ab-angle->ABCF A

Percentage Accurate: 79.4% → 79.4%

Time: 15.0s

Alternatives: 10

Speedup: 2.0×

• 36.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 79.4% 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: 79.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(angle\_m \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 2e-5)
   (+
    (* b b)
    (pow
     (*
      angle_m
      (*
       a
       (*
        PI
        (fma
         (* (* angle_m angle_m) -2.8577960676726107e-8)
         (* PI PI)
         0.005555555555555556))))
     2.0))
   (fma
    (* a (fma (cos (* PI (* angle_m 0.011111111111111112))) -0.5 0.5))
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-5) {
		tmp = (b * b) + pow((angle_m * (a * (((double) M_PI) * fma(((angle_m * angle_m) * -2.8577960676726107e-8), (((double) M_PI) * ((double) M_PI)), 0.005555555555555556)))), 2.0);
	} else {
		tmp = fma((a * fma(cos((((double) M_PI) * (angle_m * 0.011111111111111112))), -0.5, 0.5)), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-5)
		tmp = Float64(Float64(b * b) + (Float64(angle_m * Float64(a * Float64(pi * fma(Float64(Float64(angle_m * angle_m) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)))) ^ 2.0));
	else
		tmp = fma(Float64(a * fma(cos(Float64(pi * Float64(angle_m * 0.011111111111111112))), -0.5, 0.5)), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-5], N[(N[(b * b), $MachinePrecision] + N[Power[N[(angle$95$m * N[(a * N[(Pi * N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000016e-5

   1. Initial program 88.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. /-rgt-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. rem-square-sqrt N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.4

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

   10. Applied rewrites 88.4%

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

   11. Taylor expanded in angle around 0

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. associate-*r* N/A

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

       5. +-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. associate-*l* N/A

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

       10. associate-*r* N/A

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

       11. *-commutative N/A

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

   13. Applied rewrites 83.1%

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

   if 2.00000000000000016e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.4%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. /-rgt-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. rem-square-sqrt N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.2

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

   10. Applied rewrites 62.2%

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

   12. Applied rewrites 62.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 81.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. frac-times N/A

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

   7. rem-square-sqrt N/A

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

   8. lift-sqrt.f64 N/A

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

   9. lift-sqrt.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lift-/.f64 N/A

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

   12. /-rgt-identity N/A

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

   13. lower-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. associate-*l/ N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. rem-square-sqrt N/A

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

   19. div-inv N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 81.1

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

7. Applied rewrites 81.1%

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

8. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.1

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

10. Applied rewrites 81.1%

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

Alternative 3: 79.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 2e-5)
   (+ (* b b) (pow (* a (* 0.005555555555555556 (* PI angle_m))) 2.0))
   (fma
    (* a (fma (cos (* PI (* angle_m 0.011111111111111112))) -0.5 0.5))
    a
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e-5) {
		tmp = (b * b) + pow((a * (0.005555555555555556 * (((double) M_PI) * angle_m))), 2.0);
	} else {
		tmp = fma((a * fma(cos((((double) M_PI) * (angle_m * 0.011111111111111112))), -0.5, 0.5)), a, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e-5)
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(0.005555555555555556 * Float64(pi * angle_m))) ^ 2.0));
	else
		tmp = fma(Float64(a * fma(cos(Float64(pi * Float64(angle_m * 0.011111111111111112))), -0.5, 0.5)), a, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e-5], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 2.00000000000000016e-5

   1. Initial program 88.5%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. /-rgt-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. rem-square-sqrt N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 88.4

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

   7. Applied rewrites 88.4%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.4

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

   10. Applied rewrites 88.4%

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

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 84.1

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

   13. Applied rewrites 84.1%

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

   if 2.00000000000000016e-5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.4%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 62.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. /-rgt-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. rem-square-sqrt N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 62.2

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

   7. Applied rewrites 62.2%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.2

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

   10. Applied rewrites 62.2%

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

   12. Applied rewrites 62.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{-133}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 8.2e-133)
   (* (* b b) (fma 0.5 (cos (* PI (* angle_m 0.011111111111111112))) 0.5))
   (+ (* b b) (pow (* a (* 0.005555555555555556 (* PI angle_m))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 8.2e-133) {
		tmp = (b * b) * fma(0.5, cos((((double) M_PI) * (angle_m * 0.011111111111111112))), 0.5);
	} else {
		tmp = (b * b) + pow((a * (0.005555555555555556 * (((double) M_PI) * angle_m))), 2.0);
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 8.2e-133)
		tmp = Float64(Float64(b * b) * fma(0.5, cos(Float64(pi * Float64(angle_m * 0.011111111111111112))), 0.5));
	else
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(0.005555555555555556 * Float64(pi * angle_m))) ^ 2.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 8.2e-133], 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], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.20000000000000045e-133

