sin (* (* angle_m 0.005555555555555556) PI)))))
   (fma
    t_0
    t_0
    (*
     (* b b)
     (+
      0.5
      (* 0.5 (cos (* 2.0 (* (/ (sqrt PI) 180.0) (* angle_m (sqrt PI)))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = a * sin(((angle_m * 0.005555555555555556) * ((double) M_PI)));
	return fma(t_0, t_0, ((b * b) * (0.5 + (0.5 * cos((2.0 * ((sqrt(((double) M_PI)) / 180.0) * (angle_m * sqrt(((double) M_PI))))))))));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(a * sin(Float64(Float64(angle_m * 0.005555555555555556) * pi)))
	return fma(t_0, t_0, Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(sqrt(pi) / 180.0) * Float64(angle_m * sqrt(pi)))))))))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(a * N[Sin[N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[Sqrt[Pi], $MachinePrecision] / 180.0), $MachinePrecision] * N[(angle$95$m * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 83.5%

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

4. Applied egg-rr 83.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. clear-num N/A

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

   5. associate-*l/ N/A

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

   6. *-lft-identity N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. div-inv N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 83.7

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

6. Applied egg-rr 83.7%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\frac{\sqrt{\pi}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle}}\right)}\right)\right)\right) \]
7. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. associate-/r/ N/A

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

   4. /-rgt-identity N/A

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

   5. lower-*.f64 83.7

      \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{\sqrt{\pi}}{180} \cdot \color{blue}{\left(\sqrt{\pi} \cdot angle\right)}\right)\right)\right)\right) \]

8. Applied egg-rr 83.7%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{\sqrt{\pi}}{180} \cdot \color{blue}{\left(\sqrt{\pi} \cdot angle\right)}\right)\right)\right)\right) \]

9. Final simplification 83.7%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\frac{\sqrt{\pi}}{180} \cdot \left(angle \cdot \sqrt{\pi}\right)\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 5: 79.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := a \cdot \sin \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \pi\right)\\ \mathsf{fma}\left(t\_0, t\_0, b \cdot b\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* a (sin (* (* angle_m 0.005555555555555556) PI)))))
   (fma t_0 t_0 (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = a * sin(((angle_m * 0.005555555555555556) * ((double) M_PI)));
	return fma(t_0, t_0, (b * b));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(a * sin(Float64(Float64(angle_m * 0.005555555555555556) * pi)))
	return fma(t_0, t_0, Float64(b * b))
end

Wolfram

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

Derivation

1. Initial program 83.5%

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

4. Applied egg-rr 83.6%

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

5. Taylor expanded in angle around 0

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

7. Simplified 83.6%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \color{blue}{1}\right) \]

8. Final simplification 83.6%

   \[\leadsto \mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), b \cdot b\right) \]
9. Add Preprocessing

Alternative 6: 78.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \cos \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, t\_0, 0.5\right), b \cdot b, a \cdot \left(angle\_m \cdot \left(angle\_m \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, b \cdot b, a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.011111111111111112 (* angle_m PI)))))
   (if (<= (/ angle_m 180.0) 2e+29)
     (fma
      (fma 0.5 t_0 0.5)
      (* b b)
      (* a (* angle_m (* angle_m (* a (* 3.08641975308642e-5 (* PI PI)))))))
     (fma 1.0 (* b b) (* a (* a (+ 0.5 (* -0.5 t_0))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = cos((0.011111111111111112 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 2e+29) {
		tmp = fma(fma(0.5, t_0, 0.5), (b * b), (a * (angle_m * (angle_m * (a * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI))))))));
	} else {
		tmp = fma(1.0, (b * b), (a * (a * (0.5 + (-0.5 * t_0)))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = cos(Float64(0.011111111111111112 * Float64(angle_m * pi)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+29)
		tmp = fma(fma(0.5, t_0, 0.5), Float64(b * b), Float64(a * Float64(angle_m * Float64(angle_m * Float64(a * Float64(3.08641975308642e-5 * Float64(pi * pi)))))));
	else
		tmp = fma(1.0, Float64(b * b), Float64(a * Float64(a * Float64(0.5 + Float64(-0.5 * t_0)))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Cos[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+29], N[(N[(0.5 * t$95$0 + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(a * N[(angle$95$m * N[(angle$95$m * N[(a * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(0.5 + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.99999999999999983e29

