ab-angle->ABCF C

Percentage Accurate: 79.7% → 79.7%

Time: 32.0s

Alternatives: 13

Speedup: 1.3×

• 36.8% 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, 0.7× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\frac{\pi}{\frac{-180}{angle_m}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{angle_m} \cdot \left(\left(\pi \cdot -0.005555555555555556\right) \cdot \sqrt{angle_m}\right)\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (/ PI (/ -180.0 angle_m)))) 2.0)
  (pow
   (*
    b
    (sin (* (sqrt angle_m) (* (* PI -0.005555555555555556) (sqrt angle_m)))))
   2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos((((double) M_PI) / (-180.0 / angle_m)))), 2.0) + pow((b * sin((sqrt(angle_m) * ((((double) M_PI) * -0.005555555555555556) * sqrt(angle_m))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.cos((Math.PI / (-180.0 / angle_m)))), 2.0) + Math.pow((b * Math.sin((Math.sqrt(angle_m) * ((Math.PI * -0.005555555555555556) * Math.sqrt(angle_m))))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.cos((math.pi / (-180.0 / angle_m)))), 2.0) + math.pow((b * math.sin((math.sqrt(angle_m) * ((math.pi * -0.005555555555555556) * math.sqrt(angle_m))))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos(Float64(pi / Float64(-180.0 / angle_m)))) ^ 2.0) + (Float64(b * sin(Float64(sqrt(angle_m) * Float64(Float64(pi * -0.005555555555555556) * sqrt(angle_m))))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * cos((pi / (-180.0 / angle_m)))) ^ 2.0) + ((b * sin((sqrt(angle_m) * ((pi * -0.005555555555555556) * sqrt(angle_m))))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[Sqrt[angle$95$m], $MachinePrecision] * N[(N[(Pi * -0.005555555555555556), $MachinePrecision] * N[Sqrt[angle$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

   1. associate-/r/ 78.5%

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

   2. add-sqr-sqrt 36.4%

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

   3. associate-*r* 36.5%

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

   4. div-inv 36.5%

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

   5. metadata-eval 36.5%

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

6. Applied egg-rr 36.5%

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

7. Final simplification 36.5%

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

Alternative 2: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\frac{\pi \cdot -0.005555555555555556}{\frac{1}{angle_m}}\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \frac{angle_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (/ (* PI -0.005555555555555556) (/ 1.0 angle_m)))) 2.0)
  (pow (* a (cos (* PI (/ angle_m 180.0)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * sin(((((double) M_PI) * -0.005555555555555556) / (1.0 / angle_m)))), 2.0) + pow((a * cos((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((b * Math.sin(((Math.PI * -0.005555555555555556) / (1.0 / angle_m)))), 2.0) + Math.pow((a * Math.cos((Math.PI * (angle_m / 180.0)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((b * math.sin(((math.pi * -0.005555555555555556) / (1.0 / angle_m)))), 2.0) + math.pow((a * math.cos((math.pi * (angle_m / 180.0)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(Float64(pi * -0.005555555555555556) / Float64(1.0 / angle_m)))) ^ 2.0) + (Float64(a * cos(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin(((pi * -0.005555555555555556) / (1.0 / angle_m)))) ^ 2.0) + ((a * cos((pi * (angle_m / 180.0)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(N[(Pi * -0.005555555555555556), $MachinePrecision] / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

   1. add-sqr-sqrt 36.3%

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

   2. sqrt-unprod 57.0%

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

   3. associate-*r/ 56.9%

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

   4. associate-*r/ 57.0%

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

   5. frac-times 56.6%

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

   6. *-commutative 56.6%

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

   7. *-commutative 56.6%

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

   8. metadata-eval 56.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\sqrt{\frac{\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)}{\color{blue}{32400}}}\right)\right)}^{2} \]

