ab-angle->ABCF C

Percentage Accurate: 80.2% → 80.1%

Time: 39.7s

Alternatives: 12

Speedup: N/A×

• 37.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% 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: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + pow((b * sin((((double) M_PI) * (-1.0 / (-180.0 / 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((Math.PI * (-1.0 / (-180.0 / angle))))), 2.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in angle around 0 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.4%

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

   3. clear-num 78.4%

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

   4. un-div-inv 78.4%

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

7. Applied egg-rr 78.4%

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

   1. frac-2neg 78.4%

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

   2. div-inv 78.4%

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

   3. distribute-neg-frac 78.4%

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

   4. metadata-eval 78.4%

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

9. Applied egg-rr 78.4%

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

10. Final simplification 78.4%

    \[\leadsto {a}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{-1}{\frac{-180}{angle}}\right)\right)}^{2} \]
11. Add Preprocessing

Alternative 2: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in angle around 0 78.3%

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

   1. metadata-eval 78.3%

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

   2. div-inv 78.4%

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

   3. clear-num 78.4%

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

   4. un-div-inv 78.4%

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

7. Applied egg-rr 78.4%

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

8. Final simplification 78.4%

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

Alternative 3: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in angle around 0 78.3%

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

6. Taylor expanded in angle around inf 78.3%

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

   1. associate-*r* 78.3%

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

   2. *-commutative 78.3%

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

   3. associate-*r* 78.4%

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

8. Simplified 78.4%

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

9. Final simplification 78.4%

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

Alternative 4: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 58000000000000:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \left(\pi \cdot 0.005555555555555556\right) \cdot \left(\left(b \cdot angle\right) \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot {b}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 58000000000000.0)
   (+
    (pow (* a (cos (* PI (* angle 0.005555555555555556)))) 2.0)
    (*
     (* PI 0.005555555555555556)
     (* (* b angle) (* (* PI 0.005555555555555556) (* b angle)))))
   (+
    (pow a 2.0)
    (*
     angle
     (*
      PI
      (*
       0.005555555555555556
       (* (* angle (* PI 0.005555555555555556)) (pow b 2.0))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 58000000000000.0) {
		tmp = pow((a * cos((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0) + ((((double) M_PI) * 0.005555555555555556) * ((b * angle) * ((((double) M_PI) * 0.005555555555555556) * (b * angle))));
	} else {
		tmp = pow(a, 2.0) + (angle * (((double) M_PI) * (0.005555555555555556 * ((angle * (((double) M_PI) * 0.005555555555555556)) * pow(b, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 58000000000000.0) {
		tmp = Math.pow((a * Math.cos((Math.PI * (angle * 0.005555555555555556)))), 2.0) + ((Math.PI * 0.005555555555555556) * ((b * angle) * ((Math.PI * 0.005555555555555556) * (b * angle))));
	} else {
		tmp = Math.pow(a, 2.0) + (angle * (Math.PI * (0.005555555555555556 * ((angle * (Math.PI * 0.005555555555555556)) * Math.pow(b, 2.0)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 58000000000000.0:
		tmp = math.pow((a * math.cos((math.pi * (angle * 0.005555555555555556)))), 2.0) + ((math.pi * 0.005555555555555556) * ((b * angle) * ((math.pi * 0.005555555555555556) * (b * angle))))
	else:
		tmp = math.pow(a, 2.0) + (angle * (math.pi * (0.005555555555555556 * ((angle * (math.pi * 0.005555555555555556)) * math.pow(b, 2.0)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 58000000000000.0)
		tmp = Float64((Float64(a * cos(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0) + Float64(Float64(pi * 0.005555555555555556) * Float64(Float64(b * angle) * Float64(Float64(pi * 0.005555555555555556) * Float64(b * angle)))));
	else
		tmp = Float64((a ^ 2.0) + Float64(angle * Float64(pi * Float64(0.005555555555555556 * Float64(Float64(angle * Float64(pi * 0.005555555555555556)) * (b ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 58000000000000.0)
		tmp = ((a * cos((pi * (angle * 0.005555555555555556)))) ^ 2.0) + ((pi * 0.005555555555555556) * ((b * angle) * ((pi * 0.005555555555555556) * (b * angle))));
	else
		tmp = (a ^ 2.0) + (angle * (pi * (0.005555555555555556 * ((angle * (pi * 0.005555555555555556)) * (b ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 58000000000000.0], N[(N[Power[N[(a * N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[(N[(b * angle), $MachinePrecision] * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle * N[(Pi * N[(0.005555555555555556 * N[(N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision] * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 5.8e13

