ab-angle->ABCF A

Percentage Accurate: 80.0% → 80.0%

Time: 16.1s

Alternatives: 12

Speedup: 1.2×

• 36.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 80.0% 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: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left({\pi}^{0.16666666666666666} \cdot \left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (sin
     (*
      (pow PI 0.16666666666666666)
      (*
       (* 0.005555555555555556 (* angle (pow PI 0.6666666666666666)))
       (pow PI 0.16666666666666666)))))
   2.0)
  (pow (* b (cos (* PI (/ angle 180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * sin((pow(((double) M_PI), 0.16666666666666666) * ((0.005555555555555556 * (angle * pow(((double) M_PI), 0.6666666666666666))) * pow(((double) M_PI), 0.16666666666666666))))), 2.0) + pow((b * cos((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin((Math.pow(Math.PI, 0.16666666666666666) * ((0.005555555555555556 * (angle * Math.pow(Math.PI, 0.6666666666666666))) * Math.pow(Math.PI, 0.16666666666666666))))), 2.0) + Math.pow((b * Math.cos((Math.PI * (angle / 180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin((math.pow(math.pi, 0.16666666666666666) * ((0.005555555555555556 * (angle * math.pow(math.pi, 0.6666666666666666))) * math.pow(math.pi, 0.16666666666666666))))), 2.0) + math.pow((b * math.cos((math.pi * (angle / 180.0)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64((pi ^ 0.16666666666666666) * Float64(Float64(0.005555555555555556 * Float64(angle * (pi ^ 0.6666666666666666))) * (pi ^ 0.16666666666666666))))) ^ 2.0) + (Float64(b * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((pi ^ 0.16666666666666666) * ((0.005555555555555556 * (angle * (pi ^ 0.6666666666666666))) * (pi ^ 0.16666666666666666))))) ^ 2.0) + ((b * cos((pi * (angle / 180.0)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[Power[Pi, 0.16666666666666666], $MachinePrecision] * N[(N[(0.005555555555555556 * N[(angle * N[Power[Pi, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 0.16666666666666666], $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 77.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. cbrt-prod N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

5. Final simplification 77.6%

   \[\leadsto {\left(a \cdot \sin \left({\pi}^{0.16666666666666666} \cdot \left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 2: 80.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* a (sin (* angle (* 0.005555555555555556 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((angle * (0.005555555555555556 * ((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((angle * (0.005555555555555556 * 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((angle * (0.005555555555555556 * 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(angle * Float64(0.005555555555555556 * pi)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * cos((pi * (angle / 180.0)))) ^ 2.0) + ((a * sin((angle * (0.005555555555555556 * 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[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. div-inv N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 77.6

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

4. Applied rewrites 77.6%

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

5. Final simplification 77.6%

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

Alternative 3: 79.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. associate-*r* N/A

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

   5. add-sqr-sqrt N/A

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

   6. cbrt-prod N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

4. Applied rewrites 77.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   2. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   3. associate-*l* N/A

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   4. lift-*.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. pow2 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \color{blue}{{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}^{2}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   6. lift-pow.f64 N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}}^{2}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   7. pow-pow N/A

      \[\leadsto {\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} \cdot 2\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   8. metadata-eval N/A

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

   9. pow1/3 N/A

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

   10. lift-PI.f64 N/A

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

   11. associate-*l* N/A

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

6. Applied rewrites 77.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. /-rgt-identity N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/l/ N/A

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

   7. associate-*r/ N/A

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

8. Applied rewrites 77.6%

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

9. Taylor expanded in angle around 0

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

11. Applied rewrites 77.6%

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

12. Final simplification 77.6%

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

Alternative 4: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(0.005555555555555556 \cdot angle\right)\\ t_1 := a \cdot \sin t\_0\\ \mathsf{fma}\left(t\_1, t\_1, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))) (t_1 (* a (sin t_0))))
   (fma t_1 t_1 (* (* b b) (+ 0.5 (* 0.5 (cos (* 2.0 t_0))))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	double t_1 = a * sin(t_0);
	return fma(t_1, t_1, ((b * b) * (0.5 + (0.5 * cos((2.0 * t_0))))));
}

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	t_1 = Float64(a * sin(t_0))
	return fma(t_1, t_1, Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_0))))))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$1 * t$95$1 + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 77.5%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 77.5

