ab-angle->ABCF C

Percentage Accurate: 80.3% → 80.1%

Time: 28.6s

Alternatives: 7

Speedup: N/A×

• 36.1% 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: 80.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 79.0%

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

5. Taylor expanded in angle around 0 79.3%

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

   1. add-exp-log 40.6%

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

   2. metadata-eval 40.6%

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

   3. div-inv 40.6%

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

   4. *-commutative 40.6%

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

   5. div-inv 40.6%

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

   6. metadata-eval 40.6%

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

   7. associate-*l* 40.6%

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

7. Applied egg-rr 40.6%

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

   1. rem-exp-log 79.3%

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

   2. *-commutative 79.3%

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

   3. associate-*r* 79.3%

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

   4. metadata-eval 79.3%

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

   5. associate-/r/ 79.3%

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

   6. div-inv 79.3%

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

   7. associate-/r* 79.3%

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

   8. metadata-eval 79.3%

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

   9. associate-/r* 79.3%

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

9. Applied egg-rr 79.3%

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

10. Final simplification 79.3%

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

Alternative 2: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 79.0%

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

5. Taylor expanded in angle around 0 79.3%

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

6. Taylor expanded in b around 0 79.3%

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

7. Final simplification 79.3%

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

Alternative 3: 75.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 79.0%

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

5. Taylor expanded in angle around 0 79.3%

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

6. Taylor expanded in angle around 0 73.3%

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

7. Final simplification 73.3%

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

Alternative 4: 75.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 79.0%

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

5. Taylor expanded in angle around 0 79.3%

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

   1. add-exp-log 40.6%

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

   2. metadata-eval 40.6%

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

   3. div-inv 40.6%

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

   4. *-commutative 40.6%

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

   5. div-inv 40.6%

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

   6. metadata-eval 40.6%

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

   7. associate-*l* 40.6%

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

7. Applied egg-rr 40.6%

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

8. Taylor expanded in angle around 0 73.3%

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

   1. associate-*r* 73.3%

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

   2. *-commutative 73.3%

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

   3. *-commutative 73.3%

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

   4. *-commutative 73.3%

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

   5. associate-*r* 73.3%

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

   6. *-commutative 73.3%

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

   7. associate-*r* 73.3%

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

   8. associate-*l* 73.3%

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

   9. *-commutative 73.3%

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

10. Simplified 73.3%

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

11. Final simplification 73.3%

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

Alternative 5: 74.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 79.0%

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

5. Taylor expanded in angle around 0 79.3%

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

6. Taylor expanded in angle around 0 73.3%

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

   1. unpow2 73.3%

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

   2. associate-*r* 73.3%

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

   3. associate-*l* 72.6%

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

   4. *-commutative 72.6%

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

   5. *-commutative 72.6%

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

   6. *-commutative 72.6%

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

   7. associate-*l* 72.6%

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

   8. *-commutative 72.6%

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

8. Applied egg-rr 72.6%

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

   1. metadata-eval 72.6%

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

   2. div-inv 72.6%

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

   3. *-commutative 72.6%

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

   4. clear-num 72.6%

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

   5. un-div-inv 72.6%

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

   6. associate-*r* 72.6%

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

   7. *-commutative 72.6%

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

   8. associate-*l* 72.6%

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

   9. *-commutative 72.6%

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

   10. associate-*l* 72.6%

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

10. Applied egg-rr 72.6%

    \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\frac{\left(\pi \cdot \left(0.005555555555555556 \cdot b\right)\right) \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)}{\frac{180}{angle}}} \]
11. Step-by-step derivation

    1. associate-/l* 72.8%

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

    2. *-commutative 72.8%

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

    3. associate-*r* 72.8%

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

    4. *-commutative 72.8%

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

    5. associate-*r* 72.8%

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

    6. *-commutative 72.8%

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

12. Simplified 72.8%

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

13. Final simplification 72.8%

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

Alternative 6: 75.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 79.0%

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

5. Taylor expanded in angle around 0 79.3%

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

6. Taylor expanded in angle around 0 73.3%

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

   1. unpow2 73.3%

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

   2. *-commutative 73.3%

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

   3. associate-*l* 73.3%

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

   4. *-commutative 73.3%

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

   5. *-commutative 73.3%

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

   6. associate-*l* 73.3%

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

   7. *-commutative 73.3%

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

8. Applied egg-rr 73.3%

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

9. Final simplification 73.3%

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

Alternative 7: 75.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 79.0%

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

5. Taylor expanded in angle around 0 79.3%

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

6. Taylor expanded in angle around 0 73.3%

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

   1. unpow2 73.3%

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

   2. associate-*r* 73.3%

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

   3. *-commutative 73.3%

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

   4. associate-*l* 73.3%

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

   5. *-commutative 73.3%

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

   6. associate-*r* 73.3%

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

   7. *-commutative 73.3%

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

8. Applied egg-rr 73.3%

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

9. Final simplification 73.3%

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

Reproduce

herbie shell --seed 2024076 
(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)))