   1. Initial program 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in angle around 0 78.3%

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

      1. associate-*r* 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-*r* 78.3%

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

   7. Simplified 78.3%

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

      1. unpow2 78.3%

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

      2. associate-*r* 78.4%

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

      3. associate-*r* 78.4%

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

      4. associate-*r* 78.4%

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

      5. *-commutative 78.4%

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

      6. *-commutative 78.4%

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

      7. *-commutative 78.4%

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

   9. Applied egg-rr 78.4%

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

   if 5.8e13 < angle

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in angle around 0 65.2%

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

   6. Taylor expanded in angle around 0 51.0%

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

      1. *-commutative 51.0%

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

      2. associate-*r* 51.0%

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

      3. *-commutative 51.0%

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

      4. associate-*r* 51.0%

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

      5. *-commutative 51.0%

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

      6. unpow2 51.0%

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

      7. metadata-eval 51.0%

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

      8. swap-sqr 51.0%

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

      9. associate-*r* 51.0%

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

      10. unpow2 51.0%

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

      11. unpow2 51.0%

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

      12. swap-sqr 54.8%

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

      13. swap-sqr 54.8%

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

      14. unpow2 54.8%

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

   8. Simplified 54.8%

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

      1. unpow2 54.8%

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

      2. associate-*r* 54.8%

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

      3. associate-*l* 54.8%

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

      4. *-commutative 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*r* 54.8%

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

      7. *-commutative 54.8%

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

      8. associate-*l* 54.8%

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

   10. Applied egg-rr 54.8%

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

       1. associate-*r* 54.8%

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

       2. *-commutative 54.8%

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

       3. associate-*l* 61.5%

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

       4. associate-*r* 61.5%

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

       5. *-commutative 61.5%

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

       6. *-commutative 61.5%

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

       7. associate-*r* 61.5%

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

       8. *-commutative 61.5%

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

       9. associate-*l* 61.8%

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

       10. unpow2 61.8%

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

       11. *-commutative 61.8%

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

   12. Simplified 61.8%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left({b}^{2} \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 58000000000000:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \left(\pi \cdot 0.005555555555555556\right) \cdot \left(\left(b \cdot angle\right) \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot {b}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 58000000000000:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \left(b \cdot angle\right) \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot {b}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 58000000000000.0)
   (+
    (pow (* a (cos (* PI (* angle 0.005555555555555556)))) 2.0)
    (*
     (* b angle)
     (*
      (* PI 0.005555555555555556)
      (* (* PI 0.005555555555555556) (* b angle)))))
   (+
    (pow a 2.0)
    (*
     angle
     (*
      PI
      (*
       0.005555555555555556
       (* (* angle (* PI 0.005555555555555556)) (pow b 2.0))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 58000000000000.0) {
		tmp = pow((a * cos((((double) M_PI) * (angle * 0.005555555555555556)))), 2.0) + ((b * angle) * ((((double) M_PI) * 0.005555555555555556) * ((((double) M_PI) * 0.005555555555555556) * (b * angle))));
	} else {
		tmp = pow(a, 2.0) + (angle * (((double) M_PI) * (0.005555555555555556 * ((angle * (((double) M_PI) * 0.005555555555555556)) * pow(b, 2.0)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 58000000000000.0) {
		tmp = Math.pow((a * Math.cos((Math.PI * (angle * 0.005555555555555556)))), 2.0) + ((b * angle) * ((Math.PI * 0.005555555555555556) * ((Math.PI * 0.005555555555555556) * (b * angle))));
	} else {
		tmp = Math.pow(a, 2.0) + (angle * (Math.PI * (0.005555555555555556 * ((angle * (Math.PI * 0.005555555555555556)) * Math.pow(b, 2.0)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 58000000000000.0:
		tmp = math.pow((a * math.cos((math.pi * (angle * 0.005555555555555556)))), 2.0) + ((b * angle) * ((math.pi * 0.005555555555555556) * ((math.pi * 0.005555555555555556) * (b * angle))))
	else:
		tmp = math.pow(a, 2.0) + (angle * (math.pi * (0.005555555555555556 * ((angle * (math.pi * 0.005555555555555556)) * math.pow(b, 2.0)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 58000000000000.0)
		tmp = Float64((Float64(a * cos(Float64(pi * Float64(angle * 0.005555555555555556)))) ^ 2.0) + Float64(Float64(b * angle) * Float64(Float64(pi * 0.005555555555555556) * Float64(Float64(pi * 0.005555555555555556) * Float64(b * angle)))));
	else
		tmp = Float64((a ^ 2.0) + Float64(angle * Float64(pi * Float64(0.005555555555555556 * Float64(Float64(angle * Float64(pi * 0.005555555555555556)) * (b ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 58000000000000.0)
		tmp = ((a * cos((pi * (angle * 0.005555555555555556)))) ^ 2.0) + ((b * angle) * ((pi * 0.005555555555555556) * ((pi * 0.005555555555555556) * (b * angle))));
	else
		tmp = (a ^ 2.0) + (angle * (pi * (0.005555555555555556 * ((angle * (pi * 0.005555555555555556)) * (b ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 58000000000000.0], N[(N[Power[N[(a * N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(b * angle), $MachinePrecision] * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[(N[(Pi * 0.005555555555555556), $MachinePrecision] * N[(b * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle * N[(Pi * N[(0.005555555555555556 * N[(N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision] * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 5.8e13

