ab-angle->ABCF C

Percentage Accurate: 78.9% → 78.8%

Time: 16.1s

Alternatives: 12

Speedup: 1.9×

• Could not converge on a ground truth

  1. a = 1.2945207624963611e-289
  2. b = 2.3171035619447185e+282
  3. angle = 5.283837503900573e+168

• 35.6% 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: 78.9% 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: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. lift-PI.f64 N/A

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

   2. div-inv N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 76.5

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

4. Applied rewrites 76.5%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. div-inv N/A

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

   6. times-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 76.5

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

6. Applied rewrites 76.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lower-/.f64 76.5

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

8. Applied rewrites 76.5%

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

Alternative 2: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. lift-PI.f64 N/A

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

   2. div-inv N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 76.5

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

4. Applied rewrites 76.5%

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

Alternative 3: 38.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

   1. lift-PI.f64 N/A

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

   2. div-inv N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 76.5

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

4. Applied rewrites 76.5%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. div-inv N/A

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

   6. times-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 76.5

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

6. Applied rewrites 76.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lower-/.f64 76.5

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

8. Applied rewrites 76.5%

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

10. Applied rewrites 38.0%

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

11. Final simplification 38.0%

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

Alternative 4: 78.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* 0.005555555555555556 (/ 1.0 (/ 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 / (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 / (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 / (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(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 / (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[(1.0 / N[(Pi * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.5%

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

   1. lift-PI.f64 N/A

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

   2. div-inv N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. metadata-eval 76.5

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

4. Applied rewrites 76.5%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. div-inv N/A

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

   6. times-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 76.5

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

6. Applied rewrites 76.5%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-/l/ N/A

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

   7. lift-*.f64 N/A

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

   8. lower-/.f64 76.5

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

8. Applied rewrites 76.5%

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

9. Taylor expanded in angle around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 75.8

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

11. Applied rewrites 75.8%

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

12. Final simplification 75.8%

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

Alternative 5: 78.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.5%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 75.7

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

5. Applied rewrites 75.7%

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

6. Final simplification 75.7%

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

Alternative 6: 72.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(b \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 0.5, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0
         (*
          (* a a)
          (fma 0.5 (cos (* PI (* angle 0.011111111111111112))) 0.5))))
   (if (<= (/ angle 180.0) 4e-8)
     (fma b (* (* b (* PI PI)) (* (* angle angle) 3.08641975308642e-5)) t_0)
     (fma (* b b) 0.5 t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = (a * a) * fma(0.5, cos((((double) M_PI) * (angle * 0.011111111111111112))), 0.5);
	double tmp;
	if ((angle / 180.0) <= 4e-8) {
		tmp = fma(b, ((b * (((double) M_PI) * ((double) M_PI))) * ((angle * angle) * 3.08641975308642e-5)), t_0);
	} else {
		tmp = fma((b * b), 0.5, t_0);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(a * a) * fma(0.5, cos(Float64(pi * Float64(angle * 0.011111111111111112))), 0.5))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 4e-8)
		tmp = fma(b, Float64(Float64(b * Float64(pi * pi)) * Float64(Float64(angle * angle) * 3.08641975308642e-5)), t_0);
	else
		tmp = fma(Float64(b * b), 0.5, t_0);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e-8], N[(b * N[(N[(b * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(N[(angle * angle), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 0.5 + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 4.0000000000000001e-8

   1. Initial program 82.7%

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

      1. lift-PI.f64 N/A

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

      2. div-inv N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 82.8

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

   4. Applied rewrites 82.8%

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

   6. Applied rewrites 54.9%

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

   7. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(\frac{1}{32400} \cdot {angle}^{2}\right) \cdot \left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)}, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right)}, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right), \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(b, \left(b \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right), \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \left(b \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right), \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      7. lower-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \left(b \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right), \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {angle}^{2}\right), \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(b, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)}, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left({angle}^{2} \cdot \frac{1}{32400}\right)}, \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(b, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \frac{1}{32400}\right), \mathsf{fma}\left(\frac{1}{2}, \cos \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{90}\right)\right), \frac{1}{2}\right) \cdot \left(a \cdot a\right)\right) \]

      12. lower-*.f64 75.4

          \[\leadsto \mathsf{fma}\left(b, \left(b \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot 3.08641975308642 \cdot 10^{-5}\right), \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right) \cdot \left(a \cdot a\right)\right) \]