   9. metadata-eval 56.6%

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

   10. frac-times 57.0%

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

   11. associate-*r/ 57.0%

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

   12. associate-*r/ 57.0%

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

   13. sqrt-unprod 42.2%

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

   14. add-sqr-sqrt 78.5%

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

   15. *-commutative 78.5%

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

   16. associate-/r/ 78.5%

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

   17. div-inv 78.6%

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

   18. associate-/r* 78.6%

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

4. Applied egg-rr 78.6%

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

5. Final simplification 78.6%

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

Alternative 3: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(a \cdot \cos \left(\frac{\pi}{\frac{-180}{angle_m}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot -0.005555555555555556}{\frac{1}{angle_m}}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* a (cos (/ PI (/ -180.0 angle_m)))) 2.0)
  (pow (* b (sin (/ (* PI -0.005555555555555556) (/ 1.0 angle_m)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((a * cos((((double) M_PI) / (-180.0 / angle_m)))), 2.0) + pow((b * sin(((((double) M_PI) * -0.005555555555555556) / (1.0 / angle_m)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((a * Math.cos((Math.PI / (-180.0 / angle_m)))), 2.0) + Math.pow((b * Math.sin(((Math.PI * -0.005555555555555556) / (1.0 / angle_m)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((a * math.cos((math.pi / (-180.0 / angle_m)))), 2.0) + math.pow((b * math.sin(((math.pi * -0.005555555555555556) / (1.0 / angle_m)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(a * cos(Float64(pi / Float64(-180.0 / angle_m)))) ^ 2.0) + (Float64(b * sin(Float64(Float64(pi * -0.005555555555555556) / Float64(1.0 / angle_m)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * cos((pi / (-180.0 / angle_m)))) ^ 2.0) + ((b * sin(((pi * -0.005555555555555556) / (1.0 / angle_m)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(a * N[Cos[N[(Pi / N[(-180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(Pi * -0.005555555555555556), $MachinePrecision] / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

   1. associate-/r/ 78.5%

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

   2. add-sqr-sqrt 36.4%

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

   3. associate-*r* 36.5%

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

   4. div-inv 36.5%

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

   5. metadata-eval 36.5%

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

6. Applied egg-rr 36.5%

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

   1. associate-*l* 36.4%

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

   2. metadata-eval 36.4%

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

   3. div-inv 36.4%

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

   4. add-sqr-sqrt 78.5%

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

   5. associate-/r/ 78.5%

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

   6. div-inv 78.6%

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

   7. associate-/r* 78.6%

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

   8. div-inv 78.6%

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

   9. metadata-eval 78.6%

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

8. Applied egg-rr 78.6%

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

9. Final simplification 78.6%

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

Alternative 4: 79.8% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (/ PI -180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(angle$95$m * N[(Pi / -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]]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Final simplification 78.5%

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

Alternative 5: 79.7% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (/ (* PI -0.005555555555555556) (/ 1.0 angle_m)))) 2.0)
  (pow a 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(N[(Pi * -0.005555555555555556), $MachinePrecision] / N[(1.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

   1. associate-/r/ 78.5%

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

   2. add-sqr-sqrt 36.4%

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

   3. associate-*r* 36.5%

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

   4. div-inv 36.5%

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

   5. metadata-eval 36.5%

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

6. Applied egg-rr 36.5%

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

   1. associate-*l* 36.4%

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

   2. metadata-eval 36.4%

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

   3. div-inv 36.4%

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

   4. add-sqr-sqrt 78.5%

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

   5. associate-/r/ 78.5%

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

   6. div-inv 78.6%

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

   7. associate-/r* 78.6%

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

   8. div-inv 78.6%

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

   9. metadata-eval 78.6%

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

8. Applied egg-rr 78.6%

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

9. Taylor expanded in angle around 0 78.0%

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

10. Final simplification 78.0%

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

Alternative 6: 79.7% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (pow (* b (sin (* -0.005555555555555556 (* PI angle_m)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(-0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in angle around 0 78.0%

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

6. Taylor expanded in angle around inf 77.6%

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

7. Final simplification 77.6%

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

Alternative 7: 79.8% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* b (sin (* angle_m (/ PI -180.0)))) 2.0) (pow a 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in angle around 0 78.0%

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

6. Final simplification 78.0%

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

Alternative 8: 74.9% accurate, 2.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow a 2.0) (pow (* b (* angle_m (* PI -0.005555555555555556))) 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(angle$95$m * N[(Pi * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in angle around 0 78.0%

