ab-angle->ABCF C

Percentage Accurate: 78.9% → 78.7%

Time: 21.1s

Alternatives: 5

Speedup: N/A×

• 35.4% 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: 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: 78.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + ((b * sin((-0.005555555555555556 * (pi / (1.0 / angle))))) ^ 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[(Pi / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

   1. *-un-lft-identity 81.2%

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

   2. div-inv 81.2%

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

   3. times-frac 81.2%

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

   4. metadata-eval 81.2%

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

6. Applied egg-rr 81.2%

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

7. Taylor expanded in angle around 0 81.3%

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

8. Final simplification 81.3%

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

Alternative 2: 78.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.2%

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

6. Taylor expanded in angle around 0 81.3%

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

7. Final simplification 81.3%

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

Alternative 3: 72.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.2%

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

6. Taylor expanded in angle around 0 77.4%

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

   1. unpow2 77.4%

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

   2. associate-*r* 77.4%

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

   3. associate-*l* 75.4%

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

   4. *-commutative 75.4%

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

   5. *-commutative 75.4%

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

   6. *-commutative 75.4%

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

   7. associate-*l* 75.4%

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

   8. *-commutative 75.4%

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

   9. *-commutative 75.4%

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

8. Applied egg-rr 75.4%

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

9. Final simplification 75.4%

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

Alternative 4: 73.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle) {
	double t_0 = (b * ((double) M_PI)) * (-0.005555555555555556 * angle);
	return pow(a, 2.0) + (t_0 * t_0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (b * Math.PI) * (-0.005555555555555556 * angle);
	return Math.pow(a, 2.0) + (t_0 * t_0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.2%

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

6. Taylor expanded in angle around 0 77.4%

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

   1. unpow2 77.4%

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

   2. *-commutative 77.4%

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

   3. *-commutative 77.4%

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

   4. *-commutative 77.4%

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

   5. associate-*l* 77.4%

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

   6. *-commutative 77.4%

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

   7. *-commutative 77.4%

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

   8. *-commutative 77.4%

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

   9. associate-*l* 77.4%

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

   10. *-commutative 77.4%

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

   11. *-commutative 77.4%

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

8. Applied egg-rr 77.4%

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

9. Final simplification 77.4%

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

Alternative 5: 73.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 81.2%

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

5. Taylor expanded in angle around 0 81.2%

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

6. Taylor expanded in angle around 0 77.4%

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

   1. unpow2 77.4%

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

   2. associate-*r* 77.4%

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

   3. *-commutative 77.4%

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

   4. *-commutative 77.4%

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

   5. associate-*l* 77.4%

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

   6. *-commutative 77.4%

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

   7. *-commutative 77.4%

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

   8. *-commutative 77.4%

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

8. Applied egg-rr 77.4%

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

9. Final simplification 77.4%

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

Reproduce

herbie shell --seed 2024021 
(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)))