ab-angle->ABCF C

Percentage Accurate: 79.7% → 79.7%

Time: 48.6s

Alternatives: 8

Speedup: N/A×

• 36.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in angle around 0 82.1%

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

6. Taylor expanded in angle around 0 82.1%

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

   1. associate-*r* 82.1%

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

8. Simplified 82.1%

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

9. Final simplification 82.1%

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

Alternative 2: 79.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in angle around 0 82.1%

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

6. Taylor expanded in angle around 0 82.1%

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

7. Final simplification 82.1%

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

Alternative 3: 79.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in angle around 0 82.1%

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

6. Final simplification 82.1%

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

Alternative 4: 72.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in angle around 0 76.9%

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

   1. associate-*r* 76.9%

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

7. Simplified 76.9%

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

8. Taylor expanded in angle around 0 76.9%

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

   1. unpow2 76.9%

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

   2. associate-*l* 76.9%

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

   3. associate-*l* 77.0%

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

   4. *-commutative 77.0%

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

   5. associate-*r* 77.0%

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

   6. *-commutative 77.0%

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

10. Applied egg-rr 77.0%

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

    1. associate-*l* 75.7%

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

    2. associate-*r* 75.7%

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

    3. *-commutative 75.7%

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

12. Simplified 75.7%

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

13. Taylor expanded in angle around 0 75.7%

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

    1. *-commutative 75.7%

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

15. Simplified 75.7%

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

16. Final simplification 75.7%

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

Alternative 5: 72.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in angle around 0 76.9%

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

   1. associate-*r* 76.9%

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

7. Simplified 76.9%

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

8. Taylor expanded in angle around 0 76.9%

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

   1. unpow2 76.9%

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

   2. associate-*l* 76.9%

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

   3. associate-*l* 77.0%

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

   4. *-commutative 77.0%

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

   5. associate-*r* 77.0%

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

   6. *-commutative 77.0%

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

10. Applied egg-rr 77.0%

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

    1. associate-*l* 75.7%

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

    2. associate-*r* 75.7%

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

    3. *-commutative 75.7%

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

12. Simplified 75.7%

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

13. Final simplification 75.7%

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

Alternative 6: 72.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in angle around 0 76.9%

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

   1. associate-*r* 76.9%

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

7. Simplified 76.9%

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

8. Taylor expanded in angle around 0 76.9%

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

   1. unpow2 76.9%

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

   2. associate-*l* 76.9%

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

   3. associate-*l* 77.0%

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

   4. *-commutative 77.0%

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

   5. associate-*r* 77.0%

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

   6. *-commutative 77.0%

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

10. Applied egg-rr 77.0%

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

    1. associate-*l* 75.7%

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

    2. *-commutative 75.7%

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

12. Simplified 75.7%

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

13. Final simplification 75.7%

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

Alternative 7: 74.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in angle around 0 76.9%

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

   1. associate-*r* 76.9%

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

7. Simplified 76.9%

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

8. Taylor expanded in angle around 0 76.9%

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

   1. unpow2 76.9%

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

   2. associate-*l* 76.9%

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

   3. associate-*l* 77.0%

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

   4. *-commutative 77.0%

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

   5. associate-*r* 77.0%

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

   6. *-commutative 77.0%

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

10. Applied egg-rr 77.0%

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

    1. associate-*r* 77.0%

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

    2. *-commutative 77.0%

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

    3. *-commutative 77.0%

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

    4. *-commutative 77.0%

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

    5. associate-*r* 76.9%

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

    6. *-commutative 76.9%

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

12. Simplified 76.9%

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

13. Final simplification 76.9%

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

Alternative 8: 74.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Simplified 81.9%

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

5. Taylor expanded in angle around 0 76.9%

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

   1. associate-*r* 76.9%

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

7. Simplified 76.9%

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

8. Taylor expanded in angle around 0 76.9%

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

   1. unpow2 76.9%

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

   2. associate-*l* 76.9%

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

   3. associate-*l* 77.0%

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

   4. *-commutative 77.0%

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

   5. associate-*r* 77.0%

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

   6. *-commutative 77.0%

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

10. Applied egg-rr 77.0%

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

    1. associate-*l* 75.7%

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

    2. associate-*r* 75.7%

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

    3. *-commutative 75.7%

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

12. Simplified 75.7%

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

    1. pow1 75.7%

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

    2. associate-*r* 77.0%

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

    3. *-commutative 77.0%

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

14. Applied egg-rr 77.0%

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

    1. unpow1 77.0%

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

16. Simplified 77.0%

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

17. Final simplification 77.0%

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

Reproduce

herbie shell --seed 2024053 
(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)))