ab-angle->ABCF A

Percentage Accurate: 79.8% → 79.7%

Time: 20.3s

Alternatives: 28

Speedup: 1.4×

• 36.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sqrt{\pi}}\\ {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\frac{t\_0}{\frac{1}{angle}} \cdot \frac{t\_0}{\frac{1}{\sqrt{\pi}}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \left({\left(\sqrt{\pi} \cdot \left(\pi \cdot \pi\right)\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt PI))))
   (+
    (pow
     (*
      a
      (sin
       (*
        0.005555555555555556
        (* (/ t_0 (/ 1.0 angle)) (/ t_0 (/ 1.0 (sqrt PI)))))))
     2.0)
    (pow
     (*
      b
      (cos
       (*
        (/ angle 180.0)
        (*
         (pow (* (sqrt PI) (* PI PI)) 0.3333333333333333)
         (cbrt (sqrt PI))))))
     2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = sqrt(sqrt(((double) M_PI)));
	return pow((a * sin((0.005555555555555556 * ((t_0 / (1.0 / angle)) * (t_0 / (1.0 / sqrt(((double) M_PI)))))))), 2.0) + pow((b * cos(((angle / 180.0) * (pow((sqrt(((double) M_PI)) * (((double) M_PI) * ((double) M_PI))), 0.3333333333333333) * cbrt(sqrt(((double) M_PI))))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sqrt(Math.sqrt(Math.PI));
	return Math.pow((a * Math.sin((0.005555555555555556 * ((t_0 / (1.0 / angle)) * (t_0 / (1.0 / Math.sqrt(Math.PI))))))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * (Math.pow((Math.sqrt(Math.PI) * (Math.PI * Math.PI)), 0.3333333333333333) * Math.cbrt(Math.sqrt(Math.PI)))))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = sqrt(sqrt(pi))
	return Float64((Float64(a * sin(Float64(0.005555555555555556 * Float64(Float64(t_0 / Float64(1.0 / angle)) * Float64(t_0 / Float64(1.0 / sqrt(pi))))))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * Float64((Float64(sqrt(pi) * Float64(pi * pi)) ^ 0.3333333333333333) * cbrt(sqrt(pi)))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(N[(t$95$0 / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * N[(N[Power[N[(N[Sqrt[Pi], $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision] * N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.2%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 81.3

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

4. Applied egg-rr 81.3%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. unpow1 N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-up N/A

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. pow1/2 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. lower-/.f64 N/A

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

6. Applied egg-rr 81.3%

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

   1. add-cbrt-cube N/A

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

   2. pow1/3 N/A

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

   3. lift-PI.f64 N/A

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

   4. rem-square-sqrt N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*r* N/A

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

   8. unpow-prod-down N/A

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

   9. pow1/3 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-cbrt.f64 81.4

       \[\leadsto {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\frac{\sqrt{\sqrt{\pi}}}{\frac{1}{angle}} \cdot \frac{\sqrt{\sqrt{\pi}}}{\frac{1}{\sqrt{\pi}}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \left({\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\sqrt{\pi}}}\right)\right)\right)}^{2} \]

8. Applied egg-rr 81.4%

   \[\leadsto {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\frac{\sqrt{\sqrt{\pi}}}{\frac{1}{angle}} \cdot \frac{\sqrt{\sqrt{\pi}}}{\frac{1}{\sqrt{\pi}}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \color{blue}{\left({\left(\left(\pi \cdot \pi\right) \cdot \sqrt{\pi}\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt{\pi}}\right)}\right)\right)}^{2} \]

9. Final simplification 81.4%

   \[\leadsto {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\frac{\sqrt{\sqrt{\pi}}}{\frac{1}{angle}} \cdot \frac{\sqrt{\sqrt{\pi}}}{\frac{1}{\sqrt{\pi}}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \left({\left(\sqrt{\pi} \cdot \left(\pi \cdot \pi\right)\right)}^{0.3333333333333333} \cdot \sqrt[3]{\sqrt{\pi}}\right)\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 79.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt PI))))
   (+
    (pow
     (*
      a
      (sin
       (*
        0.005555555555555556
        (* (/ t_0 (/ 1.0 angle)) (/ t_0 (/ 1.0 (sqrt PI)))))))
     2.0)
    (pow (* b (cos (* PI (/ angle 180.0)))) 2.0))))

C

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

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sqrt(Math.sqrt(Math.PI));
	return Math.pow((a * Math.sin((0.005555555555555556 * ((t_0 / (1.0 / angle)) * (t_0 / (1.0 / Math.sqrt(Math.PI))))))), 2.0) + Math.pow((b * Math.cos((Math.PI * (angle / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	t_0 = math.sqrt(math.sqrt(math.pi))
	return math.pow((a * math.sin((0.005555555555555556 * ((t_0 / (1.0 / angle)) * (t_0 / (1.0 / math.sqrt(math.pi))))))), 2.0) + math.pow((b * math.cos((math.pi * (angle / 180.0)))), 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	t_0 = sqrt(sqrt(pi));
	tmp = ((a * sin((0.005555555555555556 * ((t_0 / (1.0 / angle)) * (t_0 / (1.0 / sqrt(pi))))))) ^ 2.0) + ((b * cos((pi * (angle / 180.0)))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 81.3

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

4. Applied egg-rr 81.3%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. unpow1 N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-up N/A

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. pow1/2 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. lower-/.f64 N/A

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

6. Applied egg-rr 81.3%

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

7. Final simplification 81.3%

   \[\leadsto {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\frac{\sqrt{\sqrt{\pi}}}{\frac{1}{angle}} \cdot \frac{\sqrt{\sqrt{\pi}}}{\frac{1}{\sqrt{\pi}}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 3: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. lift-/.f64 N/A

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

   2. add-sqr-sqrt N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 81.3

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

4. Applied egg-rr 81.3%

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

5. Final simplification 81.3%

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

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \sin \left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* a (sin (* 0.005555555555555556 (/ PI (/ 1.0 angle))))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * sin((0.005555555555555556 * (((double) M_PI) / (1.0 / angle))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.cos((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((a * Math.sin((0.005555555555555556 * (Math.PI / (1.0 / angle))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.cos((math.pi * (angle / 180.0)))), 2.0) + math.pow((a * math.sin((0.005555555555555556 * (math.pi / (1.0 / angle))))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(b * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * sin(Float64(0.005555555555555556 * Float64(pi / Float64(1.0 / angle))))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * cos((pi * (angle / 180.0)))) ^ 2.0) + ((a * sin((0.005555555555555556 * (pi / (1.0 / angle))))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(Pi / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 81.3