   9. Applied rewrites 75.4%

      \[\leadsto \mathsf{fma}\left(b, \color{blue}{\left(b \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)}, \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right) \cdot \left(a \cdot a\right)\right) \]

   if 4.0000000000000001e-8 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 60.5%

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

      1. lift-PI.f64 N/A

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

      2. div-inv N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 60.5

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

   4. Applied rewrites 60.5%

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.2

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

   9. Applied rewrites 59.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(b, \left(b \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(angle \cdot angle\right) \cdot 3.08641975308642 \cdot 10^{-5}\right), \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 0.5, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.4% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.55 \cdot 10^{-39}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot b, 0.5, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.55e-39)
   (fma
    (*
     (* PI angle)
     (* PI (fma b (* b 3.08641975308642e-5) (* (* a a) -3.08641975308642e-5))))
    angle
    (* 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 (a <= 1.55e-39) {
		tmp = fma(((((double) M_PI) * angle) * (((double) M_PI) * fma(b, (b * 3.08641975308642e-5), ((a * a) * -3.08641975308642e-5)))), angle, (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 (a <= 1.55e-39)
		tmp = fma(Float64(Float64(pi * angle) * Float64(pi * fma(b, Float64(b * 3.08641975308642e-5), Float64(Float64(a * a) * -3.08641975308642e-5)))), angle, Float64(a * a));
	else
		tmp = fma(Float64(b * b), 0.5, Float64(Float64(a * a) * fma(0.5, cos(Float64(pi * Float64(angle * 0.011111111111111112))), 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.55e-39], N[(N[(N[(Pi * angle), $MachinePrecision] * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(b * b), $MachinePrecision] * 0.5 + N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.54999999999999985e-39

   1. Initial program 75.6%

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

   3. Taylor expanded in angle around 0

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

      1. 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 45.8%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

   7. Applied rewrites 49.3%

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

   if 1.54999999999999985e-39 < a

   1. Initial program 78.3%

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

      1. lift-PI.f64 N/A

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

      2. div-inv N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval 78.4

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

   4. Applied rewrites 78.4%

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

   6. Applied rewrites 60.9%

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

   7. Taylor expanded in b around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 72.9

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

   9. Applied rewrites 72.9%

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

Alternative 8: 52.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{-152}:\\ \;\;\;\;angle \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.45e-152)
   (* angle (* (* PI angle) (* PI (* 3.08641975308642e-5 (* b b)))))
   (if (<= a 3.5e+144)
     (fma
      (* angle angle)
      (*
       PI
       (*
        PI
        (fma (* b b) 3.08641975308642e-5 (* (* a a) -3.08641975308642e-5))))
      (* a a))
     (* a a))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.45e-152) {
		tmp = angle * ((((double) M_PI) * angle) * (((double) M_PI) * (3.08641975308642e-5 * (b * b))));
	} else if (a <= 3.5e+144) {
		tmp = fma((angle * angle), (((double) M_PI) * (((double) M_PI) * fma((b * b), 3.08641975308642e-5, ((a * a) * -3.08641975308642e-5)))), (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.45e-152)
		tmp = Float64(angle * Float64(Float64(pi * angle) * Float64(pi * Float64(3.08641975308642e-5 * Float64(b * b)))));
	elseif (a <= 3.5e+144)
		tmp = fma(Float64(angle * angle), Float64(pi * Float64(pi * fma(Float64(b * b), 3.08641975308642e-5, Float64(Float64(a * a) * -3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.45e-152], N[(angle * N[(N[(Pi * angle), $MachinePrecision] * N[(Pi * N[(3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.5e+144], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5 + N[(N[(a * a), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.4500000000000001e-152

   1. Initial program 76.2%

      \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 45.1%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 41.0

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

   8. Applied rewrites 41.0%

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

      1. lift-PI.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 45.2

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

   10. Applied rewrites 45.2%

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

   if 1.4500000000000001e-152 < a < 3.4999999999999998e144

   1. Initial program 63.1%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   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 46.7%