   1. Initial program 91.0%

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

   4. Applied egg-rr 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-PI.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 90.1

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

   6. Applied egg-rr 90.1%

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

   8. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right), b \cdot b, a \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right)\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(\frac{1}{32400} \cdot \left(a \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right)}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right)}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(angle \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(\left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)}\right)\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(a \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)\right)}\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(a \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right)\right) \]

       15. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right)\right)\right) \]

       16. lower-PI.f64 N/A

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

       17. lower-PI.f64 88.7

           \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \color{blue}{\pi}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 88.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right), b \cdot b, a \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)}\right) \]

   if 1.99999999999999983e29 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 53.3%

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

   4. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-PI.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 48.4

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

   6. Applied egg-rr 48.4%

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

   8. Applied egg-rr 53.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right), b \cdot b, a \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right)\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, b \cdot b, a \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 53.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1}, b \cdot b, a \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right), b \cdot b, a \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, b \cdot b, a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \cos \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(angle\_m \cdot angle\_m\right)\right)\right), a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, t\_0, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, b \cdot b, a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (cos (* 0.011111111111111112 (* angle_m PI)))))
   (if (<= (/ angle_m 180.0) 2e+29)
     (fma
      (* (* a 3.08641975308642e-5) (* PI (* PI (* angle_m angle_m))))
      a
      (* (* b b) (fma 0.5 t_0 0.5)))
     (fma 1.0 (* b b) (* a (* a (+ 0.5 (* -0.5 t_0))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = cos((0.011111111111111112 * (angle_m * ((double) M_PI))));
	double tmp;
	if ((angle_m / 180.0) <= 2e+29) {
		tmp = fma(((a * 3.08641975308642e-5) * (((double) M_PI) * (((double) M_PI) * (angle_m * angle_m)))), a, ((b * b) * fma(0.5, t_0, 0.5)));
	} else {
		tmp = fma(1.0, (b * b), (a * (a * (0.5 + (-0.5 * t_0)))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = cos(Float64(0.011111111111111112 * Float64(angle_m * pi)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+29)
		tmp = fma(Float64(Float64(a * 3.08641975308642e-5) * Float64(pi * Float64(pi * Float64(angle_m * angle_m)))), a, Float64(Float64(b * b) * fma(0.5, t_0, 0.5)));
	else
		tmp = fma(1.0, Float64(b * b), Float64(a * Float64(a * Float64(0.5 + Float64(-0.5 * t_0)))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Cos[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+29], N[(N[(N[(a * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * N[(Pi * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(0.5 + N[(-0.5 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.99999999999999983e29

   1. Initial program 91.0%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 91.1

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

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      4. unpow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      7. unpow2 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\left({angle}^{2} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \frac{1}{32400}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      8. associate-*r* N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\left({angle}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(\left({angle}^{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\color{blue}{\left({angle}^{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      11. unpow2 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \mathsf{PI}\left(\right)}}\right)\right)}^{2} \]

      14. lower-PI.f64 71.4

          \[\leadsto \left(a \cdot a\right) \cdot \left(\left(\left(\left(angle \cdot angle\right) \cdot \pi\right) \cdot \color{blue}{\pi}\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} \]

   7. Simplified 71.4%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \left(\left(\left(\left(angle \cdot angle\right) \cdot \pi\right) \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} + {\left(b \cdot \cos \left(\frac{1}{\frac{180}{angle \cdot \pi}}\right)\right)}^{2} \]