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

4. Applied egg-rr 81.3%

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

5. Final simplification 81.3%

   \[\leadsto {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \sin \left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 5: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 81.3

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

4. Applied egg-rr 81.3%

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

5. Final simplification 81.3%

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

Alternative 6: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. div-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 81.2

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

4. Applied egg-rr 81.2%

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

5. Final simplification 81.2%

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

Alternative 7: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sqrt{\pi}}\\ {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(\frac{t\_0}{\frac{1}{angle}} \cdot \frac{t\_0}{\frac{1}{\sqrt{\pi}}}\right)\right)\right)}^{2} + {b}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt PI))))
   (+
    (pow
     (*
      a
      (sin
       (*
        0.005555555555555556
        (* (/ t_0 (/ 1.0 angle)) (/ t_0 (/ 1.0 (sqrt PI)))))))
     2.0)
    (pow b 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = sqrt(sqrt(((double) M_PI)));
	return pow((a * sin((0.005555555555555556 * ((t_0 / (1.0 / angle)) * (t_0 / (1.0 / sqrt(((double) M_PI)))))))), 2.0) + pow(b, 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sqrt(Math.sqrt(Math.PI));
	return Math.pow((a * Math.sin((0.005555555555555556 * ((t_0 / (1.0 / angle)) * (t_0 / (1.0 / Math.sqrt(Math.PI))))))), 2.0) + Math.pow(b, 2.0);
}

Python

def code(a, b, angle):
	t_0 = math.sqrt(math.sqrt(math.pi))
	return math.pow((a * math.sin((0.005555555555555556 * ((t_0 / (1.0 / angle)) * (t_0 / (1.0 / math.sqrt(math.pi))))))), 2.0) + math.pow(b, 2.0)

Julia

function code(a, b, angle)
	t_0 = sqrt(sqrt(pi))
	return Float64((Float64(a * sin(Float64(0.005555555555555556 * Float64(Float64(t_0 / Float64(1.0 / angle)) * Float64(t_0 / Float64(1.0 / sqrt(pi))))))) ^ 2.0) + (b ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = sqrt(sqrt(pi));
	tmp = ((a * sin((0.005555555555555556 * ((t_0 / (1.0 / angle)) * (t_0 / (1.0 / sqrt(pi))))))) ^ 2.0) + (b ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(N[(t$95$0 / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.2%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 81.3

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

4. Applied egg-rr 81.3%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
5. Step-by-step derivation

   1. lift-PI.f64 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. clear-num N/A

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

   8. un-div-inv N/A

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

   9. unpow1 N/A

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

   10. metadata-eval N/A

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

   11. pow-prod-up N/A

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

   12. div-inv N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. pow1/2 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. lower-/.f64 N/A

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

6. Applied egg-rr 81.3%

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

7. Taylor expanded in angle around 0

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

9. Simplified 81.2%

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

10. Final simplification 81.2%

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

Alternative 8: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. lift-/.f64 N/A

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

   2. add-sqr-sqrt N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 81.3

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

4. Applied egg-rr 81.3%

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

5. Taylor expanded in angle around 0

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

7. Simplified 81.2%

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

8. Final simplification 81.2%

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

Alternative 9: 79.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* 0.005555555555555556 (/ PI (/ 1.0 angle))))) 2.0)
  (pow b 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. associate-*l/ N/A

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

   4. div-inv N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 81.3

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

4. Applied egg-rr 81.3%

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

5. Taylor expanded in angle around 0

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

7. Simplified 81.2%

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

8. Final simplification 81.2%

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

Alternative 10: 79.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Taylor expanded in angle around 0

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

5. Simplified 81.1%

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

6. Final simplification 81.1%

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

Alternative 11: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ t_1 := angle \cdot \left(a \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot t\_1, \pi \cdot \pi, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right), a \cdot a, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0
         (*
          (* b b)
          (+ 0.5 (* 0.5 (cos (* 2.0 (* angle (* 0.005555555555555556 PI))))))))
        (t_1 (* angle (* a 0.005555555555555556))))
   (if (<= (/ angle 180.0) 1e-6)
     (fma (* t_1 t_1) (* PI PI) t_0)
     (fma
      (-
       0.5
       (*
        0.5
        (cos
         (* 2.0 (* angle (* 0.005555555555555556 (* (sqrt PI) (sqrt PI))))))))
      (* a a)
      t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = (b * b) * (0.5 + (0.5 * cos((2.0 * (angle * (0.005555555555555556 * ((double) M_PI)))))));
	double t_1 = angle * (a * 0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= 1e-6) {
		tmp = fma((t_1 * t_1), (((double) M_PI) * ((double) M_PI)), t_0);
	} else {
		tmp = fma((0.5 - (0.5 * cos((2.0 * (angle * (0.005555555555555556 * (sqrt(((double) M_PI)) * sqrt(((double) M_PI))))))))), (a * a), t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * pi)))))))
	t_1 = Float64(angle * Float64(a * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-6)
		tmp = fma(Float64(t_1 * t_1), Float64(pi * pi), t_0);
	else
		tmp = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * Float64(sqrt(pi) * sqrt(pi)))))))), Float64(a * a), t_0);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-6], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(0.005555555555555556 * N[(N[Sqrt[Pi], $MachinePrecision] * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999955e-7

   1. Initial program 87.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 84.3

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

   5. Simplified 84.3%

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

   7. Applied egg-rr 84.4%

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

   if 9.99999999999999955e-7 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 64.4%

