ab-angle->ABCF C

Percentage Accurate: 79.5% → 79.5%

Time: 18.5s

Alternatives: 16

Speedup: 1.2×

• 36.4% 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: 79.5% 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: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\sqrt{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{{\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt{\pi}}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (cos
     (*
      (sqrt PI)
      (*
       (* angle 0.005555555555555556)
       (sqrt
        (*
         (pow (* (* PI PI) (sqrt PI)) 0.3333333333333333)
         (cbrt (sqrt PI))))))))
   2.0)
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((sqrt(((double) M_PI)) * ((angle * 0.005555555555555556) * sqrt((pow(((((double) M_PI) * ((double) M_PI)) * sqrt(((double) M_PI))), 0.3333333333333333) * cbrt(sqrt(((double) M_PI))))))))), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.sqrt(Math.PI) * ((angle * 0.005555555555555556) * Math.sqrt((Math.pow(((Math.PI * Math.PI) * Math.sqrt(Math.PI)), 0.3333333333333333) * Math.cbrt(Math.sqrt(Math.PI)))))))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(sqrt(pi) * Float64(Float64(angle * 0.005555555555555556) * sqrt(Float64((Float64(Float64(pi * pi) * sqrt(pi)) ^ 0.3333333333333333) * cbrt(sqrt(pi)))))))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[Sqrt[N[(N[Power[N[(N[(Pi * Pi), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. add-cube-cbrt N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. div-inv N/A

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

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

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

   14. metadata-eval N/A

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

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

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

   16. PI-lowering-PI.f64 77.4

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

4. Applied egg-rr 77.4%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. add-cube-cbrt N/A

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

   5. *-commutative N/A

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

   6. add-sqr-sqrt N/A

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

   7. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f64 77.4

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

6. Applied egg-rr 77.4%

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

   1. add-cbrt-cube N/A

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

   2. pow1/3 N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. unpow-prod-down N/A

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

   6. pow1/3 N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   17. PI-lowering-PI.f64 77.5

       \[\leadsto {\left(a \cdot \cos \left(\left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{{\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt{\color{blue}{\pi}}}}\right) \cdot \sqrt{\pi}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Applied egg-rr 77.5%

   \[\leadsto {\left(a \cdot \cos \left(\left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{\color{blue}{{\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt{\pi}}}}\right) \cdot \sqrt{\pi}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

9. Final simplification 77.5%

   \[\leadsto {\left(a \cdot \cos \left(\sqrt{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{{\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt{\pi}}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\sqrt{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt{\pi}\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (pow
   (* a (cos (* (sqrt PI) (* (* angle 0.005555555555555556) (sqrt PI)))))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * cos((sqrt(((double) M_PI)) * ((angle * 0.005555555555555556) * sqrt(((double) M_PI)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * Math.cos((Math.sqrt(Math.PI) * ((angle * 0.005555555555555556) * Math.sqrt(Math.PI))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0) + math.pow((a * math.cos((math.sqrt(math.pi) * ((angle * 0.005555555555555556) * math.sqrt(math.pi))))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * cos(Float64(sqrt(pi) * Float64(Float64(angle * 0.005555555555555556) * sqrt(pi))))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin((pi * (angle / 180.0)))) ^ 2.0) + ((a * cos((sqrt(pi) * ((angle * 0.005555555555555556) * sqrt(pi))))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[Sqrt[Pi], $MachinePrecision] * N[(N[(angle * 0.005555555555555556), $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. add-cube-cbrt N/A

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

   2. associate-*l* N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. div-inv N/A

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

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

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

   14. metadata-eval N/A

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

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

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

   16. PI-lowering-PI.f64 77.4

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

4. Applied egg-rr 77.4%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\sqrt[3]{\pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \sqrt[3]{\pi}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
5. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. add-cube-cbrt N/A

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

   5. *-commutative N/A

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

   6. add-sqr-sqrt N/A

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

   7. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   14. PI-lowering-PI.f64 77.4