   9. Applied egg-rr 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot angle\right)\right)\right), a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)} \]

   if 1.99999999999999983e29 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 53.3%

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

   4. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-PI.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 48.4

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

   6. Applied egg-rr 48.4%

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

   8. Applied egg-rr 53.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right), b \cdot b, a \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right)\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, b \cdot b, a \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 53.6%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1}, b \cdot b, a \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot angle\right)\right)\right), a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, b \cdot b, a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, b \cdot b, a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 7e+23)
   (fma
    (*
     (* angle_m PI)
     (* PI (fma b (* b -3.08641975308642e-5) (* 3.08641975308642e-5 (* a a)))))
    angle_m
    (* b b))
   (fma
    1.0
    (* b b)
    (*
     a
     (* a (+ 0.5 (* -0.5 (cos (* 0.011111111111111112 (* angle_m PI))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 7e+23) {
		tmp = fma(((angle_m * ((double) M_PI)) * (((double) M_PI) * fma(b, (b * -3.08641975308642e-5), (3.08641975308642e-5 * (a * a))))), angle_m, (b * b));
	} else {
		tmp = fma(1.0, (b * b), (a * (a * (0.5 + (-0.5 * cos((0.011111111111111112 * (angle_m * ((double) M_PI)))))))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 7e+23)
		tmp = fma(Float64(Float64(angle_m * pi) * Float64(pi * fma(b, Float64(b * -3.08641975308642e-5), Float64(3.08641975308642e-5 * Float64(a * a))))), angle_m, Float64(b * b));
	else
		tmp = fma(1.0, Float64(b * b), Float64(a * Float64(a * Float64(0.5 + Float64(-0.5 * cos(Float64(0.011111111111111112 * Float64(angle_m * pi))))))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 7e+23], N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(Pi * N[(b * N[(b * -3.08641975308642e-5), $MachinePrecision] + N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[(b * b), $MachinePrecision] + N[(a * N[(a * N[(0.5 + N[(-0.5 * N[Cos[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7.0000000000000004e23

   1. Initial program 82.5%

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

   4. Applied egg-rr 82.7%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

   9. Applied egg-rr 56.4%

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

   if 7.0000000000000004e23 < b

   1. Initial program 87.2%

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

   4. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-PI.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 85.3

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

   6. Applied egg-rr 85.3%

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

   8. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right), b \cdot b, a \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right)\right)} \]

   9. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{1}, b \cdot b, a \cdot \left(\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right)\right) \cdot a\right)\right) \]
   10. Step-by-step derivation

   11. Simplified 82.2%

       \[\leadsto \mathsf{fma}\left(\color{blue}{1}, b \cdot b, a \cdot \left(\left(0.5 + -0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right) \cdot a\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, b \cdot b, a \cdot \left(a \cdot \left(0.5 + -0.5 \cdot \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.35e+29)
   (fma
    (*
     (* angle_m PI)
     (* PI (fma b (* b -3.08641975308642e-5) (* 3.08641975308642e-5 (* a a)))))
    angle_m
    (* b b))
   (* (* b b) (fma 0.5 (cos (* angle_m (* PI 0.011111111111111112))) 0.5))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.35e+29) {
		tmp = fma(((angle_m * ((double) M_PI)) * (((double) M_PI) * fma(b, (b * -3.08641975308642e-5), (3.08641975308642e-5 * (a * a))))), angle_m, (b * b));
	} else {
		tmp = (b * b) * fma(0.5, cos((angle_m * (((double) M_PI) * 0.011111111111111112))), 0.5);
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.35e+29)
		tmp = fma(Float64(Float64(angle_m * pi) * Float64(pi * fma(b, Float64(b * -3.08641975308642e-5), Float64(3.08641975308642e-5 * Float64(a * a))))), angle_m, Float64(b * b));
	else
		tmp = Float64(Float64(b * b) * fma(0.5, cos(Float64(angle_m * Float64(pi * 0.011111111111111112))), 0.5));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.35e+29], N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(Pi * N[(b * N[(b * -3.08641975308642e-5), $MachinePrecision] + N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[Cos[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35e29

