ab-angle->ABCF C

Percentage Accurate: 80.0% → 79.9%

Time: 14.4s

Alternatives: 10

Speedup: N/A×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 80.6

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

4. Applied rewrites 80.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 80.6

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

6. Applied rewrites 80.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. /-rgt-identity N/A

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

   4. associate-/r/ N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/l/ N/A

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

   12. lift-*.f64 N/A

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

   13. lower-/.f64 80.6

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

8. Applied rewrites 80.6%

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

Alternative 2: 79.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 80.6

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

4. Applied rewrites 80.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 80.6

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

6. Applied rewrites 80.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. /-rgt-identity N/A

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

   4. associate-/r/ N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-*l/ N/A

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

   8. metadata-eval N/A

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

   9. lower-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/l/ N/A

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

   12. lift-*.f64 N/A

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

   13. lower-/.f64 80.6

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

8. Applied rewrites 80.6%

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

9. Taylor expanded in angle around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 80.6

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

11. Applied rewrites 80.6%

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

12. Final simplification 80.6%

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

Alternative 3: 79.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 80.6

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

4. Applied rewrites 80.6%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 80.6

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

6. Applied rewrites 80.6%

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

7. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 80.6

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

9. Applied rewrites 80.6%

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

Alternative 4: 64.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 1.22 \cdot 10^{-194}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 0.5, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= (/ angle 180.0) 1.22e-194)
   (* a a)
   (if (<= (/ angle 180.0) 1e-9)
     (fma
      (* angle angle)
      (* PI (* PI (* b (* b 3.08641975308642e-5))))
      (* a a))
     (fma
      b
      (* b 0.5)
      (* (* a a) (fma 0.5 (cos (* (* PI angle) 0.011111111111111112)) 0.5))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if ((angle / 180.0) <= 1.22e-194) {
		tmp = a * a;
	} else if ((angle / 180.0) <= 1e-9) {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5)))), (a * a));
	} else {
		tmp = fma(b, (b * 0.5), ((a * a) * fma(0.5, cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (Float64(angle / 180.0) <= 1.22e-194)
		tmp = Float64(a * a);
	elseif (Float64(angle / 180.0) <= 1e-9)
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = fma(b, Float64(b * 0.5), Float64(Float64(a * a) * fma(0.5, cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1.22e-194], N[(a * a), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-9], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(b * N[(b * 0.5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.2200000000000001e-194

   1. Initial program 83.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.0

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

   5. Applied rewrites 57.0%

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

   if 1.2200000000000001e-194 < (/.f64 angle #s(literal 180 binary64)) < 1.00000000000000006e-9

   1. Initial program 99.7%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 91.7%

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

   if 1.00000000000000006e-9 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 65.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 65.2

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

   4. Applied rewrites 65.2%

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

   6. Applied rewrites 55.2%

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

   7. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 64.2

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

   9. Applied rewrites 64.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 1.22 \cdot 10^{-194}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b \cdot 0.5, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot b\right) \cdot \left(angle \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8.2e-129)
   (* (* a a) (fma 0.5 (cos (* (* PI angle) 0.011111111111111112)) 0.5))
   (if (<= b 1.1e+156)
     (fma
      (* angle angle)
      (* PI (* PI (* b (* b 3.08641975308642e-5))))
      (* a a))
     (* (* angle b) (* angle (* b (* 3.08641975308642e-5 (* PI PI))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.2e-129) {
		tmp = (a * a) * fma(0.5, cos(((((double) M_PI) * angle) * 0.011111111111111112)), 0.5);
	} else if (b <= 1.1e+156) {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5)))), (a * a));
	} else {
		tmp = (angle * b) * (angle * (b * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8.2e-129)
		tmp = Float64(Float64(a * a) * fma(0.5, cos(Float64(Float64(pi * angle) * 0.011111111111111112)), 0.5));
	elseif (b <= 1.1e+156)
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(Float64(angle * b) * Float64(angle * Float64(b * Float64(3.08641975308642e-5 * Float64(pi * pi)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8.2e-129], N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(Pi * angle), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e+156], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(angle * b), $MachinePrecision] * N[(angle * N[(b * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 8.1999999999999999e-129

   1. Initial program 79.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

   6. Applied rewrites 71.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, 0.5, \left(a \cdot a\right) \cdot \left(0.5 \cdot \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right)\right)\right)} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 62.6

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

   9. Applied rewrites 62.6%

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

   if 8.1999999999999999e-129 < b < 1.10000000000000002e156

   1. Initial program 71.3%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 61.8%

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

   if 1.10000000000000002e156 < b

   1. Initial program 97.4%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 77.0%

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

   10. Applied rewrites 88.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot b\right) \cdot \left(angle \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.3% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot b\right) \cdot \left(angle \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 8.2e-129)
   (* a a)
   (if (<= b 1.1e+156)
     (fma
      (* angle angle)
      (* PI (* PI (* b (* b 3.08641975308642e-5))))
      (* a a))
     (* (* angle b) (* angle (* b (* 3.08641975308642e-5 (* PI PI))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 8.2e-129) {
		tmp = a * a;
	} else if (b <= 1.1e+156) {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5)))), (a * a));
	} else {
		tmp = (angle * b) * (angle * (b * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 8.2e-129)
		tmp = Float64(a * a);
	elseif (b <= 1.1e+156)
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(Float64(angle * b) * Float64(angle * Float64(b * Float64(3.08641975308642e-5 * Float64(pi * pi)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 8.2e-129], N[(a * a), $MachinePrecision], If[LessEqual[b, 1.1e+156], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(angle * b), $MachinePrecision] * N[(angle * N[(b * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 8.1999999999999999e-129