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

      1. lift-/.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-sqrt.f64 64.7

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

   4. Applied egg-rr 64.7%

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

   6. Applied egg-rr 64.5%

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

      1. lift-PI.f64 64.5

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lower-*.f64 64.7

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

   8. Applied egg-rr 64.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right), \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \left(\sqrt{\pi} \cdot \sqrt{\pi}\right)\right)\right)\right), a \cdot a, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\\ t_1 := angle \cdot \left(a \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t\_1 \cdot t\_1, \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - t\_0, a \cdot a, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot \left(angle \cdot 2\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* 2.0 (* angle (* 0.005555555555555556 PI))))))
        (t_1 (* angle (* a 0.005555555555555556))))
   (if (<= (/ angle 180.0) 1e-6)
     (fma (* t_1 t_1) (* PI PI) (* (* b b) (+ 0.5 t_0)))
     (fma
      (- 0.5 t_0)
      (* a a)
      (*
       (* b b)
       (+ 0.5 (* 0.5 (cos (* 0.005555555555555556 (* PI (* angle 2.0)))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.5 * cos((2.0 * (angle * (0.005555555555555556 * ((double) M_PI)))));
	double t_1 = angle * (a * 0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= 1e-6) {
		tmp = fma((t_1 * t_1), (((double) M_PI) * ((double) M_PI)), ((b * b) * (0.5 + t_0)));
	} else {
		tmp = fma((0.5 - t_0), (a * a), ((b * b) * (0.5 + (0.5 * cos((0.005555555555555556 * (((double) M_PI) * (angle * 2.0))))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * pi)))))
	t_1 = Float64(angle * Float64(a * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-6)
		tmp = fma(Float64(t_1 * t_1), Float64(pi * pi), Float64(Float64(b * b) * Float64(0.5 + t_0)));
	else
		tmp = fma(Float64(0.5 - t_0), Float64(a * a), Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(0.005555555555555556 * Float64(pi * Float64(angle * 2.0))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-6], N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(0.005555555555555556 * N[(Pi * N[(angle * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999955e-7

   1. Initial program 87.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 84.3

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

   5. Simplified 84.3%

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

   7. Applied egg-rr 84.4%

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

   if 9.99999999999999955e-7 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 64.4%

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

      1. lift-/.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-sqrt.f64 64.7

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

   4. Applied egg-rr 64.7%

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

   6. Applied egg-rr 64.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 64.6

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

   8. Applied egg-rr 64.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right), \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), a \cdot a, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot \left(angle \cdot 2\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 77.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\\ t_1 := \left(b \cdot b\right) \cdot \left(0.5 + t\_0\right)\\ t_2 := angle \cdot \left(a \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t\_2 \cdot t\_2, \pi \cdot \pi, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - t\_0, a \cdot a, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* 2.0 (* angle (* 0.005555555555555556 PI))))))
        (t_1 (* (* b b) (+ 0.5 t_0)))
        (t_2 (* angle (* a 0.005555555555555556))))
   (if (<= (/ angle 180.0) 1e-6)
     (fma (* t_2 t_2) (* PI PI) t_1)
     (fma (- 0.5 t_0) (* a a) t_1))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.5 * cos((2.0 * (angle * (0.005555555555555556 * ((double) M_PI)))));
	double t_1 = (b * b) * (0.5 + t_0);
	double t_2 = angle * (a * 0.005555555555555556);
	double tmp;
	if ((angle / 180.0) <= 1e-6) {
		tmp = fma((t_2 * t_2), (((double) M_PI) * ((double) M_PI)), t_1);
	} else {
		tmp = fma((0.5 - t_0), (a * a), t_1);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * pi)))))
	t_1 = Float64(Float64(b * b) * Float64(0.5 + t_0))
	t_2 = Float64(angle * Float64(a * 0.005555555555555556))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-6)
		tmp = fma(Float64(t_2 * t_2), Float64(pi * pi), t_1);
	else
		tmp = fma(Float64(0.5 - t_0), Float64(a * a), t_1);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(b * b), $MachinePrecision] * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-6], N[(N[(t$95$2 * t$95$2), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(0.5 - t$95$0), $MachinePrecision] * N[(a * a), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999955e-7

   1. Initial program 87.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 84.3

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

   5. Simplified 84.3%

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

   7. Applied egg-rr 84.4%

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

   if 9.99999999999999955e-7 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 64.4%

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

      1. lift-/.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-sqrt.f64 64.7

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

   4. Applied egg-rr 64.7%

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

   6. Applied egg-rr 64.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right), \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right), a \cdot a, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 77.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(a \cdot 0.005555555555555556\right)\\ t_1 := \cos \left(\pi \cdot \left(2 \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot t\_0, \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(t\_1, -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, t\_1, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* a 0.005555555555555556)))
        (t_1 (cos (* PI (* 2.0 (* 0.005555555555555556 angle))))))
   (if (<= (/ angle 180.0) 1e-6)
     (fma
      (* t_0 t_0)
      (* PI PI)
      (*
       (* b b)
       (+ 0.5 (* 0.5 (cos (* 2.0 (* angle (* 0.005555555555555556 PI))))))))
     (fma (* a (fma t_1 -0.5 0.5)) a (* (* b b) (fma 0.5 t_1 0.5))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (a * 0.005555555555555556);
	double t_1 = cos((((double) M_PI) * (2.0 * (0.005555555555555556 * angle))));
	double tmp;
	if ((angle / 180.0) <= 1e-6) {
		tmp = fma((t_0 * t_0), (((double) M_PI) * ((double) M_PI)), ((b * b) * (0.5 + (0.5 * cos((2.0 * (angle * (0.005555555555555556 * ((double) M_PI)))))))));
	} else {
		tmp = fma((a * fma(t_1, -0.5, 0.5)), a, ((b * b) * fma(0.5, t_1, 0.5)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(a * 0.005555555555555556))
	t_1 = cos(Float64(pi * Float64(2.0 * Float64(0.005555555555555556 * angle))))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-6)
		tmp = fma(Float64(t_0 * t_0), Float64(pi * pi), Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * pi))))))));
	else
		tmp = fma(Float64(a * fma(t_1, -0.5, 0.5)), a, Float64(Float64(b * b) * fma(0.5, t_1, 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(Pi * N[(2.0 * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-6], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(t$95$1 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * N[(0.5 * t$95$1 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999955e-7

   1. Initial program 87.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 84.3

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

   5. Simplified 84.3%

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

   7. Applied egg-rr 84.4%

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

   if 9.99999999999999955e-7 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 64.4%

