ab-angle->ABCF C

Percentage Accurate: 78.9% → 78.8%

Time: 19.0s

Alternatives: 12

Speedup: 1.9×

• 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[(N[(a * a), $MachinePrecision] * N[Power[N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $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. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-cos.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 76.5

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

5. Simplified 76.5%

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

Alternative 2: 44.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;{\left(b \cdot \sin t\_0\right)}^{2} + {\left(a \cdot \cos t\_0\right)}^{2} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* PI (/ angle 180.0))))
   (if (<= (+ (pow (* b (sin t_0)) 2.0) (pow (* a (cos t_0)) 2.0)) 2e+305)
     (* a a)
     (* (* angle PI) (* 3.08641975308642e-5 (* a (* a a)))))))

C

double code(double a, double b, double angle) {
	double t_0 = ((double) M_PI) * (angle / 180.0);
	double tmp;
	if ((pow((b * sin(t_0)), 2.0) + pow((a * cos(t_0)), 2.0)) <= 2e+305) {
		tmp = a * a;
	} else {
		tmp = (angle * ((double) M_PI)) * (3.08641975308642e-5 * (a * (a * a)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.PI * (angle / 180.0);
	double tmp;
	if ((Math.pow((b * Math.sin(t_0)), 2.0) + Math.pow((a * Math.cos(t_0)), 2.0)) <= 2e+305) {
		tmp = a * a;
	} else {
		tmp = (angle * Math.PI) * (3.08641975308642e-5 * (a * (a * a)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = math.pi * (angle / 180.0)
	tmp = 0
	if (math.pow((b * math.sin(t_0)), 2.0) + math.pow((a * math.cos(t_0)), 2.0)) <= 2e+305:
		tmp = a * a
	else:
		tmp = (angle * math.pi) * (3.08641975308642e-5 * (a * (a * a)))
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(pi * Float64(angle / 180.0))
	tmp = 0.0
	if (Float64((Float64(b * sin(t_0)) ^ 2.0) + (Float64(a * cos(t_0)) ^ 2.0)) <= 2e+305)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(angle * pi) * Float64(3.08641975308642e-5 * Float64(a * Float64(a * a))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = pi * (angle / 180.0);
	tmp = 0.0;
	if ((((b * sin(t_0)) ^ 2.0) + ((a * cos(t_0)) ^ 2.0)) <= 2e+305)
		tmp = a * a;
	else
		tmp = (angle * pi) * (3.08641975308642e-5 * (a * (a * a)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2e+305], N[(a * a), $MachinePrecision], N[(N[(angle * Pi), $MachinePrecision] * N[(3.08641975308642e-5 * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64))) < 1.9999999999999999e305

   1. Initial program 64.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{3}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{{a}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{2}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 18.1

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

   5. Simplified 18.1%

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

   6. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 45.3

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

   8. Simplified 45.3%

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

   if 1.9999999999999999e305 < (+.f64 (pow.f64 (*.f64 a (cos.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)) (pow.f64 (*.f64 b (sin.f64 (*.f64 (PI.f64) (/.f64 angle #s(literal 180 binary64))))) #s(literal 2 binary64)))

   1. Initial program 97.9%

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

   3. Taylor expanded in angle around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. distribute-lft1-in N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-PI.f64 N/A

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

      15. lower-PI.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 40.2

          \[\leadsto \mathsf{fma}\left(angle \cdot angle, -3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right) \cdot \color{blue}{\left(a \cdot a\right)} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. Simplified 40.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, -3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right), 1\right) \cdot \left(a \cdot a\right)} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. Taylor expanded in angle around -inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{3} \cdot \left({angle}^{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot {a}^{3}\right) \cdot \left({angle}^{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({angle}^{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({angle}^{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right)} \]

      4. unpow3 N/A

         \[\leadsto \left(\color{blue}{\left(\left(angle \cdot angle\right) \cdot angle\right)} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      5. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{{angle}^{2}} \cdot angle\right) \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{\left({angle}^{2} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left({angle}^{2} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(angle \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(angle \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      11. cube-mult N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      12. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      14. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      15. unpow2 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      17. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      18. lower-PI.f64 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \cdot \left(\frac{1}{32400} \cdot {a}^{3}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \color{blue}{\left({a}^{3} \cdot \frac{1}{32400}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(angle \cdot angle\right) \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \color{blue}{\left({a}^{3} \cdot \frac{1}{32400}\right)} \]

   8. Simplified 29.5%

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

   9. Taylor expanded in angle around 0

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

       1. lower-*.f64 N/A

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

       2. lower-PI.f64 38.4

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

   11. Simplified 38.4%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle \cdot \pi\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $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. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-cos.f64 N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 76.5