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

6. Applied egg-rr 77.4%

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

7. Final simplification 77.4%

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

Alternative 3: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

   1. associate-*r/ N/A

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

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

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

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

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

   4. PI-lowering-PI.f64 77.4

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

4. Applied egg-rr 77.4%

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

5. Final simplification 77.4%

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

Alternative 4: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (sin (* 0.005555555555555556 (* angle PI)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((b * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((b * Math.sin((0.005555555555555556 * (angle * Math.PI)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.cos((math.pi * (angle / 180.0)))), 2.0) + math.pow((b * math.sin((0.005555555555555556 * (angle * math.pi)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(b * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * cos((pi * (angle / 180.0)))) ^ 2.0) + ((b * sin((0.005555555555555556 * (angle * pi)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. associate-*r/ N/A

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

   2. div-inv N/A

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

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

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

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

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

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

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

   6. metadata-eval 77.4

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

4. Applied egg-rr 77.4%

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

5. Final simplification 77.4%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 5: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (sin (* angle (* 0.005555555555555556 PI)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((b * sin((angle * (0.005555555555555556 * ((double) M_PI))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((b * Math.sin((angle * (0.005555555555555556 * Math.PI)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.cos((math.pi * (angle / 180.0)))), 2.0) + math.pow((b * math.sin((angle * (0.005555555555555556 * math.pi)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(b * sin(Float64(angle * Float64(0.005555555555555556 * pi)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * cos((pi * (angle / 180.0)))) ^ 2.0) + ((b * sin((angle * (0.005555555555555556 * pi)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

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

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

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

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

   9. metadata-eval 77.4

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

4. Applied egg-rr 77.4%

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

5. Final simplification 77.4%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 6: 79.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right) \cdot \left(a \cdot a\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
  (*
   (+ 0.5 (* 0.5 (cos (* 2.0 (* (* angle 0.005555555555555556) PI)))))
   (* a a))))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + ((0.5 + (0.5 * cos((2.0 * ((angle * 0.005555555555555556) * ((double) M_PI)))))) * (a * a));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0) + ((0.5 + (0.5 * Math.cos((2.0 * ((angle * 0.005555555555555556) * Math.PI))))) * (a * a));
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0) + ((0.5 + (0.5 * math.cos((2.0 * ((angle * 0.005555555555555556) * math.pi))))) * (a * a))

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(angle * 0.005555555555555556) * pi))))) * Float64(a * a)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin((pi * (angle / 180.0)))) ^ 2.0) + ((0.5 + (0.5 * cos((2.0 * ((angle * 0.005555555555555556) * pi))))) * (a * a));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

   1. *-commutative N/A

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

   2. unpow-prod-down N/A

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

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

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

4. Applied egg-rr 77.4%

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

5. Final simplification 77.4%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right) \cdot \left(a \cdot a\right) \]
6. Add Preprocessing