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

      1. lift-PI.f64 N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 64.5

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

   4. Applied egg-rr 64.5%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-/.f64 N/A

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

   6. Applied egg-rr 64.6%

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

   8. Applied egg-rr 64.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right), \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(2 \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(2 \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(a \cdot 0.005555555555555556\right)\\ t_1 := \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot t\_0, \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(0.5, t\_1, 0.5\right), b, \left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* a 0.005555555555555556)))
        (t_1 (cos (* PI (* angle 0.011111111111111112)))))
   (if (<= (/ angle 180.0) 1e-6)
     (fma
      (* t_0 t_0)
      (* PI PI)
      (*
       (* b b)
       (+ 0.5 (* 0.5 (cos (* 2.0 (* angle (* 0.005555555555555556 PI))))))))
     (fma (* b (fma 0.5 t_1 0.5)) b (* (* a a) (+ 0.5 (* -0.5 t_1)))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (a * 0.005555555555555556);
	double t_1 = cos((((double) M_PI) * (angle * 0.011111111111111112)));
	double tmp;
	if ((angle / 180.0) <= 1e-6) {
		tmp = fma((t_0 * t_0), (((double) M_PI) * ((double) M_PI)), ((b * b) * (0.5 + (0.5 * cos((2.0 * (angle * (0.005555555555555556 * ((double) M_PI)))))))));
	} else {
		tmp = fma((b * fma(0.5, t_1, 0.5)), b, ((a * a) * (0.5 + (-0.5 * t_1))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(a * 0.005555555555555556))
	t_1 = cos(Float64(pi * Float64(angle * 0.011111111111111112)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-6)
		tmp = fma(Float64(t_0 * t_0), Float64(pi * pi), Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * pi))))))));
	else
		tmp = fma(Float64(b * fma(0.5, t_1, 0.5)), b, Float64(Float64(a * a) * Float64(0.5 + Float64(-0.5 * t_1))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-6], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(0.5 * t$95$1 + 0.5), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(0.5 + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 9.99999999999999955e-7

   1. Initial program 87.6%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 84.3

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

   5. Simplified 84.3%

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

   7. Applied egg-rr 84.4%

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

   if 9.99999999999999955e-7 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 64.4%

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

   4. Applied egg-rr 43.4%

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

   6. Applied egg-rr 64.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right), \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right), b, \left(a \cdot a\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(a \cdot 0.005555555555555556\right)\\ \mathbf{if}\;a \leq 3.15 \cdot 10^{-80}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot t\_0, \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* a 0.005555555555555556))))
   (if (<= a 3.15e-80)
     (* b b)
     (fma
      (* t_0 t_0)
      (* PI PI)
      (*
       (* b b)
       (+ 0.5 (* 0.5 (cos (* 2.0 (* angle (* 0.005555555555555556 PI)))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (a * 0.005555555555555556);
	double tmp;
	if (a <= 3.15e-80) {
		tmp = b * b;
	} else {
		tmp = fma((t_0 * t_0), (((double) M_PI) * ((double) M_PI)), ((b * b) * (0.5 + (0.5 * cos((2.0 * (angle * (0.005555555555555556 * ((double) M_PI)))))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(a * 0.005555555555555556))
	tmp = 0.0
	if (a <= 3.15e-80)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(t_0 * t_0), Float64(pi * pi), Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * pi))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3.15e-80], N[(b * b), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.14999999999999983e-80

   1. Initial program 80.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.3

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

   5. Simplified 66.3%

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

   if 3.14999999999999983e-80 < a

   1. Initial program 82.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 79.5

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

   5. Simplified 79.5%

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

   7. Applied egg-rr 79.6%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.15 \cdot 10^{-80}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right) \cdot \left(angle \cdot \left(a \cdot 0.005555555555555556\right)\right), \pi \cdot \pi, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 67.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;a \leq 3.15 \cdot 10^{-80}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot t\_0, 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* a (* PI angle))))
   (if (<= a 3.15e-80)
     (* b b)
     (fma
      (* t_0 t_0)
      3.08641975308642e-5
      (*
       (* b b)
       (+ 0.5 (* 0.5 (cos (* 2.0 (* angle (* 0.005555555555555556 PI)))))))))))

C

double code(double a, double b, double angle) {
	double t_0 = a * (((double) M_PI) * angle);
	double tmp;
	if (a <= 3.15e-80) {
		tmp = b * b;
	} else {
		tmp = fma((t_0 * t_0), 3.08641975308642e-5, ((b * b) * (0.5 + (0.5 * cos((2.0 * (angle * (0.005555555555555556 * ((double) M_PI)))))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(a * Float64(pi * angle))
	tmp = 0.0
	if (a <= 3.15e-80)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(t_0 * t_0), 3.08641975308642e-5, Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * pi))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3.15e-80], N[(b * b), $MachinePrecision], N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 3.08641975308642e-5 + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.14999999999999983e-80

   1. Initial program 80.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.3

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

   5. Simplified 66.3%

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

   if 3.14999999999999983e-80 < a

   1. Initial program 82.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 79.5

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

   5. Simplified 79.5%

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

   7. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right), 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.15 \cdot 10^{-80}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right), 3.08641975308642 \cdot 10^{-5}, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 67.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.15 \cdot 10^{-80}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot angle\right) \cdot \left(a \cdot angle\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.15e-80)
   (* b b)
   (fma
    (* (* PI PI) 3.08641975308642e-5)
    (* (* a angle) (* a angle))
    (*
     (* 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 (a <= 3.15e-80) {
		tmp = b * b;
	} else {
		tmp = fma(((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5), ((a * angle) * (a * angle)), ((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 (a <= 3.15e-80)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(Float64(pi * pi) * 3.08641975308642e-5), Float64(Float64(a * angle) * Float64(a * angle)), Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * pi))))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.15e-80], N[(b * b), $MachinePrecision], N[(N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(N[(a * angle), $MachinePrecision] * N[(a * angle), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.14999999999999983e-80