   1. Initial program 79.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if 8.1999999999999999e-129 < b < 1.10000000000000002e156

   1. Initial program 71.3%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 61.8%

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

   if 1.10000000000000002e156 < b

   1. Initial program 97.4%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 77.0%

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

   10. Applied rewrites 88.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot b\right) \cdot \left(angle \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.3% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{if}\;b \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, b \cdot t\_0, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot b\right) \cdot \left(angle \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* b (* 3.08641975308642e-5 (* PI PI)))))
   (if (<= b 8.2e-129)
     (* a a)
     (if (<= b 1.1e+156)
       (fma (* angle angle) (* b t_0) (* a a))
       (* (* angle b) (* angle t_0))))))

C

double code(double a, double b, double angle) {
	double t_0 = b * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI)));
	double tmp;
	if (b <= 8.2e-129) {
		tmp = a * a;
	} else if (b <= 1.1e+156) {
		tmp = fma((angle * angle), (b * t_0), (a * a));
	} else {
		tmp = (angle * b) * (angle * t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(b * Float64(3.08641975308642e-5 * Float64(pi * pi)))
	tmp = 0.0
	if (b <= 8.2e-129)
		tmp = Float64(a * a);
	elseif (b <= 1.1e+156)
		tmp = fma(Float64(angle * angle), Float64(b * t_0), Float64(a * a));
	else
		tmp = Float64(Float64(angle * b) * Float64(angle * t_0));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(b * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 8.2e-129], N[(a * a), $MachinePrecision], If[LessEqual[b, 1.1e+156], N[(N[(angle * angle), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(angle * b), $MachinePrecision] * N[(angle * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if b < 8.1999999999999999e-129

   1. Initial program 79.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   if 8.1999999999999999e-129 < b < 1.10000000000000002e156

   1. Initial program 71.3%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 31.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 61.7%

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

   if 1.10000000000000002e156 < b

   1. Initial program 97.4%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 77.0%

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

   10. Applied rewrites 88.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.2 \cdot 10^{-129}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{+156}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, b \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot b\right) \cdot \left(angle \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot b\right) \cdot \left(angle \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.8e+71)
   (* a a)
   (* (* angle b) (* angle (* b (* 3.08641975308642e-5 (* PI PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+71) {
		tmp = a * a;
	} else {
		tmp = (angle * b) * (angle * (b * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+71) {
		tmp = a * a;
	} else {
		tmp = (angle * b) * (angle * (b * (3.08641975308642e-5 * (Math.PI * Math.PI))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.8e+71:
		tmp = a * a
	else:
		tmp = (angle * b) * (angle * (b * (3.08641975308642e-5 * (math.pi * math.pi))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.8e+71)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(angle * b) * Float64(angle * Float64(b * Float64(3.08641975308642e-5 * Float64(pi * pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.8e+71)
		tmp = a * a;
	else
		tmp = (angle * b) * (angle * (b * (3.08641975308642e-5 * (pi * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.8e+71], N[(a * a), $MachinePrecision], N[(N[(angle * b), $MachinePrecision] * N[(angle * N[(b * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e71

   1. Initial program 77.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   if 1.8e71 < b

   1. Initial program 92.2%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 70.6%

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

   10. Applied rewrites 77.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot b\right) \cdot \left(angle \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.8e+71)
   (* a a)
   (* angle (* angle (* b (* b (* 3.08641975308642e-5 (* PI PI))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+71) {
		tmp = a * a;
	} else {
		tmp = angle * (angle * (b * (b * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI))))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.8e+71) {
		tmp = a * a;
	} else {
		tmp = angle * (angle * (b * (b * (3.08641975308642e-5 * (Math.PI * Math.PI)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.8e+71:
		tmp = a * a
	else:
		tmp = angle * (angle * (b * (b * (3.08641975308642e-5 * (math.pi * math.pi)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.8e+71)
		tmp = Float64(a * a);
	else
		tmp = Float64(angle * Float64(angle * Float64(b * Float64(b * Float64(3.08641975308642e-5 * Float64(pi * pi))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.8e+71)
		tmp = a * a;
	else
		tmp = angle * (angle * (b * (b * (3.08641975308642e-5 * (pi * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.8e+71], N[(a * a), $MachinePrecision], N[(angle * N[(angle * N[(b * N[(b * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.8e71

   1. Initial program 77.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 62.0

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

   5. Applied rewrites 62.0%

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

   if 1.8e71 < b

   1. Initial program 92.2%

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

   3. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 70.6%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+71}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.0% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 54.0

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

5. Applied rewrites 54.0%

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

Reproduce

herbie shell --seed 2024227 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))