   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 rewrites 38.5%

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

   5. Taylor expanded in b around inf

      \[\leadsto \color{blue}{{b}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   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. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 58.3

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

   7. Applied rewrites 58.3%

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

   if 8.20000000000000045e-133 < a

   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

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

   5. Applied rewrites 83.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. /-rgt-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. rem-square-sqrt N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.4

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

   10. Applied rewrites 83.4%

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

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 80.6

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

   13. Applied rewrites 80.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8.2 \cdot 10^{-133}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.9e-17)
   (* b b)
   (+ (* b b) (pow (* a (* 0.005555555555555556 (* PI angle_m))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.9e-17) {
		tmp = b * b;
	} else {
		tmp = (b * b) + pow((a * (0.005555555555555556 * (((double) M_PI) * angle_m))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.9e-17) {
		tmp = b * b;
	} else {
		tmp = (b * b) + Math.pow((a * (0.005555555555555556 * (Math.PI * angle_m))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.9e-17:
		tmp = b * b
	else:
		tmp = (b * b) + math.pow((a * (0.005555555555555556 * (math.pi * angle_m))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.9e-17)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(0.005555555555555556 * Float64(pi * angle_m))) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.9e-17)
		tmp = b * b;
	else
		tmp = (b * b) + ((a * (0.005555555555555556 * (pi * angle_m))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.9e-17], N[(b * b), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.9000000000000001e-17

   1. Initial program 77.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 1.9000000000000001e-17 < a

   1. Initial program 88.4%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. /-rgt-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. rem-square-sqrt N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 88.5

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

   7. Applied rewrites 88.5%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.5

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

   10. Applied rewrites 88.5%

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

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-PI.f64 86.7

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

   13. Applied rewrites 86.7%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(0.005555555555555556 \cdot \left(a \cdot \left(\pi \cdot angle\_m\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.9e-17)
   (* b b)
   (+ (* b b) (pow (* 0.005555555555555556 (* a (* PI angle_m))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.9e-17) {
		tmp = b * b;
	} else {
		tmp = (b * b) + pow((0.005555555555555556 * (a * (((double) M_PI) * angle_m))), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.9e-17) {
		tmp = b * b;
	} else {
		tmp = (b * b) + Math.pow((0.005555555555555556 * (a * (Math.PI * angle_m))), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.9e-17:
		tmp = b * b
	else:
		tmp = (b * b) + math.pow((0.005555555555555556 * (a * (math.pi * angle_m))), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.9e-17)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(b * b) + (Float64(0.005555555555555556 * Float64(a * Float64(pi * angle_m))) ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.9e-17)
		tmp = b * b;
	else
		tmp = (b * b) + ((0.005555555555555556 * (a * (pi * angle_m))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.9e-17], N[(b * b), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[Power[N[(0.005555555555555556 * N[(a * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.9000000000000001e-17