   1. Initial program 80.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.3

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

   5. Simplified 66.3%

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

   if 3.14999999999999983e-80 < a

   1. Initial program 82.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 79.5

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

   5. Simplified 79.5%

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

   7. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot angle\right) \cdot \left(a \cdot angle\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 67.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;a \leq 3.15 \cdot 10^{-80}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(t\_0 \cdot t\_0\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* a (* PI angle))))
   (if (<= a 3.15e-80)
     (* b b)
     (+ (pow b 2.0) (* (* t_0 t_0) 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double t_0 = a * (((double) M_PI) * angle);
	double tmp;
	if (a <= 3.15e-80) {
		tmp = b * b;
	} else {
		tmp = pow(b, 2.0) + ((t_0 * t_0) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = a * (Math.PI * angle);
	double tmp;
	if (a <= 3.15e-80) {
		tmp = b * b;
	} else {
		tmp = Math.pow(b, 2.0) + ((t_0 * t_0) * 3.08641975308642e-5);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = a * (math.pi * angle)
	tmp = 0
	if a <= 3.15e-80:
		tmp = b * b
	else:
		tmp = math.pow(b, 2.0) + ((t_0 * t_0) * 3.08641975308642e-5)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(a * Float64(pi * angle))
	tmp = 0.0
	if (a <= 3.15e-80)
		tmp = Float64(b * b);
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(t_0 * t_0) * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = a * (pi * angle);
	tmp = 0.0;
	if (a <= 3.15e-80)
		tmp = b * b;
	else
		tmp = (b ^ 2.0) + ((t_0 * t_0) * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 3.15e-80], N[(b * b), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 3.14999999999999983e-80

   1. Initial program 80.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 66.3

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

   5. Simplified 66.3%

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

   if 3.14999999999999983e-80 < a

   1. Initial program 82.3%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 79.5

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

   5. Simplified 79.5%

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

      1. lift-PI.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow1 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. unpow1 N/A

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

      9. unpow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*r* N/A

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

   7. Applied egg-rr 79.6%

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

   8. Taylor expanded in angle around 0

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

   10. Simplified 79.6%

       \[\leadsto \left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) \cdot 3.08641975308642 \cdot 10^{-5} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.15 \cdot 10^{-80}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(\left(a \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-107}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.25e-107)
   (*
    angle
    (*
     (* a (* angle (* 0.005555555555555556 PI)))
     (* PI (* a 0.005555555555555556))))
   (if (<= b 5.5e+30)
     (fma
      (* angle angle)
      (*
       (* PI PI)
       (fma (* b b) -3.08641975308642e-5 (* (* a a) 3.08641975308642e-5)))
      (* b b))
     (* (* b b) (fma 0.5 (cos (* (* PI angle) 0.011111111111111112)) 0.5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.25e-107) {
		tmp = angle * ((a * (angle * (0.005555555555555556 * ((double) M_PI)))) * (((double) M_PI) * (a * 0.005555555555555556)));
	} else if (b <= 5.5e+30) {
		tmp = fma((angle * angle), ((((double) M_PI) * ((double) M_PI)) * fma((b * b), -3.08641975308642e-5, ((a * a) * 3.08641975308642e-5))), (b * b));
	} else {
		tmp = (b * b) * fma(0.5, cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.25e-107)
		tmp = Float64(angle * Float64(Float64(a * Float64(angle * Float64(0.005555555555555556 * pi))) * Float64(pi * Float64(a * 0.005555555555555556))));
	elseif (b <= 5.5e+30)
		tmp = fma(Float64(angle * angle), Float64(Float64(pi * pi) * fma(Float64(b * b), -3.08641975308642e-5, Float64(Float64(a * a) * 3.08641975308642e-5))), Float64(b * b));
	else
		tmp = Float64(Float64(b * b) * fma(0.5, cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.25e-107], N[(angle * N[(N[(a * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e+30], 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[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.24999999999999993e-107

   1. Initial program 80.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 36.6

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

   8. Simplified 36.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

   10. Applied egg-rr 41.6%

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

   if 1.24999999999999993e-107 < b < 5.50000000000000025e30

   1. Initial program 66.1%

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

      1. lift-PI.f64 N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 65.7

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

   4. Applied egg-rr 65.7%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-/.f64 N/A

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

   6. Applied egg-rr 66.2%

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

   7. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   9. Simplified 59.1%

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

   if 5.50000000000000025e30 < b

   1. Initial program 91.6%

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

   4. Applied egg-rr 52.3%

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

   5. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 83.1

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

   7. Simplified 83.1%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-107}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 51.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-107}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.25e-107)
   (*
    angle
    (*
     (* a (* angle (* 0.005555555555555556 PI)))
     (* PI (* a 0.005555555555555556))))
   (if (<= b 5.5e+30)
     (fma
      (* angle angle)
      (*
       (* PI PI)
       (fma (* b b) -3.08641975308642e-5 (* (* a a) 3.08641975308642e-5)))
      (* b b))
     (* (* b b) (fma 0.5 (cos (* angle (* PI 0.011111111111111112))) 0.5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.25e-107) {
		tmp = angle * ((a * (angle * (0.005555555555555556 * ((double) M_PI)))) * (((double) M_PI) * (a * 0.005555555555555556)));
	} else if (b <= 5.5e+30) {
		tmp = fma((angle * angle), ((((double) M_PI) * ((double) M_PI)) * fma((b * b), -3.08641975308642e-5, ((a * a) * 3.08641975308642e-5))), (b * b));
	} else {
		tmp = (b * b) * fma(0.5, cos((angle * (((double) M_PI) * 0.011111111111111112))), 0.5);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.25e-107)
		tmp = Float64(angle * Float64(Float64(a * Float64(angle * Float64(0.005555555555555556 * pi))) * Float64(pi * Float64(a * 0.005555555555555556))));
	elseif (b <= 5.5e+30)
		tmp = fma(Float64(angle * angle), Float64(Float64(pi * pi) * fma(Float64(b * b), -3.08641975308642e-5, Float64(Float64(a * a) * 3.08641975308642e-5))), Float64(b * b));
	else
		tmp = Float64(Float64(b * b) * fma(0.5, cos(Float64(angle * Float64(pi * 0.011111111111111112))), 0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.25e-107], N[(angle * N[(N[(a * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e+30], 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[(b * b), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.24999999999999993e-107

   1. Initial program 80.0%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 73.2

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

   5. Simplified 73.2%

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

   6. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 36.6

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

   8. Simplified 36.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

   10. Applied egg-rr 41.6%

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

   if 1.24999999999999993e-107 < b < 5.50000000000000025e30

   1. Initial program 66.1%

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

      1. lift-PI.f64 N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 65.7

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

   4. Applied egg-rr 65.7%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-/.f64 N/A

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

   6. Applied egg-rr 66.2%

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

   7. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   9. Simplified 59.1%

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

   if 5.50000000000000025e30 < b

   1. Initial program 91.6%

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

      1. lift-/.f64 N/A

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

      2. add-sqr-sqrt N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. div-inv N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-sqrt.f64 91.6