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

   if 3.4999999999999998e144 < a

   1. Initial program 97.8%

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

   3. 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 97.8

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

   5. Applied rewrites 97.8%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.45 \cdot 10^{-152}:\\ \;\;\;\;angle \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right)\right)\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.5 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot \mathsf{fma}\left(b, b \cdot 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle, a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 3.5e+144)
   (fma
    (*
     (* PI angle)
     (* PI (fma b (* b 3.08641975308642e-5) (* (* a a) -3.08641975308642e-5))))
    angle
    (* a a))
   (* a a)))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 3.5e+144) {
		tmp = fma(((((double) M_PI) * angle) * (((double) M_PI) * fma(b, (b * 3.08641975308642e-5), ((a * a) * -3.08641975308642e-5)))), angle, (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 3.5e+144)
		tmp = fma(Float64(Float64(pi * angle) * Float64(pi * fma(b, Float64(b * 3.08641975308642e-5), Float64(Float64(a * a) * -3.08641975308642e-5)))), angle, Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 3.5e+144], N[(N[(N[(Pi * angle), $MachinePrecision] * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision] + N[(N[(a * a), $MachinePrecision] * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.4999999999999998e144

   1. Initial program 72.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   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 45.6%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

   7. Applied rewrites 48.4%

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

   if 3.4999999999999998e144 < a

   1. Initial program 97.8%

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

   3. 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 97.8

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

   5. Applied rewrites 97.8%

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

Alternative 10: 60.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{+67}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9.8e+67)
   (* a a)
   (* angle (* (* PI angle) (* PI (* 3.08641975308642e-5 (* b b)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.8e+67) {
		tmp = a * a;
	} else {
		tmp = angle * ((((double) M_PI) * angle) * (((double) M_PI) * (3.08641975308642e-5 * (b * b))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.8e+67) {
		tmp = a * a;
	} else {
		tmp = angle * ((Math.PI * angle) * (Math.PI * (3.08641975308642e-5 * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 9.8e+67:
		tmp = a * a
	else:
		tmp = angle * ((math.pi * angle) * (math.pi * (3.08641975308642e-5 * (b * b))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 9.8e+67)
		tmp = Float64(a * a);
	else
		tmp = Float64(angle * Float64(Float64(pi * angle) * Float64(pi * Float64(3.08641975308642e-5 * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 9.8e+67)
		tmp = a * a;
	else
		tmp = angle * ((pi * angle) * (pi * (3.08641975308642e-5 * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 9.8e+67], N[(a * a), $MachinePrecision], N[(angle * N[(N[(Pi * angle), $MachinePrecision] * N[(Pi * N[(3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.7999999999999998e67

   1. Initial program 75.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if 9.7999999999999998e67 < b

   1. Initial program 79.7%

      \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 28.9%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 53.6

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

   8. Applied rewrites 53.6%

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

      1. lift-PI.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-*.f64 57.8

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

   10. Applied rewrites 57.8%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{+67}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(\pi \cdot angle\right) \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{+67}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 9.8e+67)
   (* a a)
   (* (* angle angle) (* PI (* PI (* 3.08641975308642e-5 (* b b)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.8e+67) {
		tmp = a * a;
	} else {
		tmp = (angle * angle) * (((double) M_PI) * (((double) M_PI) * (3.08641975308642e-5 * (b * b))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 9.8e+67) {
		tmp = a * a;
	} else {
		tmp = (angle * angle) * (Math.PI * (Math.PI * (3.08641975308642e-5 * (b * b))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 9.8e+67:
		tmp = a * a
	else:
		tmp = (angle * angle) * (math.pi * (math.pi * (3.08641975308642e-5 * (b * b))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 9.8e+67)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(angle * angle) * Float64(pi * Float64(pi * Float64(3.08641975308642e-5 * Float64(b * b)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 9.8e+67)
		tmp = a * a;
	else
		tmp = (angle * angle) * (pi * (pi * (3.08641975308642e-5 * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 9.8e+67], N[(a * a), $MachinePrecision], N[(N[(angle * angle), $MachinePrecision] * N[(Pi * N[(Pi * N[(3.08641975308642e-5 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 9.7999999999999998e67

   1. Initial program 75.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 57.6

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

   5. Applied rewrites 57.6%

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

   if 9.7999999999999998e67 < b

   1. Initial program 79.7%

      \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   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 28.9%

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

   6. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f64 53.6

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

   8. Applied rewrites 53.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9.8 \cdot 10^{+67}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.9% 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 76.5%

   \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
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 54.8

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

5. Applied rewrites 54.8%

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

Reproduce

herbie shell --seed 2024214 
(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)))