ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.0%

Time: 35.6s

Alternatives: 8

Speedup: N/A×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 80.9%

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

6. Applied egg-rr 80.9%

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

7. Taylor expanded in angle around 0 81.8%

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

8. Final simplification 81.8%

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

Alternative 2: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 80.9%

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

6. Applied egg-rr 80.9%

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

7. Taylor expanded in angle around 0 81.8%

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

8. Taylor expanded in angle around inf 70.5%

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

   1. unpow2 70.5%

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

   2. *-commutative 70.5%

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

   3. associate-*r* 70.5%

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

   4. unpow2 70.5%

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

   5. swap-sqr 81.8%

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

   6. unpow2 81.8%

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

   7. associate-*r* 81.8%

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

   8. *-commutative 81.8%

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

10. Simplified 81.8%

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

11. Final simplification 81.8%

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

Alternative 3: 79.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 80.9%

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

6. Applied egg-rr 80.9%

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

7. Taylor expanded in angle around 0 81.8%

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

   1. associate-*l* 81.8%

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

   2. pow2 81.8%

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

   3. *-commutative 81.8%

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

9. Applied egg-rr 81.8%

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

10. Final simplification 81.8%

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

Alternative 4: 72.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 80.9%

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

5. Taylor expanded in angle around 0 76.6%

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

   1. associate-*r* 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.7%

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

   4. *-commutative 76.7%

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

7. Simplified 76.7%

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

8. Taylor expanded in angle around 0 77.3%

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

   1. unpow2 77.3%

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

   2. associate-*r* 77.3%

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

   3. metadata-eval 77.3%

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

   4. div-inv 77.3%

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

   5. associate-*l* 74.9%

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

   6. div-inv 74.9%

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

   7. metadata-eval 74.9%

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

   8. *-commutative 74.9%

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

   9. associate-*l* 74.9%

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

10. Applied egg-rr 74.9%

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

    1. associate-*l* 74.9%

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

    2. associate-*l* 74.9%

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

    3. *-commutative 74.9%

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

12. Simplified 74.9%

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

13. Taylor expanded in b around 0 74.9%

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

    1. *-commutative 74.9%

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

    2. associate-*r* 74.9%

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

    3. *-commutative 74.9%

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

15. Simplified 74.9%

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

16. Final simplification 74.9%

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

Alternative 5: 72.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 80.9%

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

5. Taylor expanded in angle around 0 76.6%

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

   1. associate-*r* 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.7%

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

   4. *-commutative 76.7%

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

7. Simplified 76.7%

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

8. Taylor expanded in angle around 0 77.3%

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

   1. unpow2 77.3%

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

   2. associate-*r* 77.3%

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

   3. metadata-eval 77.3%

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

   4. div-inv 77.3%

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

   5. associate-*l* 74.9%

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

   6. div-inv 74.9%

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

   7. metadata-eval 74.9%

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

   8. *-commutative 74.9%

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

   9. associate-*l* 74.9%

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

10. Applied egg-rr 74.9%

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

    1. associate-*l* 74.9%

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

    2. associate-*l* 74.9%

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

    3. *-commutative 74.9%

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

12. Simplified 74.9%

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

    1. add-log-exp 67.2%

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

    2. exp-prod 67.2%

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

    3. associate-*r* 67.2%

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

    4. *-commutative 67.2%

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

    5. associate-*r* 67.2%

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

    6. associate-*l* 67.2%

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

14. Applied egg-rr 67.2%

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

    1. log-pow 74.5%

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

    2. rem-log-exp 74.9%

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

    3. *-commutative 74.9%

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

    4. associate-*l* 74.9%

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

    5. associate-*r* 75.0%

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

    6. *-commutative 75.0%

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

    7. associate-*r* 74.9%

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

    8. *-commutative 74.9%

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

16. Simplified 74.9%

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

17. Final simplification 74.9%

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

Alternative 6: 73.0% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 80.9%

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

5. Taylor expanded in angle around 0 76.6%

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

   1. associate-*r* 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.7%

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

   4. *-commutative 76.7%

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

7. Simplified 76.7%

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

8. Taylor expanded in angle around 0 77.3%

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

   1. unpow2 77.3%

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

   2. associate-*r* 77.3%

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

   3. metadata-eval 77.3%

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

   4. div-inv 77.3%

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

   5. associate-*l* 74.9%

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

   6. div-inv 74.9%

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

   7. metadata-eval 74.9%

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

   8. *-commutative 74.9%

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

   9. associate-*l* 74.9%

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

10. Applied egg-rr 74.9%

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

    1. associate-*l* 74.9%

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

    2. associate-*l* 74.9%

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

    3. *-commutative 74.9%

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

12. Simplified 74.9%

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

    1. expm1-log1p-u 57.2%

       \[\leadsto {a}^{2} + angle \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-0.005555555555555556 \cdot \left(\pi \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\right)\right)} \]

