ab-angle->ABCF C

Percentage Accurate: 79.1% → 79.0%

Time: 5.0s

Alternatives: 10

Speedup: N/A×

• 36.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\_m\right) \cdot \left(\left(angle\_m \cdot b\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)}^{2}, b \cdot b, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 3.2e-18)
   (+
    (* a a)
    (* (* (* (* b PI) angle_m) (* (* angle_m b) PI)) 3.08641975308642e-5))
   (fma
    (pow (sin (* (* angle_m PI) 0.005555555555555556)) 2.0)
    (* b b)
    (* a a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.2e-18) {
		tmp = (a * a) + ((((b * ((double) M_PI)) * angle_m) * ((angle_m * b) * ((double) M_PI))) * 3.08641975308642e-5);
	} else {
		tmp = fma(pow(sin(((angle_m * ((double) M_PI)) * 0.005555555555555556)), 2.0), (b * b), (a * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 3.2e-18)
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(Float64(b * pi) * angle_m) * Float64(Float64(angle_m * b) * pi)) * 3.08641975308642e-5));
	else
		tmp = fma((sin(Float64(Float64(angle_m * pi) * 0.005555555555555556)) ^ 2.0), Float64(b * b), Float64(a * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[angle$95$m, 3.2e-18], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(N[(b * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(N[(angle$95$m * b), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(b * b), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.1999999999999999e-18

   1. Initial program 87.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 84.0

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

   8. Applied rewrites 84.0%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

          \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      12. lift-*.f64 84.0

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

   10. Applied rewrites 84.0%

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

       1. lift-*.f64 N/A

          \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 84.1

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

   12. Applied rewrites 84.1%

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

   if 3.1999999999999999e-18 < angle

   1. Initial program 61.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. pow-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-PI.f64 N/A

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

      13. lift-sin.f64 61.7

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

      14. lift-PI.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-PI.f64 61.7

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

   7. Applied rewrites 61.7%

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

   8. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 62.1

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

   10. Applied rewrites 62.1%

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

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-pow.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. unpow-prod-down N/A

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

       7. lower-fma.f64 N/A

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

   12. Applied rewrites 62.1%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left({\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}^{2}, b \cdot b, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a + {\left(b \cdot \frac{1}{{\sin \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{-1}}\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (* a a)
  (pow
   (* b (/ 1.0 (pow (sin (* angle_m (* 0.005555555555555556 PI))) -1.0)))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return (a * a) + pow((b * (1.0 / pow(sin((angle_m * (0.005555555555555556 * ((double) M_PI)))), -1.0))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return (a * a) + Math.pow((b * (1.0 / Math.pow(Math.sin((angle_m * (0.005555555555555556 * Math.PI))), -1.0))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return (a * a) + math.pow((b * (1.0 / math.pow(math.sin((angle_m * (0.005555555555555556 * math.pi))), -1.0))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(Float64(a * a) + (Float64(b * Float64(1.0 / (sin(Float64(angle_m * Float64(0.005555555555555556 * pi))) ^ -1.0))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a * a) + ((b * (1.0 / (sin((angle_m * (0.005555555555555556 * pi))) ^ -1.0))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[(1.0 / N[Power[N[Sin[N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. pow-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-sin.f64 81.4

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

   14. lift-PI.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-PI.f64 81.4

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

7. Applied rewrites 81.4%

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

8. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 81.5

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

10. Applied rewrites 81.5%

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

    1. lift-*.f64 N/A

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

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. associate-*l* N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lift-PI.f64 81.5

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

12. Applied rewrites 81.5%

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

13. Final simplification 81.5%

    \[\leadsto a \cdot a + {\left(b \cdot \frac{1}{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{-1}}\right)}^{2} \]
14. Add Preprocessing

Alternative 3: 79.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a + {\left(b \cdot \frac{1}{\frac{1}{\sin \left(\left(angle\_m \cdot \pi\right) \cdot 0.005555555555555556\right)}}\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (* a a)
  (pow
   (* b (/ 1.0 (/ 1.0 (sin (* (* angle_m PI) 0.005555555555555556)))))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return (a * a) + pow((b * (1.0 / (1.0 / sin(((angle_m * ((double) M_PI)) * 0.005555555555555556))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return (a * a) + Math.pow((b * (1.0 / (1.0 / Math.sin(((angle_m * Math.PI) * 0.005555555555555556))))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return (a * a) + math.pow((b * (1.0 / (1.0 / math.sin(((angle_m * math.pi) * 0.005555555555555556))))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(Float64(a * a) + (Float64(b * Float64(1.0 / Float64(1.0 / sin(Float64(Float64(angle_m * pi) * 0.005555555555555556))))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a * a) + ((b * (1.0 / (1.0 / sin(((angle_m * pi) * 0.005555555555555556))))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(a * a), $MachinePrecision] + N[Power[N[(b * N[(1.0 / N[(1.0 / N[Sin[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. pow-neg N/A