   1. Initial program 82.5%

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

   4. Applied egg-rr 82.7%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

   9. Applied egg-rr 56.4%

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

   if 1.35e29 < b

   1. Initial program 87.2%

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

   4. Applied egg-rr 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), a \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-PI.f64 N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 85.3

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

   6. Applied egg-rr 85.3%

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

   7. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(\left(angle \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{90}\right)}, \frac{1}{2}\right) \]

      8. associate-*l* N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)\right)}, \frac{1}{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{90}\right)}\right), \frac{1}{2}\right) \]

      11. lower-PI.f64 80.0

          \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\color{blue}{\pi} \cdot 0.011111111111111112\right)\right), 0.5\right) \]

   9. Simplified 80.0%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle\_m \cdot \pi\right)\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.35e+29)
   (fma
    (*
     (* angle_m PI)
     (* PI (fma b (* b -3.08641975308642e-5) (* 3.08641975308642e-5 (* a a)))))
    angle_m
    (* b b))
   (* (* b b) (fma 0.5 (cos (* 0.011111111111111112 (* angle_m PI))) 0.5))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.35e+29) {
		tmp = fma(((angle_m * ((double) M_PI)) * (((double) M_PI) * fma(b, (b * -3.08641975308642e-5), (3.08641975308642e-5 * (a * a))))), angle_m, (b * b));
	} else {
		tmp = (b * b) * fma(0.5, cos((0.011111111111111112 * (angle_m * ((double) M_PI)))), 0.5);
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.35e+29)
		tmp = fma(Float64(Float64(angle_m * pi) * Float64(pi * fma(b, Float64(b * -3.08641975308642e-5), Float64(3.08641975308642e-5 * Float64(a * a))))), angle_m, Float64(b * b));
	else
		tmp = Float64(Float64(b * b) * fma(0.5, cos(Float64(0.011111111111111112 * Float64(angle_m * pi))), 0.5));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.35e+29], N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(Pi * N[(b * N[(b * -3.08641975308642e-5), $MachinePrecision] + N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[Cos[N[(0.011111111111111112 * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35e29

   1. Initial program 82.5%

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

   4. Applied egg-rr 82.7%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

   9. Applied egg-rr 56.4%

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

   if 1.35e29 < b

   1. Initial program 87.2%

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

   4. Applied egg-rr 87.2%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{1}{2}\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), \frac{1}{2}\right)} \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \color{blue}{\left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \frac{1}{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\frac{1}{90} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right), \frac{1}{2}\right) \]

      9. lower-PI.f64 80.0

         \[\leadsto \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \color{blue}{\pi}\right)\right), 0.5\right) \]

   7. Simplified 80.0%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.35e+29)
   (fma
    (*
     (* angle_m PI)
     (* PI (fma b (* b -3.08641975308642e-5) (* 3.08641975308642e-5 (* a a)))))
    angle_m
    (* b b))
   (* b b)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.35e+29) {
		tmp = fma(((angle_m * ((double) M_PI)) * (((double) M_PI) * fma(b, (b * -3.08641975308642e-5), (3.08641975308642e-5 * (a * a))))), angle_m, (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.35e+29)
		tmp = fma(Float64(Float64(angle_m * pi) * Float64(pi * fma(b, Float64(b * -3.08641975308642e-5), Float64(3.08641975308642e-5 * Float64(a * a))))), angle_m, Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.35e+29], N[(N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(Pi * N[(b * N[(b * -3.08641975308642e-5), $MachinePrecision] + N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.35e29

   1. Initial program 82.5%

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

   4. Applied egg-rr 82.7%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 47.5%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