Alternative 7: 79.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 76.5

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

5. Simplified 76.5%

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

6. Final simplification 76.5%

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

Alternative 8: 65.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(angle \cdot \mathsf{fma}\left(\pi \cdot \left(\pi \cdot \pi\right), -2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), 0.005555555555555556 \cdot \pi\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 5000000.0)
   (fma
    (*
     (* b (* 0.005555555555555556 (* angle PI)))
     (*
      angle
      (fma
       (* PI (* PI PI))
       (* -2.8577960676726107e-8 (* angle angle))
       (* 0.005555555555555556 PI))))
    b
    (* (* a a) (fma 0.5 (cos (* (* angle PI) 0.011111111111111112)) 0.5)))
   (fma
    a
    a
    (*
     (* b b)
     (- 0.5 (* 0.5 (cos (* 2.0 (* (* angle 0.005555555555555556) PI)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 5000000.0) {
		tmp = fma(((b * (0.005555555555555556 * (angle * ((double) M_PI)))) * (angle * fma((((double) M_PI) * (((double) M_PI) * ((double) M_PI))), (-2.8577960676726107e-8 * (angle * angle)), (0.005555555555555556 * ((double) M_PI))))), b, ((a * a) * fma(0.5, cos(((angle * ((double) M_PI)) * 0.011111111111111112)), 0.5)));
	} else {
		tmp = fma(a, a, ((b * b) * (0.5 - (0.5 * cos((2.0 * ((angle * 0.005555555555555556) * ((double) M_PI))))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5000000.0)
		tmp = fma(Float64(Float64(b * Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(angle * fma(Float64(pi * Float64(pi * pi)), Float64(-2.8577960676726107e-8 * Float64(angle * angle)), Float64(0.005555555555555556 * pi)))), b, Float64(Float64(a * a) * fma(0.5, cos(Float64(Float64(angle * pi) * 0.011111111111111112)), 0.5)));
	else
		tmp = fma(a, a, Float64(Float64(b * b) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(angle * 0.005555555555555556) * pi)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5000000.0], N[(N[(N[(b * N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle * N[(N[(Pi * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(-2.8577960676726107e-8 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] + N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * a + N[(N[(b * b), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5e6

   1. Initial program 82.5%

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

      1. add-cube-cbrt N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. div-inv N/A

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

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

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

      14. metadata-eval N/A

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

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

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

      16. PI-lowering-PI.f64 82.5

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

   4. Applied egg-rr 82.5%

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

   6. Applied egg-rr 81.2%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\left(b \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\left(b \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

      3. PI-lowering-PI.f64 71.3

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

   9. Simplified 71.3%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \frac{-1}{34992000} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \color{blue}{\left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \mathsf{PI}\left(\right) + \left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)\right)}, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(angle \cdot \color{blue}{\left(\left(\frac{-1}{34992000} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2} + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(angle \cdot \left(\color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \frac{-1}{34992000}\right)} \cdot {angle}^{2} + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

       7. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(angle \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{3} \cdot \left(\frac{-1}{34992000} \cdot {angle}^{2}\right)} + \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

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

          \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(angle \cdot \color{blue}{\mathsf{fma}\left({\mathsf{PI}\left(\right)}^{3}, \frac{-1}{34992000} \cdot {angle}^{2}, \frac{1}{180} \cdot \mathsf{PI}\left(\right)\right)}\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

   12. Simplified 63.7%

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

   if 5e6 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 60.8%

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

      1. rem-exp-log N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. log-prod N/A

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

      6. exp-sum N/A

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

      7. rem-exp-log N/A

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

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

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

   4. Applied egg-rr 40.1%

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

   5. Taylor expanded in angle around 0

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

   7. Simplified 60.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5000000:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(angle \cdot \mathsf{fma}\left(\pi \cdot \left(\pi \cdot \pi\right), -2.8577960676726107 \cdot 10^{-8} \cdot \left(angle \cdot angle\right), 0.005555555555555556 \cdot \pi\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(angle \cdot 0.005555555555555556\right) \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \sin t\_0, b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, 1, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* angle 0.005555555555555556) PI)))
   (if (<= (/ angle 180.0) 5e-7)
     (fma
      (* (* b (* 0.005555555555555556 (* angle PI))) (sin t_0))
      b
      (* (* a a) (fma 0.5 1.0 0.5)))
     (fma a a (* (* b b) (- 0.5 (* 0.5 (cos (* 2.0 t_0)))))))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle * 0.005555555555555556) * ((double) M_PI);
	double tmp;
	if ((angle / 180.0) <= 5e-7) {
		tmp = fma(((b * (0.005555555555555556 * (angle * ((double) M_PI)))) * sin(t_0)), b, ((a * a) * fma(0.5, 1.0, 0.5)));
	} else {
		tmp = fma(a, a, ((b * b) * (0.5 - (0.5 * cos((2.0 * t_0))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle * 0.005555555555555556) * pi)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 5e-7)
		tmp = fma(Float64(Float64(b * Float64(0.005555555555555556 * Float64(angle * pi))) * sin(t_0)), b, Float64(Float64(a * a) * fma(0.5, 1.0, 0.5)));
	else
		tmp = fma(a, a, Float64(Float64(b * b) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 5e-7], N[(N[(N[(b * N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(0.5 * 1.0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * a + N[(N[(b * b), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.99999999999999977e-7