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lower-*.f64 N/A

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

   8. metadata-eval 76.5

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 77.6

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

4. Applied rewrites 77.5%

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

5. Final simplification 77.5%

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

Alternative 5: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.5%

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

3. Taylor expanded in angle around 0

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

5. Applied rewrites 77.5%

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

6. Final simplification 77.5%

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

Alternative 6: 73.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sqrt{\pi}} \cdot \sqrt{\pi \cdot \sqrt{\pi}}\\ \mathbf{if}\;\frac{angle}{180} \leq 10^{-22}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) + {\left(b \cdot \cos \left(\frac{t\_0}{\frac{180}{angle}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot 1\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot t\_0\right)\right)\right) \cdot \left(a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (sqrt (sqrt PI)) (sqrt (* PI (sqrt PI))))))
   (if (<= (/ angle 180.0) 1e-22)
     (+
      (* (* a (* a (* angle angle))) (* PI (* PI 3.08641975308642e-5)))
      (pow (* b (cos (/ t_0 (/ 180.0 angle)))) 2.0))
     (+
      (pow (* b 1.0) 2.0)
      (*
       (- 0.5 (* 0.5 (cos (* 2.0 (* (* 0.005555555555555556 angle) t_0)))))
       (* a a))))))

C

double code(double a, double b, double angle) {
	double t_0 = sqrt(sqrt(((double) M_PI))) * sqrt((((double) M_PI) * sqrt(((double) M_PI))));
	double tmp;
	if ((angle / 180.0) <= 1e-22) {
		tmp = ((a * (a * (angle * angle))) * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))) + pow((b * cos((t_0 / (180.0 / angle)))), 2.0);
	} else {
		tmp = pow((b * 1.0), 2.0) + ((0.5 - (0.5 * cos((2.0 * ((0.005555555555555556 * angle) * t_0))))) * (a * a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.sqrt(Math.sqrt(Math.PI)) * Math.sqrt((Math.PI * Math.sqrt(Math.PI)));
	double tmp;
	if ((angle / 180.0) <= 1e-22) {
		tmp = ((a * (a * (angle * angle))) * (Math.PI * (Math.PI * 3.08641975308642e-5))) + Math.pow((b * Math.cos((t_0 / (180.0 / angle)))), 2.0);
	} else {
		tmp = Math.pow((b * 1.0), 2.0) + ((0.5 - (0.5 * Math.cos((2.0 * ((0.005555555555555556 * angle) * t_0))))) * (a * a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.sqrt(math.sqrt(math.pi)) * math.sqrt((math.pi * math.sqrt(math.pi)))
	tmp = 0
	if (angle / 180.0) <= 1e-22:
		tmp = ((a * (a * (angle * angle))) * (math.pi * (math.pi * 3.08641975308642e-5))) + math.pow((b * math.cos((t_0 / (180.0 / angle)))), 2.0)
	else:
		tmp = math.pow((b * 1.0), 2.0) + ((0.5 - (0.5 * math.cos((2.0 * ((0.005555555555555556 * angle) * t_0))))) * (a * a))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(sqrt(sqrt(pi)) * sqrt(Float64(pi * sqrt(pi))))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1e-22)
		tmp = Float64(Float64(Float64(a * Float64(a * Float64(angle * angle))) * Float64(pi * Float64(pi * 3.08641975308642e-5))) + (Float64(b * cos(Float64(t_0 / Float64(180.0 / angle)))) ^ 2.0));
	else
		tmp = Float64((Float64(b * 1.0) ^ 2.0) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(0.005555555555555556 * angle) * t_0))))) * Float64(a * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = sqrt(sqrt(pi)) * sqrt((pi * sqrt(pi)));
	tmp = 0.0;
	if ((angle / 180.0) <= 1e-22)
		tmp = ((a * (a * (angle * angle))) * (pi * (pi * 3.08641975308642e-5))) + ((b * cos((t_0 / (180.0 / angle)))) ^ 2.0);
	else
		tmp = ((b * 1.0) ^ 2.0) + ((0.5 - (0.5 * cos((2.0 * ((0.005555555555555556 * angle) * t_0))))) * (a * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-22], N[(N[(N[(a * N[(a * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Cos[N[(t$95$0 / N[(180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(b * 1.0), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(0.005555555555555556 * angle), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e-22

   1. Initial program 84.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 84.0

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

   4. Applied rewrites 84.0%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{\frac{1}{4}}}}{\frac{180}{angle}}\right)\right)}^{2} \]

      11. metadata-eval N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{\left(\frac{1}{6} + \frac{1}{12}\right)}}}{\frac{180}{angle}}\right)\right)}^{2} \]

      12. metadata-eval N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} + \color{blue}{\frac{\frac{1}{6}}{2}}\right)}}{\frac{180}{angle}}\right)\right)}^{2} \]

      13. pow-prod-up N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)}}{\frac{180}{angle}}\right)\right)}^{2} \]

      14. lift-pow.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)}{\frac{180}{angle}}\right)\right)}^{2} \]

