ab-angle->ABCF C

Percentage Accurate: 79.7% → 79.7%

Time: 35.9s

Alternatives: 8

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Simplified 79.8%

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

5. Taylor expanded in angle around inf 79.9%

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

6. Final simplification 79.9%

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

Alternative 2: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Simplified 79.8%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Final simplification 79.9%

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

Alternative 3: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Simplified 79.8%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around inf 79.9%

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

7. Final simplification 79.9%

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

Alternative 4: 74.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Simplified 79.8%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around 0 63.7%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 63.2%

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

   2. *-commutative 63.2%

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

   3. *-commutative 63.2%

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

   4. associate-*r* 63.2%

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

   5. *-commutative 63.2%

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

   6. metadata-eval 63.2%

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

   7. unpow2 63.2%

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

   8. swap-sqr 63.2%

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

   9. unpow2 63.2%

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

   10. swap-sqr 63.6%

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

   11. *-commutative 63.6%

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

   12. *-commutative 63.6%

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

   13. unpow2 63.6%

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

   14. swap-sqr 74.7%

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

   15. unpow2 74.7%

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

8. Simplified 74.8%

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

9. Final simplification 74.8%

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

Alternative 5: 74.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Simplified 79.8%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around 0 63.7%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 63.2%

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

   2. *-commutative 63.2%

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

   3. *-commutative 63.2%

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

   4. associate-*r* 63.2%

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

   5. *-commutative 63.2%

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

   6. metadata-eval 63.2%

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

   7. unpow2 63.2%

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

   8. swap-sqr 63.2%

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

   9. unpow2 63.2%

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

   10. swap-sqr 63.6%

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

   11. *-commutative 63.6%

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

   12. *-commutative 63.6%

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

   13. unpow2 63.6%

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

   14. swap-sqr 74.7%

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

   15. unpow2 74.7%

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

8. Simplified 74.8%

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

9. Taylor expanded in b around 0 74.8%

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

10. Final simplification 74.8%

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

Alternative 6: 74.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Simplified 79.8%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around 0 63.7%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 63.2%

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

   2. *-commutative 63.2%

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

   3. *-commutative 63.2%

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

   4. associate-*r* 63.2%

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

   5. *-commutative 63.2%

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

   6. metadata-eval 63.2%

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

   7. unpow2 63.2%

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

   8. swap-sqr 63.2%

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

   9. unpow2 63.2%

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

   10. swap-sqr 63.6%

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

   11. *-commutative 63.6%

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

   12. *-commutative 63.6%

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

   13. unpow2 63.6%

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

   14. swap-sqr 74.7%

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

   15. unpow2 74.7%

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

8. Simplified 74.8%

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

9. Taylor expanded in angle around 0 74.8%

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

    1. associate-*r* 74.8%

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

11. Simplified 74.8%

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

12. Final simplification 74.8%

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

Alternative 7: 74.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Simplified 79.8%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around 0 63.7%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 63.2%

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

   2. *-commutative 63.2%

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

   3. *-commutative 63.2%

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

   4. associate-*r* 63.2%

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

   5. *-commutative 63.2%

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

   6. metadata-eval 63.2%

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

   7. unpow2 63.2%

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

   8. swap-sqr 63.2%

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

   9. unpow2 63.2%

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

   10. swap-sqr 63.6%

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

   11. *-commutative 63.6%

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

   12. *-commutative 63.6%

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

   13. unpow2 63.6%

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

   14. swap-sqr 74.7%

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

   15. unpow2 74.7%

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

8. Simplified 74.8%

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

9. Taylor expanded in angle around 0 74.8%

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

10. Final simplification 74.8%

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

Alternative 8: 73.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

3. Simplified 79.8%

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

5. Taylor expanded in angle around 0 79.9%

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

6. Taylor expanded in angle around 0 63.7%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. associate-*r* 63.2%

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

   2. *-commutative 63.2%

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

   3. *-commutative 63.2%

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

   4. associate-*r* 63.2%

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

   5. *-commutative 63.2%

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

   6. metadata-eval 63.2%

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

   7. unpow2 63.2%

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

   8. swap-sqr 63.2%

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

   9. unpow2 63.2%

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

   10. swap-sqr 63.6%

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

   11. *-commutative 63.6%

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

   12. *-commutative 63.6%

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

   13. unpow2 63.6%

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

   14. swap-sqr 74.7%

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

   15. unpow2 74.7%

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

8. Simplified 74.8%

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

   1. unpow2 74.8%

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

   2. associate-*r* 74.8%

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

   3. *-commutative 74.8%

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

   4. associate-*r* 74.8%

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

   5. metadata-eval 74.8%

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

   6. associate-/r/ 74.8%

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

   7. associate-*l* 73.4%

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

   8. associate-/r/ 73.4%

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

   9. metadata-eval 73.4%

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

   10. *-commutative 73.4%

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

   11. *-commutative 73.4%

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

   12. *-commutative 73.4%

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

   13. associate-*l* 73.4%

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

   14. associate-*r* 73.4%

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

   15. *-commutative 73.4%

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

10. Applied egg-rr 73.4%

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

11. Taylor expanded in angle around 0 73.4%

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

12. Final simplification 73.4%

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

Reproduce

herbie shell --seed 2024089 
(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)))