   1. Initial program 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in angle around 0 78.3%

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

      1. associate-*r* 78.3%

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

      2. *-commutative 78.3%

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

      3. associate-*r* 78.3%

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

   7. Simplified 78.3%

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

      1. unpow2 78.3%

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

      2. associate-*r* 78.4%

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

      3. associate-*l* 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*r* 78.3%

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

      6. *-commutative 78.3%

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

      7. *-commutative 78.3%

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

   9. Applied egg-rr 78.3%

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

   if 5.8e13 < angle

   1. Initial program 63.8%

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

   3. Simplified 63.8%

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

   5. Taylor expanded in angle around 0 65.2%

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

   6. Taylor expanded in angle around 0 51.0%

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

      1. *-commutative 51.0%

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

      2. associate-*r* 51.0%

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

      3. *-commutative 51.0%

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

      4. associate-*r* 51.0%

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

      5. *-commutative 51.0%

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

      6. unpow2 51.0%

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

      7. metadata-eval 51.0%

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

      8. swap-sqr 51.0%

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

      9. associate-*r* 51.0%

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

      10. unpow2 51.0%

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

      11. unpow2 51.0%

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

      12. swap-sqr 54.8%

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

      13. swap-sqr 54.8%

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

      14. unpow2 54.8%

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

   8. Simplified 54.8%

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

      1. unpow2 54.8%

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

      2. associate-*r* 54.8%

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

      3. associate-*l* 54.8%

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

      4. *-commutative 54.8%

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

      5. *-commutative 54.8%

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

      6. associate-*r* 54.8%

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

      7. *-commutative 54.8%

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

      8. associate-*l* 54.8%

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

   10. Applied egg-rr 54.8%

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

       1. associate-*r* 54.8%

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

       2. *-commutative 54.8%

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

       3. associate-*l* 61.5%

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

       4. associate-*r* 61.5%

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

       5. *-commutative 61.5%

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

       6. *-commutative 61.5%

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

       7. associate-*r* 61.5%

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

       8. *-commutative 61.5%

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

       9. associate-*l* 61.8%

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

       10. unpow2 61.8%

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

       11. *-commutative 61.8%

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

   12. Simplified 61.8%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left({b}^{2} \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 58000000000000:\\ \;\;\;\;{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + \left(b \cdot angle\right) \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(b \cdot angle\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot {b}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := b \cdot t\_0\\ \mathbf{if}\;b \leq 4 \cdot 10^{-91}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(t\_0 \cdot {b}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + t\_1 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI 0.005555555555555556))) (t_1 (* b t_0)))
   (if (<= b 4e-91)
     (+
      (pow a 2.0)
      (* angle (* PI (* 0.005555555555555556 (* t_0 (pow b 2.0))))))
     (+ (pow a 2.0) (* t_1 t_1)))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_1 = b * t_0;
	double tmp;
	if (b <= 4e-91) {
		tmp = pow(a, 2.0) + (angle * (((double) M_PI) * (0.005555555555555556 * (t_0 * pow(b, 2.0)))));
	} else {
		tmp = pow(a, 2.0) + (t_1 * t_1);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = angle * (Math.PI * 0.005555555555555556);
	double t_1 = b * t_0;
	double tmp;
	if (b <= 4e-91) {
		tmp = Math.pow(a, 2.0) + (angle * (Math.PI * (0.005555555555555556 * (t_0 * Math.pow(b, 2.0)))));
	} else {
		tmp = Math.pow(a, 2.0) + (t_1 * t_1);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = angle * (math.pi * 0.005555555555555556)
	t_1 = b * t_0
	tmp = 0
	if b <= 4e-91:
		tmp = math.pow(a, 2.0) + (angle * (math.pi * (0.005555555555555556 * (t_0 * math.pow(b, 2.0)))))
	else:
		tmp = math.pow(a, 2.0) + (t_1 * t_1)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * 0.005555555555555556))
	t_1 = Float64(b * t_0)
	tmp = 0.0
	if (b <= 4e-91)
		tmp = Float64((a ^ 2.0) + Float64(angle * Float64(pi * Float64(0.005555555555555556 * Float64(t_0 * (b ^ 2.0))))));
	else
		tmp = Float64((a ^ 2.0) + Float64(t_1 * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = angle * (pi * 0.005555555555555556);
	t_1 = b * t_0;
	tmp = 0.0;
	if (b <= 4e-91)
		tmp = (a ^ 2.0) + (angle * (pi * (0.005555555555555556 * (t_0 * (b ^ 2.0)))));
	else
		tmp = (a ^ 2.0) + (t_1 * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * t$95$0), $MachinePrecision]}, If[LessEqual[b, 4e-91], N[(N[Power[a, 2.0], $MachinePrecision] + N[(angle * N[(Pi * N[(0.005555555555555556 * N[(t$95$0 * N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if b < 4.00000000000000009e-91