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

5. Simplified 76.5%

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

6. Final simplification 76.5%

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

Alternative 4: 78.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle) {
	double t_0 = 0.005555555555555556 * (angle * ((double) M_PI));
	return ((a * a) * pow(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 = 0.005555555555555556 * (angle * Math.PI);
	return ((a * a) * Math.pow(Math.cos(t_0), 2.0)) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(a * a), $MachinePrecision] * N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $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. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-cos.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 76.5

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

5. Simplified 76.5%

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

   1. lift-PI.f64 N/A

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

   2. associate-*r/ N/A

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

   3. lift-*.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 76.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 76.5

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

7. Applied egg-rr 76.5%

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

8. Final simplification 76.5%

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

Alternative 5: 78.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ \mathsf{fma}\left(t\_0, t\_0, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* b (sin (* angle (* 0.005555555555555556 PI))))))
   (fma
    t_0
    t_0
    (* (* a a) (fma 0.5 (cos (* (* angle PI) 0.011111111111111112)) 0.5)))))

C

double code(double a, double b, double angle) {
	double t_0 = b * sin((angle * (0.005555555555555556 * ((double) M_PI))));
	return fma(t_0, t_0, ((a * a) * fma(0.5, cos(((angle * ((double) M_PI)) * 0.011111111111111112)), 0.5)));
}

Julia

function code(a, b, angle)
	t_0 = Float64(b * sin(Float64(angle * Float64(0.005555555555555556 * pi))))
	return fma(t_0, t_0, Float64(Float64(a * a) * fma(0.5, cos(Float64(Float64(angle * pi) * 0.011111111111111112)), 0.5)))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(b * N[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $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. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-cos.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 76.5

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

5. Simplified 76.5%

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

7. Applied egg-rr 76.4%

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

8. Final simplification 76.4%

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), b \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right), \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)\right) \]
9. Add Preprocessing

Alternative 6: 78.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(N[(a * a), $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. 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. Simplified 75.7%

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

Alternative 7: 75.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{if}\;\frac{angle}{180} \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, t\_0, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(t\_0, -0.5, 0.5\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, 1, 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cos (* (* angle PI) 0.011111111111111112))))
   (if (<= (/ angle 180.0) 500000.0)
     (fma
      (* angle (* angle (* b (* (* PI PI) 3.08641975308642e-5))))
      b
      (* (* a a) (fma 0.5 t_0 0.5)))
     (fma (* b (fma t_0 -0.5 0.5)) b (* (* a a) (fma 0.5 1.0 0.5))))))

C

double code(double a, double b, double angle) {
	double t_0 = cos(((angle * ((double) M_PI)) * 0.011111111111111112));
	double tmp;
	if ((angle / 180.0) <= 500000.0) {
		tmp = fma((angle * (angle * (b * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5)))), b, ((a * a) * fma(0.5, t_0, 0.5)));
	} else {
		tmp = fma((b * fma(t_0, -0.5, 0.5)), b, ((a * a) * fma(0.5, 1.0, 0.5)));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = cos(Float64(Float64(angle * pi) * 0.011111111111111112))
	tmp = 0.0
	if (Float64(angle / 180.0) <= 500000.0)
		tmp = fma(Float64(angle * Float64(angle * Float64(b * Float64(Float64(pi * pi) * 3.08641975308642e-5)))), b, Float64(Float64(a * a) * fma(0.5, t_0, 0.5)));
	else
		tmp = fma(Float64(b * fma(t_0, -0.5, 0.5)), b, Float64(Float64(a * a) * fma(0.5, 1.0, 0.5)));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], 500000.0], N[(N[(angle * N[(angle * N[(b * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(0.5 * t$95$0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(t$95$0 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(0.5 * 1.0 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5e5

   1. Initial program 82.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 82.9

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

   5. Simplified 82.9%

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

      1. lift-PI.f64 N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 82.8

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

   7. Applied egg-rr 82.8%

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

   9. Applied egg-rr 67.9%

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

   10. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. unpow2 N/A

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

       5. associate-*l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. *-commutative N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. unpow2 N/A

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

       15. lower-*.f64 N/A

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

       16. lower-PI.f64 N/A

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

       17. lower-PI.f64 78.4

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

   12. Simplified 78.4%

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

   if 5e5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 59.7%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 59.7