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

   4. Applied egg-rr 91.6%

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

   6. Applied egg-rr 78.1%

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

   7. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 83.1

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

   9. Simplified 83.1%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-107}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 51.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.62 \cdot 10^{-108}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.62e-108)
   (*
    angle
    (*
     (* a (* angle (* 0.005555555555555556 PI)))
     (* PI (* a 0.005555555555555556))))
   (if (<= b 2.7e+125)
     (fma
      (* angle angle)
      (*
       (* PI PI)
       (fma (* b b) -3.08641975308642e-5 (* (* a a) 3.08641975308642e-5)))
      (* b b))
     (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.62e-108) {
		tmp = angle * ((a * (angle * (0.005555555555555556 * ((double) M_PI)))) * (((double) M_PI) * (a * 0.005555555555555556)));
	} else if (b <= 2.7e+125) {
		tmp = fma((angle * angle), ((((double) M_PI) * ((double) M_PI)) * fma((b * b), -3.08641975308642e-5, ((a * a) * 3.08641975308642e-5))), (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.62e-108)
		tmp = Float64(angle * Float64(Float64(a * Float64(angle * Float64(0.005555555555555556 * pi))) * Float64(pi * Float64(a * 0.005555555555555556))));
	elseif (b <= 2.7e+125)
		tmp = fma(Float64(angle * angle), Float64(Float64(pi * pi) * fma(Float64(b * b), -3.08641975308642e-5, Float64(Float64(a * a) * 3.08641975308642e-5))), Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.62e-108], N[(angle * N[(N[(a * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e+125], 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[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.62e-108

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 73.6

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

   5. Simplified 73.6%

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

   6. Taylor expanded in angle around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-PI.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 36.7

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

   8. Simplified 36.7%

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

      1. lift-PI.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

   10. Applied egg-rr 41.8%

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

   if 1.62e-108 < b < 2.6999999999999999e125

   1. Initial program 73.4%

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

      1. lift-PI.f64 N/A

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

      2. clear-num N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 73.1

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

   4. Applied egg-rr 73.1%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \frac{\pi}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-PI.f64 N/A

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

      2. rem-square-sqrt N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

      8. un-div-inv N/A

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

      9. unpow1 N/A

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

      10. metadata-eval N/A

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

      11. pow-prod-up N/A

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

      12. div-inv N/A

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

      13. times-frac N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. pow1/2 N/A

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

      17. lower-sqrt.f64 N/A

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

      18. lower-/.f64 N/A

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

   6. Applied egg-rr 73.5%

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

   7. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   9. Simplified 60.3%

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

   if 2.6999999999999999e125 < b

   1. Initial program 93.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 93.7

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

   5. Simplified 93.7%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.62 \cdot 10^{-108}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 51.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.62 \cdot 10^{-108}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.62e-108)
   (*
    angle
    (*
     (* a (* angle (* 0.005555555555555556 PI)))
     (* PI (* a 0.005555555555555556))))
   (if (<= b 2.7e+125)
     (fma
      (* angle angle)
      (*
       PI
       (*
        PI
        (fma (* b b) -3.08641975308642e-5 (* (* a a) 3.08641975308642e-5))))
      (* b b))
     (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.62e-108) {
		tmp = angle * ((a * (angle * (0.005555555555555556 * ((double) M_PI)))) * (((double) M_PI) * (a * 0.005555555555555556)));
	} else if (b <= 2.7e+125) {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * fma((b * b), -3.08641975308642e-5, ((a * a) * 3.08641975308642e-5)))), (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.62e-108)
		tmp = Float64(angle * Float64(Float64(a * Float64(angle * Float64(0.005555555555555556 * pi))) * Float64(pi * Float64(a * 0.005555555555555556))));
	elseif (b <= 2.7e+125)
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * fma(Float64(b * b), -3.08641975308642e-5, Float64(Float64(a * a) * 3.08641975308642e-5)))), Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.62e-108], N[(angle * N[(N[(a * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(a * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.7e+125], 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[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.62e-108

   1. Initial program 80.4%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. lower-*.f64 73.6

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 73.6%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \cdot {angle}^{2} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {a}^{2}\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot {a}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      18. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      19. lower-*.f64 36.7

          \[\leadsto \left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 36.7%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)}\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)} \]

   10. Applied egg-rr 41.8%

       \[\leadsto \color{blue}{\left(\left(\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right) \cdot angle} \]

   if 1.62e-108 < b < 2.6999999999999999e125

   1. Initial program 73.4%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\frac{angle}{180}} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. add-sqr-sqrt N/A

         \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. associate-*r* N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{angle}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{angle}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lift-/.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\frac{angle}{180}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. div-inv N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. associate-*l* N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. lower-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      10. metadata-eval N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \left(\color{blue}{\frac{1}{180}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      11. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \left(\frac{1}{180} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      12. lower-sqrt.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \left(\frac{1}{180} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      13. lift-PI.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      14. lower-sqrt.f64 73.4

          \[\leadsto {\left(a \cdot \sin \left(\left(angle \cdot \left(0.005555555555555556 \cdot \sqrt{\pi}\right)\right) \cdot \color{blue}{\sqrt{\pi}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   4. Applied egg-rr 73.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(angle \cdot \left(0.005555555555555556 \cdot \sqrt{\pi}\right)\right) \cdot \sqrt{\pi}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {b}^{2}\right)} \]