   9. Applied egg-rr 56.4%

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

   if 1.35e29 < b

   1. Initial program 87.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 78.3

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

   5. Simplified 78.3%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right)\right), angle, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.4% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 9.4 \cdot 10^{-123}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, \left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(a \cdot \left(angle\_m \cdot \left(\pi \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 9.4e-123)
   (* b b)
   (if (<= a 1.4e+154)
     (fma
      (* angle_m angle_m)
      (* (* PI PI) (* 3.08641975308642e-5 (* a a)))
      (* b b))
     (* (* a 3.08641975308642e-5) (* a (* angle_m (* PI (* angle_m PI))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 9.4e-123) {
		tmp = b * b;
	} else if (a <= 1.4e+154) {
		tmp = fma((angle_m * angle_m), ((((double) M_PI) * ((double) M_PI)) * (3.08641975308642e-5 * (a * a))), (b * b));
	} else {
		tmp = (a * 3.08641975308642e-5) * (a * (angle_m * (((double) M_PI) * (angle_m * ((double) M_PI)))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 9.4e-123)
		tmp = Float64(b * b);
	elseif (a <= 1.4e+154)
		tmp = fma(Float64(angle_m * angle_m), Float64(Float64(pi * pi) * Float64(3.08641975308642e-5 * Float64(a * a))), Float64(b * b));
	else
		tmp = Float64(Float64(a * 3.08641975308642e-5) * Float64(a * Float64(angle_m * Float64(pi * Float64(angle_m * pi)))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 9.4e-123], N[(b * b), $MachinePrecision], If[LessEqual[a, 1.4e+154], N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(a * 3.08641975308642e-5), $MachinePrecision] * N[(a * N[(angle$95$m * N[(Pi * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 9.4000000000000004e-123

   1. Initial program 83.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.6

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

   5. Simplified 65.6%

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

   if 9.4000000000000004e-123 < a < 1.4e154

   1. Initial program 77.2%

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

   4. Applied egg-rr 77.2%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 45.9%

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

   8. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 70.4

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

   10. Simplified 70.4%

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

   if 1.4e154 < a

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 17.9%

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

   8. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-PI.f64 37.9

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

   10. Simplified 37.9%

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

       1. associate-*r* N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 61.3

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. lift-*.f64 N/A

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

       16. lower-*.f64 61.3

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

   12. Applied egg-rr 61.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 9.4 \cdot 10^{-123}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.5% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot \pi\right)\\ \mathbf{if}\;a \leq 3.7 \cdot 10^{+117}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+201}:\\ \;\;\;\;angle\_m \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(a \cdot \left(angle\_m \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m PI))))
   (if (<= a 3.7e+117)
     (* b b)
     (if (<= a 3e+201)
       (* angle_m (* (* 3.08641975308642e-5 (* a a)) t_0))
       (* (* a 3.08641975308642e-5) (* a (* angle_m t_0)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * ((double) M_PI));
	double tmp;
	if (a <= 3.7e+117) {
		tmp = b * b;
	} else if (a <= 3e+201) {
		tmp = angle_m * ((3.08641975308642e-5 * (a * a)) * t_0);
	} else {
		tmp = (a * 3.08641975308642e-5) * (a * (angle_m * t_0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * Math.PI);
	double tmp;
	if (a <= 3.7e+117) {
		tmp = b * b;
	} else if (a <= 3e+201) {
		tmp = angle_m * ((3.08641975308642e-5 * (a * a)) * t_0);
	} else {
		tmp = (a * 3.08641975308642e-5) * (a * (angle_m * t_0));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (angle_m * math.pi)
	tmp = 0
	if a <= 3.7e+117:
		tmp = b * b
	elif a <= 3e+201:
		tmp = angle_m * ((3.08641975308642e-5 * (a * a)) * t_0)
	else:
		tmp = (a * 3.08641975308642e-5) * (a * (angle_m * t_0))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * pi))
	tmp = 0.0
	if (a <= 3.7e+117)
		tmp = Float64(b * b);
	elseif (a <= 3e+201)
		tmp = Float64(angle_m * Float64(Float64(3.08641975308642e-5 * Float64(a * a)) * t_0));
	else
		tmp = Float64(Float64(a * 3.08641975308642e-5) * Float64(a * Float64(angle_m * t_0)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = pi * (angle_m * pi);
	tmp = 0.0;
	if (a <= 3.7e+117)
		tmp = b * b;
	elseif (a <= 3e+201)
		tmp = angle_m * ((3.08641975308642e-5 * (a * a)) * t_0);
	else
		tmp = (a * 3.08641975308642e-5) * (a * (angle_m * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3.7e+117], N[(b * b), $MachinePrecision], If[LessEqual[a, 3e+201], N[(angle$95$m * N[(N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(a * 3.08641975308642e-5), $MachinePrecision] * N[(a * N[(angle$95$m * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.6999999999999999e117