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

   5. Simplified 59.7%

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

      1. lift-PI.f64 N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 59.7

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

   7. Applied egg-rr 59.7%

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

   9. Applied egg-rr 59.4%

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

   10. Taylor expanded in angle around 0

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

   12. Simplified 59.5%

       \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{1}, 0.5\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 500000:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(b \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, 1, 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(1, -0.5, 0.5\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.05e+139)
   (fma
    (* b (fma 1.0 -0.5 0.5))
    b
    (* (* a a) (fma 0.5 (cos (* (* angle PI) 0.011111111111111112)) 0.5)))
   (*
    (* b (* b b))
    (fma (cos (* angle (* PI 0.011111111111111112))) -0.5 0.5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.05e+139) {
		tmp = fma((b * fma(1.0, -0.5, 0.5)), b, ((a * a) * fma(0.5, cos(((angle * ((double) M_PI)) * 0.011111111111111112)), 0.5)));
	} else {
		tmp = (b * (b * b)) * fma(cos((angle * (((double) M_PI) * 0.011111111111111112))), -0.5, 0.5);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.05e+139)
		tmp = fma(Float64(b * fma(1.0, -0.5, 0.5)), b, Float64(Float64(a * a) * fma(0.5, cos(Float64(Float64(angle * pi) * 0.011111111111111112)), 0.5)));
	else
		tmp = Float64(Float64(b * Float64(b * b)) * fma(cos(Float64(angle * Float64(pi * 0.011111111111111112))), -0.5, 0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.05e+139], N[(N[(b * N[(1.0 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0499999999999999e139

   1. Initial program 73.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 73.8

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

   5. Simplified 73.8%

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

      1. lift-PI.f64 N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 73.8

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

   7. Applied egg-rr 73.8%

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

   9. Applied egg-rr 63.6%

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

   10. Taylor expanded in angle around 0

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

   12. Simplified 55.7%

       \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(\color{blue}{1}, -0.5, 0.5\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right) \]

   if 1.0499999999999999e139 < b

   1. Initial program 96.6%

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

   4. Applied egg-rr 56.5%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{3} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. metadata-eval 63.9

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

   7. Simplified 63.9%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \mathsf{fma}\left(1, -0.5, 0.5\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(b \cdot \mathsf{fma}\left(\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, 1, 0.5\right)\right) \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (fma
  (* b (fma (cos (* (* angle PI) 0.011111111111111112)) -0.5 0.5))
  b
  (* (* a a) (fma 0.5 1.0 0.5))))

C

double code(double a, double b, double angle) {
	return fma((b * fma(cos(((angle * ((double) M_PI)) * 0.011111111111111112)), -0.5, 0.5)), b, ((a * a) * fma(0.5, 1.0, 0.5)));
}

Julia

function code(a, b, angle)
	return fma(Float64(b * fma(cos(Float64(Float64(angle * pi) * 0.011111111111111112)), -0.5, 0.5)), b, Float64(Float64(a * a) * fma(0.5, 1.0, 0.5)))
end

Wolfram

code[a_, b_, angle_] := N[(N[(b * N[(N[Cos[N[(N[(angle * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * b + N[(N[(a * a), $MachinePrecision] * N[(0.5 * 1.0 + 0.5), $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. Taylor expanded in a around 0

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

   1. lower-*.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-cos.f64 N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 76.5

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

5. Simplified 76.5%

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

   1. lift-PI.f64 N/A

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

   2. clear-num N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. div-inv N/A

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

   6. times-frac N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 76.5

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

7. Applied egg-rr 76.5%

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

9. Applied egg-rr 65.6%

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

10. Taylor expanded in angle around 0

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

12. Simplified 64.8%

    \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \color{blue}{1}, 0.5\right)\right) \]

13. Final simplification 64.8%

    \[\leadsto \mathsf{fma}\left(b \cdot \mathsf{fma}\left(\cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), -0.5, 0.5\right), b, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, 1, 0.5\right)\right) \]
14. Add Preprocessing

Alternative 10: 57.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.05 \cdot 10^{+139}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot \left(b \cdot b\right)\right) \cdot \mathsf{fma}\left(\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.05e+139)
   (* a a)
   (*
    (* b (* b b))
    (fma (cos (* angle (* PI 0.011111111111111112))) -0.5 0.5))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.05e+139) {
		tmp = a * a;
	} else {
		tmp = (b * (b * b)) * fma(cos((angle * (((double) M_PI) * 0.011111111111111112))), -0.5, 0.5);
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.05e+139)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(b * Float64(b * b)) * fma(cos(Float64(angle * Float64(pi * 0.011111111111111112))), -0.5, 0.5));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.05e+139], N[(a * a), $MachinePrecision], N[(N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.0499999999999999e139

   1. Initial program 73.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{3}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{{a}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{2}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 27.9

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

   5. Simplified 27.9%

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

   6. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.2

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

   8. Simplified 55.2%

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

   if 1.0499999999999999e139 < b

   1. Initial program 96.6%

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

   4. Applied egg-rr 56.5%

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

   5. Taylor expanded in a around 0

      \[\leadsto \color{blue}{{b}^{3} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-PI.f64 N/A