   1. Initial program 79.0%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in angle around 0 79.4%

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

   6. Taylor expanded in angle around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. associate-*r* 61.6%

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

      3. *-commutative 61.6%

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

      4. associate-*r* 61.6%

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

      5. *-commutative 61.6%

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

      6. unpow2 61.6%

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

      7. metadata-eval 61.6%

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

      8. swap-sqr 61.6%

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

      9. associate-*r* 61.6%

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

      10. unpow2 61.6%

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

      11. unpow2 61.6%

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

      12. swap-sqr 72.4%

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

      13. swap-sqr 72.4%

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

      14. unpow2 72.4%

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

   8. Simplified 72.4%

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

      1. unpow2 72.4%

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

      2. associate-*r* 72.4%

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

      3. associate-*l* 70.8%

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

      4. *-commutative 70.8%

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

      5. *-commutative 70.8%

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

      6. associate-*r* 70.8%

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

      7. *-commutative 70.8%

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

      8. associate-*l* 70.8%

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

   10. Applied egg-rr 70.8%

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

       1. associate-*r* 72.4%

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

       2. *-commutative 72.4%

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

       3. associate-*l* 73.3%

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

       4. associate-*r* 73.2%

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

       5. *-commutative 73.2%

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

       6. *-commutative 73.2%

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

       7. associate-*r* 73.2%

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

       8. *-commutative 73.2%

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

       9. associate-*l* 70.6%

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

       10. unpow2 70.6%

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

       11. *-commutative 70.6%

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

   12. Simplified 70.6%

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

   if 4.00000000000000009e-91 < b

   1. Initial program 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in angle around 0 75.9%

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

   6. Taylor expanded in angle around 0 58.0%

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

      1. *-commutative 58.0%

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

      2. associate-*r* 58.0%

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

      3. *-commutative 58.0%

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

      4. associate-*r* 59.2%

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

      5. *-commutative 59.2%

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

      6. unpow2 59.2%

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

      7. metadata-eval 59.2%

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

      8. swap-sqr 59.2%

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

      9. associate-*r* 59.2%

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

      10. unpow2 59.2%

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

      11. unpow2 59.2%

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

      12. swap-sqr 72.2%

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

      13. swap-sqr 72.3%

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

      14. unpow2 72.3%

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

   8. Simplified 72.2%

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

      1. unpow2 72.2%

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

      2. *-commutative 72.2%

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

      3. associate-*r* 72.2%

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

      4. *-commutative 72.2%

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

      5. associate-*l* 72.2%

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

      6. *-commutative 72.2%

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

      7. associate-*r* 72.3%

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

      8. *-commutative 72.3%

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

      9. associate-*l* 72.2%

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