   1. Initial program 82.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.6

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

   5. Simplified 65.6%

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

   if 3.6999999999999999e117 < a < 3.00000000000000025e201

   1. Initial program 79.5%

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

   4. Applied egg-rr 79.6%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 27.8%

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

   8. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-PI.f64 34.3

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

   10. Simplified 34.3%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 53.0

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. lift-*.f64 N/A

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

       16. lower-*.f64 53.0

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

   12. Applied egg-rr 53.0%

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

   if 3.00000000000000025e201 < a

   1. Initial program 99.5%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 18.7%

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

   8. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-PI.f64 40.9

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

   10. Simplified 40.9%

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

       1. associate-*r* N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 62.7

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. lift-*.f64 N/A

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

       16. lower-*.f64 62.7

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

   12. Applied egg-rr 62.7%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{+117}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 3 \cdot 10^{+201}:\\ \;\;\;\;angle \cdot \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\pi \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.66 \cdot 10^{+118}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(a \cdot \left(angle\_m \cdot \left(\pi \cdot \left(angle\_m \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.66e+118)
   (* b b)
   (* (* a 3.08641975308642e-5) (* a (* angle_m (* PI (* angle_m PI)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.66e+118) {
		tmp = b * b;
	} else {
		tmp = (a * 3.08641975308642e-5) * (a * (angle_m * (((double) M_PI) * (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 (a <= 1.66e+118) {
		tmp = b * b;
	} else {
		tmp = (a * 3.08641975308642e-5) * (a * (angle_m * (Math.PI * (angle_m * Math.PI))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.66e+118:
		tmp = b * b
	else:
		tmp = (a * 3.08641975308642e-5) * (a * (angle_m * (math.pi * (angle_m * math.pi))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.66e+118)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(a * 3.08641975308642e-5) * Float64(a * Float64(angle_m * Float64(pi * 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 (a <= 1.66e+118)
		tmp = b * b;
	else
		tmp = (a * 3.08641975308642e-5) * (a * (angle_m * (pi * (angle_m * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.66e+118], N[(b * b), $MachinePrecision], N[(N[(a * 3.08641975308642e-5), $MachinePrecision] * N[(a * N[(angle$95$m * N[(Pi * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.66e118

   1. Initial program 82.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 65.6

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

   5. Simplified 65.6%

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

   if 1.66e118 < a

   1. Initial program 90.1%

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

   4. Applied egg-rr 90.2%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 23.0%

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

   8. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-PI.f64 37.8

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

   10. Simplified 37.8%

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

       1. associate-*r* N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*l* N/A

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

       8. lower-*.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 54.9

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

       11. lift-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 N/A

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

       14. associate-*l* N/A

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

       15. lift-*.f64 N/A

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

       16. lower-*.f64 54.9

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

   12. Applied egg-rr 54.9%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.66 \cdot 10^{+118}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(a \cdot \left(angle \cdot \left(\pi \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.3% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 61.4

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

5. Simplified 61.4%

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

Reproduce

herbie shell --seed 2024211 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))