   1. Initial program 82.2%

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

      1. add-cube-cbrt N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. div-inv N/A

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

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

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

      14. metadata-eval N/A

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

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

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

      16. PI-lowering-PI.f64 82.1

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

   4. Applied egg-rr 82.1%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(\left(b \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]
   8. Step-by-step derivation

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

         \[\leadsto \mathsf{fma}\left(\left(b \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

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

         \[\leadsto \mathsf{fma}\left(\left(b \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(\frac{1}{2}, \cos \left(\left(\mathsf{PI}\left(\right) \cdot angle\right) \cdot \frac{1}{90}\right), \frac{1}{2}\right)\right) \]

      3. PI-lowering-PI.f64 71.4

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

   9. Simplified 71.4%

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

   10. Taylor expanded in angle around 0

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

   12. Simplified 70.7%

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

   if 4.99999999999999977e-7 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 63.7%

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

      1. rem-exp-log N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. log-prod N/A

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

      6. exp-sum N/A

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

      7. rem-exp-log N/A

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

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

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

   4. Applied egg-rr 44.5%

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

   5. Taylor expanded in angle around 0

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

   7. Simplified 61.8%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \sin \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, 1, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.4e+69)
   (fma
    (*
     (* angle PI)
     (* PI (fma b (* b 3.08641975308642e-5) (* (* a a) -3.08641975308642e-5))))
    angle
    (* a a))
   (* (* a a) (fma 0.5 (cos (* angle (* PI 0.011111111111111112))) 0.5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e+69) {
		tmp = fma(((angle * ((double) M_PI)) * (((double) M_PI) * fma(b, (b * 3.08641975308642e-5), ((a * a) * -3.08641975308642e-5)))), angle, (a * a));
	} else {
		tmp = (a * a) * fma(0.5, cos((angle * (((double) M_PI) * 0.011111111111111112))), 0.5);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.4e+69)
		tmp = fma(Float64(Float64(angle * pi) * Float64(pi * fma(b, Float64(b * 3.08641975308642e-5), Float64(Float64(a * a) * -3.08641975308642e-5)))), angle, Float64(a * a));
	else
		tmp = Float64(Float64(a * a) * fma(0.5, cos(Float64(angle * Float64(pi * 0.011111111111111112))), 0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.4e+69], N[(N[(N[(angle * Pi), $MachinePrecision] * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.39999999999999991e69

   1. Initial program 75.6%

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

   3. Taylor expanded in angle around 0

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

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

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

   5. Simplified 43.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

   7. Applied egg-rr 47.9%

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

   if 1.39999999999999991e69 < a

   1. Initial program 84.3%

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

      1. rem-exp-log N/A

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

      2. unpow2 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. log-prod N/A

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

      6. exp-sum N/A

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

      7. rem-exp-log N/A

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

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

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

   4. Applied egg-rr 79.5%

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

   5. Taylor expanded in a around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. +-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. PI-lowering-PI.f64 84.2

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

   7. Simplified 84.2%

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

Alternative 11: 52.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{-135}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.7e-135)
   (* angle (* (* angle PI) (* PI (* (* b b) 3.08641975308642e-5))))
   (if (<= a 2.4e+69)
     (fma
      (* angle angle)
      (*
       (* PI PI)
       (fma (* b b) 3.08641975308642e-5 (* (* a a) -3.08641975308642e-5)))
      (* a a))
     (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.7e-135) {
		tmp = angle * ((angle * ((double) M_PI)) * (((double) M_PI) * ((b * b) * 3.08641975308642e-5)));
	} else if (a <= 2.4e+69) {
		tmp = fma((angle * angle), ((((double) M_PI) * ((double) M_PI)) * fma((b * b), 3.08641975308642e-5, ((a * a) * -3.08641975308642e-5))), (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.7e-135)
		tmp = Float64(angle * Float64(Float64(angle * pi) * Float64(pi * Float64(Float64(b * b) * 3.08641975308642e-5))));
	elseif (a <= 2.4e+69)
		tmp = fma(Float64(angle * angle), Float64(Float64(pi * pi) * fma(Float64(b * b), 3.08641975308642e-5, Float64(Float64(a * a) * -3.08641975308642e-5))), Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.7e-135], N[(angle * N[(N[(angle * Pi), $MachinePrecision] * N[(Pi * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.4e+69], N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5 + N[(N[(a * a), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 3.6999999999999997e-135