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

   8. lower-/.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-sin.f64 81.4

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

   14. lift-PI.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-PI.f64 81.4

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

7. Applied rewrites 81.4%

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

8. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 81.5

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

10. Applied rewrites 81.5%

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

    1. lift-pow.f64 N/A

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

    2. metadata-eval N/A

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

    3. pow-neg N/A

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

    4. metadata-eval N/A

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

    5. pow-flip N/A

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

    6. lift-pow.f64 N/A

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

    7. lift-/.f64 N/A

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

    8. lower-/.f64 81.5

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

    9. lift-/.f64 N/A

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

    10. lift-pow.f64 N/A

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

    11. pow-flip N/A

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

    12. metadata-eval N/A

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

12. Applied rewrites 81.5%

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

13. Final simplification 81.5%

    \[\leadsto a \cdot a + {\left(b \cdot \frac{1}{\frac{1}{\sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}}\right)}^{2} \]
14. Add Preprocessing

Alternative 4: 79.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

6. Final simplification 81.4%

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

Alternative 5: 79.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 0.00036:\\ \;\;\;\;a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\_m\right) \cdot \left(\left(angle\_m \cdot b\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle\_m}{180} \cdot \pi\right)\right)\right) \cdot b, b, a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 0.00036)
   (+
    (* a a)
    (* (* (* (* b PI) angle_m) (* (* angle_m b) PI)) 3.08641975308642e-5))
   (fma
    (* (- 0.5 (* 0.5 (cos (* 2.0 (* (/ angle_m 180.0) PI))))) b)
    b
    (* a a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 0.00036) {
		tmp = (a * a) + ((((b * ((double) M_PI)) * angle_m) * ((angle_m * b) * ((double) M_PI))) * 3.08641975308642e-5);
	} else {
		tmp = fma(((0.5 - (0.5 * cos((2.0 * ((angle_m / 180.0) * ((double) M_PI)))))) * b), b, (a * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 0.00036)
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(Float64(b * pi) * angle_m) * Float64(Float64(angle_m * b) * pi)) * 3.08641975308642e-5));
	else
		tmp = fma(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(angle_m / 180.0) * pi))))) * b), b, Float64(a * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[angle$95$m, 0.00036], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(N[(b * Pi), $MachinePrecision] * angle$95$m), $MachinePrecision] * N[(N[(angle$95$m * b), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.60000000000000023e-4

   1. Initial program 87.4%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow-prod-down N/A

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

      4. pow-prod-down N/A

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

      5. lower-pow.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 84.2

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

   8. Applied rewrites 84.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-PI.f64 N/A

          \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

      12. lift-*.f64 84.2

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

   10. Applied rewrites 84.2%

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

       1. lift-*.f64 N/A

          \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-*r* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift-PI.f64 84.2

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

   12. Applied rewrites 84.2%

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

   if 3.60000000000000023e-4 < angle

   1. Initial program 59.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 60.4

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

   5. Applied rewrites 60.4%

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

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow-prod-down N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

   7. Applied rewrites 60.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. unpow2 N/A

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

      7. sqr-sin-a N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-PI.f64 60.4

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

   9. Applied rewrites 60.4%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \pi\right)\right)\right)} \cdot b, b, a \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.00036:\\ \;\;\;\;a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \pi\right)\right)\right) \cdot b, b, a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 11.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\_m\right) \cdot \left(\left(angle\_m \cdot b\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \end{array} \]

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return (a * a) + ((((b * ((double) M_PI)) * angle_m) * ((angle_m * b) * ((double) M_PI))) * 3.08641975308642e-5);
}

Java

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

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return (a * a) + ((((b * math.pi) * angle_m) * ((angle_m * b) * math.pi)) * 3.08641975308642e-5)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(Float64(a * a) + Float64(Float64(Float64(Float64(b * pi) * angle_m) * Float64(Float64(angle_m * b) * pi)) * 3.08641975308642e-5))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a * a) + ((((b * pi) * angle_m) * ((angle_m * b) * pi)) * 3.08641975308642e-5);
end

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow-prod-down N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 77.7

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

8. Applied rewrites 77.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

       \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

   12. lift-*.f64 77.7

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

10. Applied rewrites 77.7%

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

    1. lift-*.f64 N/A

       \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lift-PI.f64 77.8

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

12. Applied rewrites 77.8%

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

13. Final simplification 77.8%

    \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]
14. Add Preprocessing

Alternative 7: 73.9% accurate, 11.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(angle\_m \cdot b\right) \cdot \pi\\ a \cdot a + \left(t\_0 \cdot t\_0\right) \cdot 3.08641975308642 \cdot 10^{-5} \end{array} \end{array} \]