   7. Simplified 60.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b \cdot b\right)} \]

   if 2.6999999999999999e125 < b

   1. Initial program 93.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 93.7

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Simplified 93.7%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.62 \cdot 10^{-108}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(\pi \cdot \left(a \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+125}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, -3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 63.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ \mathbf{if}\;a \leq 5.6 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* a (* angle (* 0.005555555555555556 PI)))))
   (if (<= a 5.6e+137) (* b b) (* t_0 t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = a * (angle * (0.005555555555555556 * ((double) M_PI)));
	double tmp;
	if (a <= 5.6e+137) {
		tmp = b * b;
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = a * (angle * (0.005555555555555556 * Math.PI));
	double tmp;
	if (a <= 5.6e+137) {
		tmp = b * b;
	} else {
		tmp = t_0 * t_0;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = a * (angle * (0.005555555555555556 * math.pi))
	tmp = 0
	if a <= 5.6e+137:
		tmp = b * b
	else:
		tmp = t_0 * t_0
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(a * Float64(angle * Float64(0.005555555555555556 * pi)))
	tmp = 0.0
	if (a <= 5.6e+137)
		tmp = Float64(b * b);
	else
		tmp = Float64(t_0 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = a * (angle * (0.005555555555555556 * pi));
	tmp = 0.0;
	if (a <= 5.6e+137)
		tmp = b * b;
	else
		tmp = t_0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(a * N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 5.6e+137], N[(b * b), $MachinePrecision], N[(t$95$0 * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 5.60000000000000002e137

   1. Initial program 78.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 63.5

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Simplified 63.5%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 5.60000000000000002e137 < a

   1. Initial program 97.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\left(\frac{1}{180} \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {\left(\frac{1}{180} \cdot \color{blue}{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(\left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. *-commutative N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\left(\frac{1}{180} \cdot a\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. *-commutative N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot a\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot a\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. lower-PI.f64 N/A

         \[\leadsto {\left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{180} \cdot a\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      10. lower-*.f64 97.6

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 97.6%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \cdot {angle}^{2} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {a}^{2}\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot {a}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      18. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      19. lower-*.f64 71.7

          \[\leadsto \left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 71.7%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)}\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)} \]

   10. Applied egg-rr 86.0%

       \[\leadsto \color{blue}{\left(\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right) \cdot \left(\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 62.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.6e+137)
   (* b b)
   (* (* (* PI angle) 3.08641975308642e-5) (* a (* a (* PI angle))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.6e+137) {
		tmp = b * b;
	} else {
		tmp = ((((double) M_PI) * angle) * 3.08641975308642e-5) * (a * (a * (((double) M_PI) * angle)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.6e+137) {
		tmp = b * b;
	} else {
		tmp = ((Math.PI * angle) * 3.08641975308642e-5) * (a * (a * (Math.PI * angle)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5.6e+137:
		tmp = b * b
	else:
		tmp = ((math.pi * angle) * 3.08641975308642e-5) * (a * (a * (math.pi * angle)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.6e+137)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(Float64(pi * angle) * 3.08641975308642e-5) * Float64(a * Float64(a * Float64(pi * angle))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.6e+137)
		tmp = b * b;
	else
		tmp = ((pi * angle) * 3.08641975308642e-5) * (a * (a * (pi * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.6e+137], N[(b * b), $MachinePrecision], N[(N[(N[(Pi * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision] * N[(a * N[(a * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.60000000000000002e137

   1. Initial program 78.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 63.5

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Simplified 63.5%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 5.60000000000000002e137 < a

   1. Initial program 97.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\left(\frac{1}{180} \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {\left(\frac{1}{180} \cdot \color{blue}{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(\left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. *-commutative N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\left(\frac{1}{180} \cdot a\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. *-commutative N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot a\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot a\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. lower-PI.f64 N/A

         \[\leadsto {\left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{180} \cdot a\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      10. lower-*.f64 97.6

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 97.6%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \cdot {angle}^{2} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {a}^{2}\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot {a}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      18. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      19. lower-*.f64 71.7

          \[\leadsto \left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 71.7%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)}\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)} \]

   10. Applied egg-rr 74.5%

       \[\leadsto \color{blue}{angle \cdot \left(angle \cdot \left(\left(a \cdot a\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)} \]

   12. Applied egg-rr 83.9%

       \[\leadsto \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot angle\right)\right) \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 62.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 5.6e+137)
   (* b b)
   (* angle (* (* a angle) (* a (* (* PI PI) 3.08641975308642e-5))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.6e+137) {
		tmp = b * b;
	} else {
		tmp = angle * ((a * angle) * (a * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 5.6e+137) {
		tmp = b * b;
	} else {
		tmp = angle * ((a * angle) * (a * ((Math.PI * Math.PI) * 3.08641975308642e-5)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 5.6e+137:
		tmp = b * b
	else:
		tmp = angle * ((a * angle) * (a * ((math.pi * math.pi) * 3.08641975308642e-5)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 5.6e+137)
		tmp = Float64(b * b);
	else
		tmp = Float64(angle * Float64(Float64(a * angle) * Float64(a * Float64(Float64(pi * pi) * 3.08641975308642e-5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 5.6e+137)
		tmp = b * b;
	else
		tmp = angle * ((a * angle) * (a * ((pi * pi) * 3.08641975308642e-5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 5.6e+137], N[(b * b), $MachinePrecision], N[(angle * N[(N[(a * angle), $MachinePrecision] * N[(a * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 5.60000000000000002e137

   1. Initial program 78.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 63.5

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Simplified 63.5%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 5.60000000000000002e137 < a

   1. Initial program 97.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\left(\frac{1}{180} \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {\left(\frac{1}{180} \cdot \color{blue}{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(\left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. *-commutative N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\left(\frac{1}{180} \cdot a\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. *-commutative N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot a\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot a\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. lower-PI.f64 N/A

         \[\leadsto {\left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{180} \cdot a\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      10. lower-*.f64 97.6

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 97.6%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \cdot {angle}^{2} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {a}^{2}\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot {a}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      18. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      19. lower-*.f64 71.7

          \[\leadsto \left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 71.7%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)}\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)} \]

   10. Applied egg-rr 74.5%

       \[\leadsto \color{blue}{angle \cdot \left(angle \cdot \left(\left(a \cdot a\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)} \]
   11. Step-by-step derivation

       1. lift-PI.f64 N/A

          \[\leadsto angle \cdot \left(angle \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       2. lift-PI.f64 N/A

          \[\leadsto angle \cdot \left(angle \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

       3. lift-*.f64 N/A

          \[\leadsto angle \cdot \left(angle \cdot \left(\left(a \cdot a\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       4. lift-*.f64 N/A

          \[\leadsto angle \cdot \left(angle \cdot \left(\left(a \cdot a\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto angle \cdot \left(angle \cdot \color{blue}{\left(a \cdot \left(a \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right) \]