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

      18. metadata-eval 63.9

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

   7. Simplified 63.9%

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

Alternative 11: 56.8% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+257}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot a, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(a \cdot \left(a \cdot a\right), -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 5.2e+257)
   (* a a)
   (fma
    a
    (* a a)
    (*
     (* angle angle)
     (*
      (* PI PI)
      (fma
       (* a (* a a))
       -3.08641975308642e-5
       (* 3.08641975308642e-5 (* b (* b b)))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 5.2e+257) {
		tmp = a * a;
	} else {
		tmp = fma(a, (a * a), ((angle * angle) * ((((double) M_PI) * ((double) M_PI)) * fma((a * (a * a)), -3.08641975308642e-5, (3.08641975308642e-5 * (b * (b * b)))))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 5.2e+257)
		tmp = Float64(a * a);
	else
		tmp = fma(a, Float64(a * a), Float64(Float64(angle * angle) * Float64(Float64(pi * pi) * fma(Float64(a * Float64(a * a)), -3.08641975308642e-5, Float64(3.08641975308642e-5 * Float64(b * Float64(b * b)))))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 5.2e+257], N[(a * a), $MachinePrecision], N[(a * N[(a * a), $MachinePrecision] + N[(N[(angle * angle), $MachinePrecision] * N[(N[(Pi * Pi), $MachinePrecision] * N[(N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision] * -3.08641975308642e-5 + N[(3.08641975308642e-5 * N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.20000000000000042e257

   1. Initial program 75.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{a}^{3}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

         \[\leadsto a \cdot \color{blue}{{a}^{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{a \cdot {a}^{2}} \]

      4. unpow2 N/A

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

      5. lower-*.f64 26.7

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

   5. Simplified 26.7%

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

   6. Taylor expanded in a around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 55.5

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

   8. Simplified 55.5%

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

   if 5.20000000000000042e257 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lower-cos.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 100.0

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

   5. Simplified 100.0%

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

      1. lift-PI.f64 N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. *-commutative N/A

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

      5. div-inv N/A

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

      6. times-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 99.7

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

   7. Applied egg-rr 99.7%

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

   9. Applied egg-rr 89.3%

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

   10. Taylor expanded in angle around 0

       \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{3}} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \color{blue}{{a}^{3} + {angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       2. cube-mult N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot a\right)} + {angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       3. unpow2 N/A

          \[\leadsto a \cdot \color{blue}{{a}^{2}} + {angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(a, {a}^{2}, {angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       5. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{a \cdot a}, {angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{a \cdot a}, {angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot a, \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot a, \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot a, \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       10. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(a, a \cdot a, \left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\frac{-1}{32400} \cdot {a}^{3}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + \frac{1}{32400} \cdot \left({b}^{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       11. associate-*r* N/A

           \[\leadsto \mathsf{fma}\left(a, a \cdot a, \left(angle \cdot angle\right) \cdot \left(\left(\frac{-1}{32400} \cdot {a}^{3}\right) \cdot {\mathsf{PI}\left(\right)}^{2} + \color{blue}{\left(\frac{1}{32400} \cdot {b}^{3}\right) \cdot {\mathsf{PI}\left(\right)}^{2}}\right)\right) \]

       12. distribute-rgt-out N/A

           \[\leadsto \mathsf{fma}\left(a, a \cdot a, \left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{32400} \cdot {a}^{3} + \frac{1}{32400} \cdot {b}^{3}\right)\right)}\right) \]

       13. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(a, a \cdot a, \left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{-1}{32400} \cdot {a}^{3} + \frac{1}{32400} \cdot {b}^{3}\right)\right)}\right) \]

   12. Simplified 55.6%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{+257}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot a, \left(angle \cdot angle\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(a \cdot \left(a \cdot a\right), -3.08641975308642 \cdot 10^{-5}, 3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)\right)\\ \end{array} \]
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

   \[\leadsto \color{blue}{{a}^{3}} \]
4. Step-by-step derivation

   1. cube-mult N/A

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

   2. unpow2 N/A

      \[\leadsto a \cdot \color{blue}{{a}^{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{a \cdot {a}^{2}} \]

   4. unpow2 N/A

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

   5. lower-*.f64 26.7

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

5. Simplified 26.7%

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

6. Taylor expanded in a around inf

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

   1. unpow2 N/A

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

   2. lower-*.f64 54.8

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

8. Simplified 54.8%

   \[\leadsto \color{blue}{a \cdot a} \]
9. 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)))