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (angle_m * b) * ((double) M_PI);
	return (a * a) + ((t_0 * t_0) * 3.08641975308642e-5);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (angle_m * b) * Math.PI;
	return (a * a) + ((t_0 * t_0) * 3.08641975308642e-5);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (angle_m * b) * math.pi
	return (a * a) + ((t_0 * t_0) * 3.08641975308642e-5)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(angle_m * b) * pi)
	return Float64(Float64(a * a) + Float64(Float64(t_0 * t_0) * 3.08641975308642e-5))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = (angle_m * b) * pi;
	tmp = (a * a) + ((t_0 * t_0) * 3.08641975308642e-5);
end

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow-prod-down N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 77.7

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

8. Applied rewrites 77.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

       \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

   12. lift-*.f64 77.7

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

10. Applied rewrites 77.7%

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

    1. lift-*.f64 N/A

       \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

    2. lift-PI.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. lower-*.f64 N/A

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

    7. lower-*.f64 N/A

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

    8. lift-PI.f64 77.8

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

    9. lift-*.f64 N/A

       \[\leadsto a \cdot a + \left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

    10. lift-PI.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. *-commutative N/A

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

    13. associate-*r* N/A

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

    14. lower-*.f64 N/A

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

    15. lower-*.f64 N/A

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

    16. lift-PI.f64 77.8

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

12. Applied rewrites 77.8%

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

13. Final simplification 77.8%

    \[\leadsto a \cdot a + \left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(angle \cdot b\right) \cdot \pi\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} \]
14. Add Preprocessing

Alternative 8: 74.0% accurate, 11.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \left(angle\_m \cdot b\right) \cdot \pi\\ a \cdot a + t\_0 \cdot \left(t\_0 \cdot 3.08641975308642 \cdot 10^{-5}\right) \end{array} \end{array} \]

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = (angle_m * b) * ((double) M_PI);
	return (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = (angle_m * b) * Math.PI;
	return (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = (angle_m * b) * math.pi
	return (a * a) + (t_0 * (t_0 * 3.08641975308642e-5))

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(Float64(angle_m * b) * pi)
	return Float64(Float64(a * a) + Float64(t_0 * Float64(t_0 * 3.08641975308642e-5)))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = (angle_m * b) * pi;
	tmp = (a * a) + (t_0 * (t_0 * 3.08641975308642e-5));
end

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

6. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. pow-prod-down N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 77.7

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

8. Applied rewrites 77.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-PI.f64 N/A

       \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

   12. lift-*.f64 77.7

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

10. Applied rewrites 77.7%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

       \[\leadsto a \cdot a + \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \left(\left(b \cdot \pi\right) \cdot angle\right)\right) \cdot \frac{1}{32400} \]

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-PI.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. associate-*r* N/A

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

    10. lower-*.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lift-PI.f64 N/A

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

    13. lower-*.f64 77.8

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

    14. lift-*.f64 N/A

        \[\leadsto a \cdot a + \left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(\left(b \cdot \pi\right) \cdot angle\right) \cdot \frac{1}{32400}\right) \]

    15. lift-PI.f64 N/A

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

    16. lift-*.f64 N/A

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

    17. *-commutative N/A

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

    18. associate-*r* N/A

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

    19. lower-*.f64 N/A

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

    20. lower-*.f64 N/A

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

    21. lift-PI.f64 77.8

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

12. Applied rewrites 77.8%

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

13. Final simplification 77.8%

    \[\leadsto a \cdot a + \left(\left(angle \cdot b\right) \cdot \pi\right) \cdot \left(\left(\left(angle \cdot b\right) \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \]
14. Add Preprocessing

Alternative 9: 69.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \mathsf{fma}\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot angle\_m\right)\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot b\right), b, a \cdot a\right) \end{array} \]

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return fma(((3.08641975308642e-5 * (angle_m * angle_m)) * ((((double) M_PI) * ((double) M_PI)) * b)), b, (a * a));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return fma(Float64(Float64(3.08641975308642e-5 * Float64(angle_m * angle_m)) * Float64(Float64(pi * pi) * b)), b, Float64(a * a))
end

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 81.4

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

5. Applied rewrites 81.4%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. +-commutative N/A

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

   9. unpow-prod-down N/A

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

   10. *-commutative N/A

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

   11. pow2 N/A

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

   12. associate-*r* N/A

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

7. Applied rewrites 77.2%

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

8. Taylor expanded in angle around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 73.5

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

10. Applied rewrites 73.5%

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

Alternative 10: 56.7% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

angle_m =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 61.2

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

5. Applied rewrites 61.2%

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

Reproduce

herbie shell --seed 2025077 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))