    2. expm1-udef 57.3%

       \[\leadsto {a}^{2} + angle \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-0.005555555555555556 \cdot \left(\pi \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot b\right)\right)\right)\right)\right)\right)} - 1\right)} \]

    3. associate-*r* 57.3%

       \[\leadsto {a}^{2} + angle \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-0.005555555555555556 \cdot \pi\right) \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot b\right)\right)\right)\right)}\right)} - 1\right) \]

    4. *-commutative 57.3%

       \[\leadsto {a}^{2} + angle \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot -0.005555555555555556\right)} \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot b\right)\right)\right)\right)\right)} - 1\right) \]

    5. associate-*l* 57.3%

       \[\leadsto {a}^{2} + angle \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot b\right)\right)\right)\right)\right)}\right)} - 1\right) \]

    6. *-commutative 57.3%

       \[\leadsto {a}^{2} + angle \cdot \left(e^{\mathsf{log1p}\left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \color{blue}{\left(\left(\pi \cdot \left(-0.005555555555555556 \cdot b\right)\right) \cdot angle\right)}\right)\right)\right)} - 1\right) \]

    7. associate-*r* 57.3%

       \[\leadsto {a}^{2} + angle \cdot \left(e^{\mathsf{log1p}\left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \left(\color{blue}{\left(\left(\pi \cdot -0.005555555555555556\right) \cdot b\right)} \cdot angle\right)\right)\right)\right)} - 1\right) \]

    8. associate-*l* 57.3%

       \[\leadsto {a}^{2} + angle \cdot \left(e^{\mathsf{log1p}\left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \color{blue}{\left(\left(\pi \cdot -0.005555555555555556\right) \cdot \left(b \cdot angle\right)\right)}\right)\right)\right)} - 1\right) \]

14. Applied egg-rr 57.3%

    \[\leadsto {a}^{2} + angle \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \left(\left(\pi \cdot -0.005555555555555556\right) \cdot \left(b \cdot angle\right)\right)\right)\right)\right)} - 1\right)} \]
15. Step-by-step derivation

    1. expm1-def 57.2%

       \[\leadsto {a}^{2} + angle \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(-0.005555555555555556 \cdot \left(b \cdot \left(\left(\pi \cdot -0.005555555555555556\right) \cdot \left(b \cdot angle\right)\right)\right)\right)\right)\right)} \]

    2. expm1-log1p 75.0%

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

    3. *-commutative 75.0%

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

16. Simplified 75.0%

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

17. Final simplification 75.0%

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

Alternative 7: 74.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 80.9%

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

5. Taylor expanded in angle around 0 76.6%

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

   1. associate-*r* 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.7%

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

   4. *-commutative 76.7%

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

7. Simplified 76.7%

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

8. Taylor expanded in angle around 0 77.3%

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

   1. unpow-prod-down 66.7%

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

   2. add-sqr-sqrt 66.7%

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

   3. unpow-prod-down 66.7%

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

   4. sqrt-pow1 56.8%

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

   5. metadata-eval 56.8%

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

   6. pow1 56.8%

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

   7. *-commutative 56.8%

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

   8. associate-*l* 56.8%

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

   9. unpow-prod-down 63.8%

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

   10. sqrt-pow1 77.3%

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

   11. metadata-eval 77.3%

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

   12. pow1 77.3%

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

   13. *-commutative 77.3%

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

   14. associate-*l* 77.3%

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

10. Applied egg-rr 77.3%

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

11. Final simplification 77.3%

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

Alternative 8: 73.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.8%

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

3. Simplified 80.9%

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

5. Taylor expanded in angle around 0 76.6%

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

   1. associate-*r* 76.6%

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

   2. *-commutative 76.6%

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

   3. associate-*l* 76.7%

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

   4. *-commutative 76.7%

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

7. Simplified 76.7%

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

8. Taylor expanded in angle around 0 77.3%

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

   1. unpow2 77.3%

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

   2. associate-*r* 77.3%

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

   3. metadata-eval 77.3%

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

   4. div-inv 77.3%

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

   5. associate-*r* 77.3%

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

   6. *-commutative 77.3%

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

   7. associate-*l* 77.3%

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

   8. div-inv 77.3%

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

   9. metadata-eval 77.3%

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

10. Applied egg-rr 77.3%

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

11. Final simplification 77.3%

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

Reproduce

herbie shell --seed 2024024 
(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)))