       6. associate-*r* N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot a\right) \cdot \left(a \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot a\right) \cdot \left(a \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \]

       8. lower-*.f64 N/A

          \[\leadsto angle \cdot \left(\color{blue}{\left(angle \cdot a\right)} \cdot \left(a \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       9. lower-*.f64 83.8

          \[\leadsto angle \cdot \left(\left(angle \cdot a\right) \cdot \color{blue}{\left(a \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto angle \cdot \left(\left(angle \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto angle \cdot \left(\left(angle \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)}\right)\right) \]

       12. lower-*.f64 83.8

           \[\leadsto angle \cdot \left(\left(angle \cdot a\right) \cdot \left(a \cdot \color{blue}{\left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right)\right) \]

   12. Applied egg-rr 83.8%

       \[\leadsto angle \cdot \color{blue}{\left(\left(angle \cdot a\right) \cdot \left(a \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{+137}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 61.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{+145}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.15e+145)
   (* b b)
   (* a (* a (* PI (* PI (* 3.08641975308642e-5 (* angle angle))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.15e+145) {
		tmp = b * b;
	} else {
		tmp = a * (a * (((double) M_PI) * (((double) M_PI) * (3.08641975308642e-5 * (angle * angle)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.15e+145) {
		tmp = b * b;
	} else {
		tmp = a * (a * (Math.PI * (Math.PI * (3.08641975308642e-5 * (angle * angle)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 2.15e+145:
		tmp = b * b
	else:
		tmp = a * (a * (math.pi * (math.pi * (3.08641975308642e-5 * (angle * angle)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.15e+145)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * Float64(a * Float64(pi * Float64(pi * Float64(3.08641975308642e-5 * Float64(angle * angle))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.15e+145)
		tmp = b * b;
	else
		tmp = a * (a * (pi * (pi * (3.08641975308642e-5 * (angle * angle)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 2.15e+145], N[(b * b), $MachinePrecision], N[(a * N[(a * N[(Pi * N[(Pi * N[(3.08641975308642e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.14999999999999999e145

   1. Initial program 78.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{b}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \color{blue}{b \cdot b} \]

      2. lower-*.f64 63.5

         \[\leadsto \color{blue}{b \cdot b} \]

   5. Simplified 63.5%

      \[\leadsto \color{blue}{b \cdot b} \]

   if 2.14999999999999999e145 < a

   1. Initial program 97.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\left(\frac{1}{180} \cdot \left(a \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot angle\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. associate-*r* N/A

         \[\leadsto {\left(\frac{1}{180} \cdot \color{blue}{\left(\left(a \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto {\color{blue}{\left(\left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. *-commutative N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\left(\frac{1}{180} \cdot a\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. *-commutative N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot a\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. lower-*.f64 N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{180} \cdot a\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. lower-PI.f64 N/A

         \[\leadsto {\left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{180} \cdot a\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      10. lower-*.f64 97.6

          \[\leadsto {\left(angle \cdot \left(\pi \cdot \color{blue}{\left(0.005555555555555556 \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   5. Simplified 97.6%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Taylor expanded in angle around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{32400} \cdot \left({a}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

      2. associate-*r* N/A

         \[\leadsto \frac{1}{32400} \cdot \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \cdot {angle}^{2} \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {a}^{2}\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot {a}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      11. lower-PI.f64 N/A

          \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {a}^{2}\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      13. lower-PI.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {a}^{2}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      17. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)} \]

      18. unpow2 N/A

          \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      19. lower-*.f64 71.7

          \[\leadsto \left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   8. Simplified 71.7%

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
   9. Step-by-step derivation

      1. lift-PI.f64 N/A

         \[\leadsto \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      2. lift-PI.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)}\right) \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right)} \cdot \left(\left(angle \cdot angle\right) \cdot \frac{1}{32400}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right)} \]

      8. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(a \cdot a\right)\right)\right) \cdot \frac{1}{32400}\right) \cdot \left(angle \cdot angle\right)} \]

   10. Applied egg-rr 74.5%

       \[\leadsto \color{blue}{angle \cdot \left(angle \cdot \left(\left(a \cdot a\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)} \]

   11. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       4. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       6. lower-*.f64 N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       7. lower-*.f64 N/A

          \[\leadsto a \cdot \color{blue}{\left(a \cdot \left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       8. associate-*r* N/A

          \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \]

       9. unpow2 N/A

          \[\leadsto a \cdot \left(a \cdot \left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto a \cdot \left(a \cdot \left(\color{blue}{\left(\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \]

       13. lower-*.f64 N/A

           \[\leadsto a \cdot \left(a \cdot \left(\left(\color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto a \cdot \left(a \cdot \left(\left(\left(\frac{1}{32400} \cdot \color{blue}{\left(angle \cdot angle\right)}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       15. lower-*.f64 N/A

           \[\leadsto a \cdot \left(a \cdot \left(\left(\left(\frac{1}{32400} \cdot \color{blue}{\left(angle \cdot angle\right)}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       16. lower-PI.f64 N/A

           \[\leadsto a \cdot \left(a \cdot \left(\left(\left(\frac{1}{32400} \cdot \left(angle \cdot angle\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

       17. lower-PI.f64 79.0

           \[\leadsto a \cdot \left(a \cdot \left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \pi\right) \cdot \color{blue}{\pi}\right)\right) \]

   13. Simplified 79.0%

       \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\left(\left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right) \cdot \pi\right) \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.15 \cdot 10^{+145}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot angle\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 57.6% accurate, 74.7× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* b b))

C

double code(double a, double b, double angle) {
	return b * b;
}

Fortran

real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = b * b
end function

Java

public static double code(double a, double b, double angle) {
	return b * b;
}

Python

def code(a, b, angle):
	return b * b

Julia

function code(a, b, angle)
	return Float64(b * b)
end

MATLAB

function tmp = code(a, b, angle)
	tmp = b * b;
end

Wolfram

code[a_, b_, angle_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 81.2%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Add Preprocessing

3. Taylor expanded in angle around 0

   \[\leadsto \color{blue}{{b}^{2}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \color{blue}{b \cdot b} \]

   2. lower-*.f64 59.1

      \[\leadsto \color{blue}{b \cdot b} \]

5. Simplified 59.1%

   \[\leadsto \color{blue}{b \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024212 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))