   10. Applied egg-rr 72.2%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot b\right) \cdot \left(\left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4 \cdot 10^{-91}:\\ \;\;\;\;{a}^{2} + angle \cdot \left(\pi \cdot \left(0.005555555555555556 \cdot \left(\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot {b}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in angle around 0 78.3%

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

6. Taylor expanded in angle around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. associate-*r* 60.5%

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

   3. *-commutative 60.5%

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

   4. associate-*r* 60.9%

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

   5. *-commutative 60.9%

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

   6. unpow2 60.9%

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

   7. metadata-eval 60.9%

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

   8. swap-sqr 60.9%

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

   9. associate-*r* 60.9%

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

   10. unpow2 60.9%

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

   11. unpow2 60.9%

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

   12. swap-sqr 72.3%

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

   13. swap-sqr 72.4%

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

   14. unpow2 72.4%

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

8. Simplified 72.3%

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

   1. unpow2 72.3%

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

   2. associate-*r* 72.4%

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

   3. associate-*r* 72.6%

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

   4. *-commutative 72.6%

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

   5. associate-*r* 72.6%

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

   6. *-commutative 72.6%

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

   7. associate-*l* 72.6%

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

   8. *-commutative 72.6%

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

10. Applied egg-rr 72.6%

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

11. Final simplification 72.6%

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

Alternative 8: 73.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in angle around 0 78.3%

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

6. Taylor expanded in angle around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. associate-*r* 60.5%

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

   3. *-commutative 60.5%

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

   4. associate-*r* 60.9%

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

   5. *-commutative 60.9%

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

   6. unpow2 60.9%

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

   7. metadata-eval 60.9%

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

   8. swap-sqr 60.9%

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

   9. associate-*r* 60.9%

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

   10. unpow2 60.9%

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

   11. unpow2 60.9%

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

   12. swap-sqr 72.3%

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

   13. swap-sqr 72.4%

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

   14. unpow2 72.4%

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

8. Simplified 72.3%

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

   1. unpow2 72.3%

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

   2. *-commutative 72.3%

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

   3. metadata-eval 72.3%

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

   4. div-inv 72.3%

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

   5. *-commutative 72.3%

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

   6. associate-*l/ 72.4%

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

   7. associate-/r/ 72.3%

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

   8. associate-*r* 72.6%

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

   9. *-commutative 72.6%

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

   10. associate-*r* 72.6%

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

   11. *-commutative 72.6%

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

   12. associate-*l* 72.6%

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

   13. clear-num 72.6%

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

   14. associate-/r/ 72.6%

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

   15. clear-num 72.5%

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

   16. div-inv 72.6%

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

   17. metadata-eval 72.6%

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

   18. associate-*l* 72.5%

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

10. Applied egg-rr 72.5%

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

11. Final simplification 72.5%

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

Alternative 9: 75.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in angle around 0 78.3%