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

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

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

   5. Simplified 37.0%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 36.0

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

   8. Simplified 36.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 44.0

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

   10. Applied egg-rr 44.0%

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

   if 3.6999999999999997e-135 < a < 2.4000000000000002e69

   1. Initial program 76.0%

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

      1. add-cube-cbrt N/A

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

      2. associate-*l* N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. div-inv N/A

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

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

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

      14. metadata-eval N/A

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

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

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

      16. PI-lowering-PI.f64 76.0

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

   4. Applied egg-rr 76.0%

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

   5. Taylor expanded in angle around 0

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

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

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

   7. Simplified 64.4%

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

   if 2.4000000000000002e69 < a

   1. Initial program 84.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 81.8

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

   5. Simplified 81.8%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{-135}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.4 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.96 \cdot 10^{-134}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.96e-134)
   (* angle (* (* angle PI) (* PI (* (* b b) 3.08641975308642e-5))))
   (if (<= a 4.8e+70)
     (fma
      (* angle angle)
      (*
       PI
       (*
        PI
        (fma (* b b) 3.08641975308642e-5 (* (* a a) -3.08641975308642e-5))))
      (* a a))
     (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.96e-134) {
		tmp = angle * ((angle * ((double) M_PI)) * (((double) M_PI) * ((b * b) * 3.08641975308642e-5)));
	} else if (a <= 4.8e+70) {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * fma((b * b), 3.08641975308642e-5, ((a * a) * -3.08641975308642e-5)))), (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.96e-134)
		tmp = Float64(angle * Float64(Float64(angle * pi) * Float64(pi * Float64(Float64(b * b) * 3.08641975308642e-5))));
	elseif (a <= 4.8e+70)
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * fma(Float64(b * b), 3.08641975308642e-5, Float64(Float64(a * a) * -3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.96e-134], N[(angle * N[(N[(angle * Pi), $MachinePrecision] * N[(Pi * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 4.8e+70], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5 + N[(N[(a * a), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.9600000000000001e-134

   1. Initial program 75.4%

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

   3. Taylor expanded in angle around 0

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

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

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

   5. Simplified 37.0%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 36.0

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

   8. Simplified 36.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 44.0

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

   10. Applied egg-rr 44.0%

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

   if 1.9600000000000001e-134 < a < 4.79999999999999974e70

   1. Initial program 76.0%

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

   3. Taylor expanded in angle around 0

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

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

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

   5. Simplified 64.4%

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

   if 4.79999999999999974e70 < a

   1. Initial program 84.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 81.8

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

   5. Simplified 81.8%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.96 \cdot 10^{-134}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.5% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8.6 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 8.6e+71)
   (fma
    (*
     (* angle PI)
     (* PI (fma b (* b 3.08641975308642e-5) (* (* a a) -3.08641975308642e-5))))
    angle
    (* a a))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 8.6e+71) {
		tmp = fma(((angle * ((double) M_PI)) * (((double) M_PI) * fma(b, (b * 3.08641975308642e-5), ((a * a) * -3.08641975308642e-5)))), angle, (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 8.6e+71)
		tmp = fma(Float64(Float64(angle * pi) * Float64(pi * fma(b, Float64(b * 3.08641975308642e-5), Float64(Float64(a * a) * -3.08641975308642e-5)))), angle, Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 8.6e+71], N[(N[(N[(angle * Pi), $MachinePrecision] * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8.59999999999999967e71