      15. lower-*.f64 N/A

          \[\leadsto {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{\color{blue}{\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)}}{\frac{180}{angle}}\right)\right)}^{2} \]

   6. Applied rewrites 84.1%

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

   7. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. unpow2 N/A

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

      16. associate-*l* N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. lower-*.f64 N/A

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

      19. lower-PI.f64 N/A

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

      20. distribute-rgt-neg-in N/A

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

      21. metadata-eval N/A

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

   9. Applied rewrites 74.2%

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

   if 1e-22 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 60.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right) \cdot \left(a \cdot a\right)} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lift-PI.f64 N/A

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

      4. add-sqr-sqrt N/A

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

      5. associate-*r* N/A

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

      6. sqrt-prod N/A

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

      7. lift-PI.f64 N/A

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

      8. pow1/2 N/A

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

      9. sqrt-pow1 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{\left(\frac{1}{6} + \frac{1}{12}\right)}}\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      12. metadata-eval N/A

          \[\leadsto \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} + \color{blue}{\frac{\frac{1}{6}}{2}}\right)}\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      13. pow-prod-up N/A

          \[\leadsto \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)}\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      14. lift-pow.f64 N/A

          \[\leadsto \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\color{blue}{{\mathsf{PI}\left(\right)}^{\frac{1}{6}}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      15. lower-*.f64 N/A

          \[\leadsto \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\left(\frac{\frac{1}{6}}{2}\right)}\right)\right)}\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   6. Applied rewrites 60.6%

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

   7. Taylor expanded in angle around 0

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

   9. Applied rewrites 60.5%

      \[\leadsto \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \left(\sqrt{\pi \cdot \sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 10^{-22}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) + {\left(b \cdot \cos \left(\frac{\sqrt{\sqrt{\pi}} \cdot \sqrt{\pi \cdot \sqrt{\pi}}}{\frac{180}{angle}}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot 1\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(0.005555555555555556 \cdot angle\right) \cdot \left(\sqrt{\sqrt{\pi}} \cdot \sqrt{\pi \cdot \sqrt{\pi}}\right)\right)\right)\right) \cdot \left(a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\\ \mathbf{if}\;a \leq 1.05 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 - t\_0\right), \left(b \cdot b\right) \cdot \left(0.5 + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos (* 2.0 (* angle (* 0.005555555555555556 PI)))))))
   (if (<= a 1.05e+166)
     (fma a (* a (- 0.5 t_0)) (* (* b b) (+ 0.5 t_0)))
     (* (* PI (* PI 3.08641975308642e-5)) (* (* a angle) (* a angle))))))

C

double code(double a, double b, double angle) {
	double t_0 = 0.5 * cos((2.0 * (angle * (0.005555555555555556 * ((double) M_PI)))));
	double tmp;
	if (a <= 1.05e+166) {
		tmp = fma(a, (a * (0.5 - t_0)), ((b * b) * (0.5 + t_0)));
	} else {
		tmp = (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * ((a * angle) * (a * angle));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(0.5 * cos(Float64(2.0 * Float64(angle * Float64(0.005555555555555556 * pi)))))
	tmp = 0.0
	if (a <= 1.05e+166)
		tmp = fma(a, Float64(a * Float64(0.5 - t_0)), Float64(Float64(b * b) * Float64(0.5 + t_0)));
	else
		tmp = Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(Float64(a * angle) * Float64(a * angle)));
	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]}, If[LessEqual[a, 1.05e+166], N[(a * N[(a * N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(N[(a * angle), $MachinePrecision] * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.05e166

   1. Initial program 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*r* N/A

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

      5. add-sqr-sqrt N/A

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

      6. cbrt-prod N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 75.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. lift-*.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right)} \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. pow2 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \color{blue}{{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}^{2}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. lift-pow.f64 N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\color{blue}{\left({\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)}}^{2}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. pow-pow N/A

         \[\leadsto {\left(a \cdot \sin \left(\left(\frac{1}{180} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} \cdot 2\right)}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. metadata-eval N/A

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

      9. pow1/3 N/A

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

      10. lift-PI.f64 N/A

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

      11. associate-*l* N/A

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

   6. Applied rewrites 75.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. /-rgt-identity N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. associate-*r/ N/A

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

   8. Applied rewrites 75.3%

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

   10. Applied rewrites 69.0%

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

   if 1.05e166 < a

   1. Initial program 96.1%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 40.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 82.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.05 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;a \leq 1.05 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(t\_0, -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, t\_0, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cos (* PI (* angle 0.011111111111111112)))))
   (if (<= a 1.05e+166)
     (fma (* a (fma t_0 -0.5 0.5)) a (* (* b b) (fma 0.5 t_0 0.5)))
     (* (* PI (* PI 3.08641975308642e-5)) (* (* a angle) (* a angle))))))