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

6. Taylor expanded in angle around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. associate-*r* 60.5%

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

   3. *-commutative 60.5%

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

   4. associate-*r* 60.9%

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

   5. *-commutative 60.9%

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

   6. unpow2 60.9%

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

   7. metadata-eval 60.9%

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

   8. swap-sqr 60.9%

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

   9. associate-*r* 60.9%

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

   10. unpow2 60.9%

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

   11. unpow2 60.9%

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

   12. swap-sqr 72.3%

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

   13. swap-sqr 72.4%

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

   14. unpow2 72.4%

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

8. Simplified 72.3%

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

   1. unpow2 72.3%

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

   2. *-commutative 72.3%

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

   3. associate-*r* 72.4%

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

   4. *-commutative 72.4%

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

   5. associate-*l* 72.4%

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

   6. *-commutative 72.4%

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

   7. associate-*r* 72.4%

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

   8. *-commutative 72.4%

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

   9. associate-*l* 72.4%

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

10. Applied egg-rr 72.4%

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

11. Final simplification 72.4%

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

Alternative 10: 74.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in angle around 0 78.3%

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

6. Taylor expanded in angle around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. associate-*r* 60.5%

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

   3. *-commutative 60.5%

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

   4. associate-*r* 60.9%

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

   5. *-commutative 60.9%

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

   6. unpow2 60.9%

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

   7. metadata-eval 60.9%

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

   8. swap-sqr 60.9%

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

   9. associate-*r* 60.9%

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

   10. unpow2 60.9%

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

   11. unpow2 60.9%

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

   12. swap-sqr 72.3%

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

   13. swap-sqr 72.4%

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

   14. unpow2 72.4%

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

8. Simplified 72.3%

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

   1. unpow2 72.3%

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

   2. associate-*r* 72.4%

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

   3. associate-*l* 70.9%

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

   4. *-commutative 70.9%

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

   5. *-commutative 70.9%

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

   6. associate-*r* 70.9%

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

   7. *-commutative 70.9%

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

   8. associate-*l* 70.9%

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

10. Applied egg-rr 70.9%

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

11. Final simplification 70.9%

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

Alternative 11: 74.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in angle around 0 78.3%

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

6. Taylor expanded in angle around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. associate-*r* 60.5%

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

   3. *-commutative 60.5%

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

   4. associate-*r* 60.9%

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

   5. *-commutative 60.9%

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

   6. unpow2 60.9%

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

   7. metadata-eval 60.9%

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

   8. swap-sqr 60.9%

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

   9. associate-*r* 60.9%

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

   10. unpow2 60.9%

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

   11. unpow2 60.9%

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

   12. swap-sqr 72.3%

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

   13. swap-sqr 72.4%

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

   14. unpow2 72.4%

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

8. Simplified 72.3%

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

   1. unpow2 72.3%

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

   2. associate-*r* 72.4%

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

   3. associate-*l* 70.9%

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

   4. *-commutative 70.9%

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

   5. *-commutative 70.9%

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

   6. associate-*r* 70.9%

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

   7. *-commutative 70.9%

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

   8. associate-*l* 70.9%

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

10. Applied egg-rr 70.9%

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

11. Taylor expanded in angle around 0 70.9%

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

    1. associate-*r* 70.9%

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

13. Simplified 70.9%

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

14. Final simplification 70.9%

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

Alternative 12: 74.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Simplified 78.1%

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

5. Taylor expanded in angle around 0 78.3%

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

6. Taylor expanded in angle around 0 60.5%

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

   1. *-commutative 60.5%

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

   2. associate-*r* 60.5%

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

   3. *-commutative 60.5%

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

   4. associate-*r* 60.9%

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

   5. *-commutative 60.9%

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

   6. unpow2 60.9%

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

   7. metadata-eval 60.9%

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

   8. swap-sqr 60.9%

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

   9. associate-*r* 60.9%

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

   10. unpow2 60.9%

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

   11. unpow2 60.9%

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

   12. swap-sqr 72.3%

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

   13. swap-sqr 72.4%

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

   14. unpow2 72.4%

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

8. Simplified 72.3%

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

   1. unpow2 72.3%

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

   2. associate-*r* 72.4%

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

   3. associate-*l* 70.9%

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

   4. *-commutative 70.9%

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

   5. *-commutative 70.9%

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

   6. associate-*r* 70.9%

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

   7. *-commutative 70.9%

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

   8. associate-*l* 70.9%

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

10. Applied egg-rr 70.9%

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

11. Taylor expanded in angle around 0 70.9%

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

12. Final simplification 70.9%

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

Reproduce

herbie shell --seed 2024096 
(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)))