   1. Initial program 75.6%

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

   3. Taylor expanded in angle around 0

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

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

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

   5. Simplified 43.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

   7. Applied egg-rr 47.9%

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

   if 8.59999999999999967e71 < a

   1. Initial program 84.3%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 81.8

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

   5. Simplified 81.8%

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

Alternative 14: 61.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{+83}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9.4e+83)
   (* a a)
   (* angle (* (* angle PI) (* PI (* (* b b) 3.08641975308642e-5))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.4e+83) {
		tmp = a * a;
	} else {
		tmp = angle * ((angle * ((double) M_PI)) * (((double) M_PI) * ((b * b) * 3.08641975308642e-5)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.4e+83) {
		tmp = a * a;
	} else {
		tmp = angle * ((angle * Math.PI) * (Math.PI * ((b * b) * 3.08641975308642e-5)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 9.4e+83:
		tmp = a * a
	else:
		tmp = angle * ((angle * math.pi) * (math.pi * ((b * b) * 3.08641975308642e-5)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 9.4e+83)
		tmp = Float64(a * a);
	else
		tmp = Float64(angle * Float64(Float64(angle * pi) * Float64(pi * Float64(Float64(b * b) * 3.08641975308642e-5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 9.4e+83)
		tmp = a * a;
	else
		tmp = angle * ((angle * pi) * (pi * ((b * b) * 3.08641975308642e-5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 9.4e+83], N[(a * a), $MachinePrecision], N[(angle * N[(N[(angle * Pi), $MachinePrecision] * N[(Pi * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.3999999999999997e83

   1. Initial program 75.6%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 60.2

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

   5. Simplified 60.2%

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

   if 9.3999999999999997e83 < b

   1. Initial program 87.0%

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

   3. Taylor expanded in angle around 0

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

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

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

   5. Simplified 51.5%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 56.6

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

   8. Simplified 56.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 64.1

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

   10. Applied egg-rr 64.1%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.4 \cdot 10^{+83}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(angle \cdot \pi\right) \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.2% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.32 \cdot 10^{+112}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.32e+112)
   (* a a)
   (* (* angle angle) (* PI (* PI (* b (* b 3.08641975308642e-5)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.32e+112) {
		tmp = a * a;
	} else {
		tmp = (angle * angle) * (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.32e+112) {
		tmp = a * a;
	} else {
		tmp = (angle * angle) * (Math.PI * (Math.PI * (b * (b * 3.08641975308642e-5))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.32e+112:
		tmp = a * a
	else:
		tmp = (angle * angle) * (math.pi * (math.pi * (b * (b * 3.08641975308642e-5))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.32e+112)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(angle * angle) * Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.32e+112)
		tmp = a * a;
	else
		tmp = (angle * angle) * (pi * (pi * (b * (b * 3.08641975308642e-5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.32e+112], N[(a * a), $MachinePrecision], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.32e112

   1. Initial program 75.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 60.0

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

   5. Simplified 60.0%

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

   if 1.32e112 < b

   1. Initial program 90.6%

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

   3. Taylor expanded in angle around 0

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

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

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

   5. Simplified 54.2%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

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

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

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

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

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

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

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

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

      15. *-commutative N/A

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

      16. unpow2 N/A

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

      17. associate-*l* N/A

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

      18. *-commutative N/A

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

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

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

      20. *-commutative N/A

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

      21. *-lowering-*.f64 59.8

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

   8. Simplified 59.8%

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

Alternative 16: 56.4% accurate, 74.7× speedup

Math

\[\begin{array}{l} \\ a \cdot a \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* a a))

C

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

Fortran

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

Java

public static double code(double a, double b, double angle) {
	return a * a;
}

Python

def code(a, b, angle):
	return a * a

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 56.6

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

5. Simplified 56.6%

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

Reproduce

herbie shell --seed 2024207 
(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)))