C

double code(double a, double b, double angle) {
	double t_0 = cos((((double) M_PI) * (angle * 0.011111111111111112)));
	double tmp;
	if (a <= 1.05e+166) {
		tmp = fma((a * fma(t_0, -0.5, 0.5)), a, ((b * b) * fma(0.5, t_0, 0.5)));
	} else {
		tmp = (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * ((a * angle) * (a * angle));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cos(Float64(pi * Float64(angle * 0.011111111111111112)))
	tmp = 0.0
	if (a <= 1.05e+166)
		tmp = fma(Float64(a * fma(t_0, -0.5, 0.5)), a, Float64(Float64(b * b) * fma(0.5, t_0, 0.5)));
	else
		tmp = Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(Float64(a * angle) * Float64(a * angle)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[a, 1.05e+166], N[(N[(a * N[(t$95$0 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(N[(b * b), $MachinePrecision] * N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(N[(a * angle), $MachinePrecision] * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 1.05e166

   1. Initial program 75.3%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*r* N/A

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

      5. add-sqr-sqrt N/A

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

      6. cbrt-prod N/A

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

      7. associate-*r* N/A

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

      8. lower-*.f64 N/A

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

   4. Applied rewrites 75.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\left(0.005555555555555556 \cdot \left(angle \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   6. Applied rewrites 68.9%

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

   if 1.05e166 < a

   1. Initial program 96.1%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 40.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.8%

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

   10. Applied rewrites 82.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.05 \cdot 10^{+166}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, \left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.7% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.4e+154)
   (* b b)
   (* (* PI (* PI 3.08641975308642e-5)) (* (* a angle) (* a angle)))))

C

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

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e+154) {
		tmp = b * b;
	} else {
		tmp = (Math.PI * (Math.PI * 3.08641975308642e-5)) * ((a * angle) * (a * angle));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.4e+154:
		tmp = b * b
	else:
		tmp = (math.pi * (math.pi * 3.08641975308642e-5)) * ((a * angle) * (a * angle))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.4e+154)
		tmp = b * b;
	else
		tmp = (pi * (pi * 3.08641975308642e-5)) * ((a * angle) * (a * angle));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.4e+154], N[(b * b), $MachinePrecision], N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(N[(a * angle), $MachinePrecision] * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4e154

   1. Initial program 74.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 1.4e154 < a

   1. Initial program 96.5%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 57.4%

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

   10. Applied rewrites 78.0%

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

4. Final simplification 60.9%

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

Alternative 10: 62.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(a \cdot \left(angle \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.4e+154)
   (* b b)
   (* (* PI (* PI 3.08641975308642e-5)) (* a (* angle (* a angle))))))

C

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

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e+154) {
		tmp = b * b;
	} else {
		tmp = (Math.PI * (Math.PI * 3.08641975308642e-5)) * (a * (angle * (a * angle)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.4e+154:
		tmp = b * b
	else:
		tmp = (math.pi * (math.pi * 3.08641975308642e-5)) * (a * (angle * (a * angle)))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.4e+154)
		tmp = b * b;
	else
		tmp = (pi * (pi * 3.08641975308642e-5)) * (a * (angle * (a * angle)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.4e+154], N[(b * b), $MachinePrecision], N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(a * N[(angle * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4e154

   1. Initial program 74.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 1.4e154 < a

   1. Initial program 96.5%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 57.4%

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

   10. Applied rewrites 77.9%

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

4. Final simplification 60.9%

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

Alternative 11: 61.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.4e+154)
   (* b b)
   (* (* a (* a (* angle angle))) (* PI (* PI 3.08641975308642e-5)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.4e+154) {
		tmp = b * b;
	} else {
		tmp = (a * (a * (angle * angle))) * (((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 <= 1.4e+154) {
		tmp = b * b;
	} else {
		tmp = (a * (a * (angle * angle))) * (Math.PI * (Math.PI * 3.08641975308642e-5));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.4e+154:
		tmp = b * b
	else:
		tmp = (a * (a * (angle * angle))) * (math.pi * (math.pi * 3.08641975308642e-5))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.4e+154)
		tmp = b * b;
	else
		tmp = (a * (a * (angle * angle))) * (pi * (pi * 3.08641975308642e-5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.4e+154], N[(b * b), $MachinePrecision], N[(N[(a * N[(a * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.4e154

   1. Initial program 74.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 1.4e154 < a

   1. Initial program 96.5%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 57.4%

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

Alternative 12: 56.9% 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 77.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 55.2

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

5. Applied rewrites 55.2%

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

Reproduce

herbie shell --seed 2024234 
(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)))