   1. Initial program 77.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 1.9000000000000001e-17 < a

   1. Initial program 88.4%

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

   3. Taylor expanded in angle around 0

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

   5. Applied rewrites 88.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. associate-*l/ N/A

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

      11. lift-/.f64 N/A

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

      12. /-rgt-identity N/A

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

      13. lower-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-sqrt.f64 N/A

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

      17. lift-sqrt.f64 N/A

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

      18. rem-square-sqrt N/A

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

      19. div-inv N/A

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

      20. metadata-eval N/A

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

      21. lower-*.f64 88.5

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

   7. Applied rewrites 88.5%

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

   8. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.5

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

   10. Applied rewrites 88.5%

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

   11. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 86.6

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

   13. Applied rewrites 86.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(0.005555555555555556 \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.85 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle\_m \cdot \left(\pi \cdot \pi\right)\right) \cdot \mathsf{fma}\left(b, b \cdot -3.08641975308642 \cdot 10^{-5}, a \cdot \left(a \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), angle\_m, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.85e+117)
   (fma
    (*
     (* angle_m (* PI PI))
     (fma b (* b -3.08641975308642e-5) (* a (* a 3.08641975308642e-5))))
    angle_m
    (* b b))
   (* b b)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.85e+117) {
		tmp = fma(((angle_m * (((double) M_PI) * ((double) M_PI))) * fma(b, (b * -3.08641975308642e-5), (a * (a * 3.08641975308642e-5)))), 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 <= 1.85e+117)
		tmp = fma(Float64(Float64(angle_m * Float64(pi * pi)) * fma(b, Float64(b * -3.08641975308642e-5), Float64(a * Float64(a * 3.08641975308642e-5)))), 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, 1.85e+117], N[(N[(N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(b * N[(b * -3.08641975308642e-5), $MachinePrecision] + N[(a * N[(a * 3.08641975308642e-5), $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 < 1.8499999999999999e117

   1. Initial program 77.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   7. Applied rewrites 44.8%

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

   9. Applied rewrites 50.7%

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

   if 1.8499999999999999e117 < b

   1. Initial program 96.2%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

Alternative 8: 62.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(angle\_m \cdot angle\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.9e-17)
   (* b b)
   (if (<= a 1.12e+173)
     (fma
      (* angle_m (* angle_m (* PI PI)))
      (* 3.08641975308642e-5 (* a a))
      (* b b))
     (* a (* (* a 3.08641975308642e-5) (* PI (* PI (* angle_m angle_m))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.9e-17) {
		tmp = b * b;
	} else if (a <= 1.12e+173) {
		tmp = fma((angle_m * (angle_m * (((double) M_PI) * ((double) M_PI)))), (3.08641975308642e-5 * (a * a)), (b * b));
	} else {
		tmp = a * ((a * 3.08641975308642e-5) * (((double) M_PI) * (((double) M_PI) * (angle_m * angle_m))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.9e-17)
		tmp = Float64(b * b);
	elseif (a <= 1.12e+173)
		tmp = fma(Float64(angle_m * Float64(angle_m * Float64(pi * pi))), Float64(3.08641975308642e-5 * Float64(a * a)), Float64(b * b));
	else
		tmp = Float64(a * Float64(Float64(a * 3.08641975308642e-5) * Float64(pi * Float64(pi * Float64(angle_m * angle_m)))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.9e-17], N[(b * b), $MachinePrecision], If[LessEqual[a, 1.12e+173], N[(N[(angle$95$m * N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * N[(Pi * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.9000000000000001e-17

   1. Initial program 77.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 1.9000000000000001e-17 < a < 1.12e173

   1. Initial program 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 72.7%

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

   if 1.12e173 < a

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   7. Applied rewrites 45.6%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 56.2%

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

   12. Applied rewrites 71.6%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;b \cdot b\\ \mathbf{elif}\;a \leq 1.12 \cdot 10^{+173}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot angle\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+159}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(angle\_m \cdot angle\_m\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.3e+159)
   (* b b)
   (* a (* (* a 3.08641975308642e-5) (* PI (* PI (* angle_m angle_m)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.3e+159) {
		tmp = b * b;
	} else {
		tmp = a * ((a * 3.08641975308642e-5) * (((double) M_PI) * (((double) M_PI) * (angle_m * angle_m))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.3e+159) {
		tmp = b * b;
	} else {
		tmp = a * ((a * 3.08641975308642e-5) * (Math.PI * (Math.PI * (angle_m * angle_m))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 1.3e+159:
		tmp = b * b
	else:
		tmp = a * ((a * 3.08641975308642e-5) * (math.pi * (math.pi * (angle_m * angle_m))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.3e+159)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * Float64(Float64(a * 3.08641975308642e-5) * Float64(pi * Float64(pi * Float64(angle_m * angle_m)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 1.3e+159)
		tmp = b * b;
	else
		tmp = a * ((a * 3.08641975308642e-5) * (pi * (pi * (angle_m * angle_m))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.3e+159], N[(b * b), $MachinePrecision], N[(a * N[(N[(a * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * N[(Pi * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.3e159

   1. Initial program 77.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   if 1.3e159 < a

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   7. Applied rewrites 45.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 58.4%

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

   12. Applied rewrites 70.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{+159}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot angle\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.5% 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 80.9%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 54.1

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

5. Applied rewrites 54.1%

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

Reproduce

herbie shell --seed 2024219 
(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)))