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

6. Taylor expanded in angle around 0 72.1%

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

   1. associate-*r* 72.1%

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

   2. *-commutative 72.1%

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

   3. *-commutative 72.1%

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

   4. associate-*r* 72.1%

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

   5. associate-*l* 72.1%

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

   6. *-commutative 72.1%

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

8. Simplified 72.1%

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

9. Final simplification 72.1%

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

Alternative 9: 73.2% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   (* angle_m -0.005555555555555556)
   (* (* PI b) (* angle_m (* PI (* b -0.005555555555555556)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(angle$95$m * -0.005555555555555556), $MachinePrecision] * N[(N[(Pi * b), $MachinePrecision] * N[(angle$95$m * N[(Pi * N[(b * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in angle around 0 78.0%

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

6. Taylor expanded in angle around 0 72.1%

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

   1. associate-*r* 72.1%

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

   2. *-commutative 72.1%

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

   3. associate-*l* 72.1%

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

   4. *-commutative 72.1%

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

8. Simplified 72.1%

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

   1. unpow2 72.1%

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

   2. associate-*r* 72.1%

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

   3. metadata-eval 72.1%

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

   4. div-inv 72.1%

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

   5. associate-*l* 71.3%

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

   6. div-inv 71.3%

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

   7. metadata-eval 71.3%

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

   8. *-commutative 71.3%

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

   9. *-commutative 71.3%

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

   10. associate-*r* 71.4%

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

10. Applied egg-rr 71.4%

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

11. Taylor expanded in angle around 0 71.4%

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

    1. *-commutative 71.4%

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

    2. *-commutative 71.4%

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

    3. associate-*l* 71.3%

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

    4. associate-*l* 71.4%

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

13. Simplified 71.4%

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

14. Final simplification 71.4%

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

Alternative 10: 73.2% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   (* angle_m -0.005555555555555556)
   (* (* PI b) (* -0.005555555555555556 (* angle_m (* PI b)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(angle$95$m * -0.005555555555555556), $MachinePrecision] * N[(N[(Pi * b), $MachinePrecision] * N[(-0.005555555555555556 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in angle around 0 78.0%

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

6. Taylor expanded in angle around 0 72.1%

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

   1. associate-*r* 72.1%

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

   2. *-commutative 72.1%

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

   3. associate-*l* 72.1%

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

   4. *-commutative 72.1%

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

8. Simplified 72.1%

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

   1. unpow2 72.1%

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

   2. associate-*r* 72.1%

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

   3. metadata-eval 72.1%

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

   4. div-inv 72.1%

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

   5. associate-*l* 71.3%

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

   6. div-inv 71.3%

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

   7. metadata-eval 71.3%

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

   8. *-commutative 71.3%

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

   9. *-commutative 71.3%

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

   10. associate-*r* 71.4%

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

10. Applied egg-rr 71.4%

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

11. Final simplification 71.4%

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

Alternative 11: 74.9% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* -0.005555555555555556 (* angle_m (* PI b)))))
   (+ (pow a 2.0) (* t_0 t_0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in angle around 0 78.0%

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

6. Taylor expanded in angle around 0 72.1%

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

   1. associate-*r* 72.1%

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

   2. *-commutative 72.1%

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

   3. associate-*l* 72.1%

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

   4. *-commutative 72.1%

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

8. Simplified 72.1%

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

   1. unpow2 72.1%

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

   2. *-commutative 72.1%

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

   3. associate-*r* 72.1%

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

   4. *-commutative 72.1%

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

   5. associate-*r* 72.1%

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

10. Applied egg-rr 72.1%

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

11. Final simplification 72.1%

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

Alternative 12: 73.8% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   (* PI b)
   (*
    (* -0.005555555555555556 (* angle_m (* PI b)))
    (* angle_m -0.005555555555555556)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(Pi * b), $MachinePrecision] * N[(N[(-0.005555555555555556 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in angle around 0 78.0%

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

6. Taylor expanded in angle around 0 72.1%

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

   1. associate-*r* 72.1%

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

   2. *-commutative 72.1%

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

   3. associate-*l* 72.1%

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

   4. *-commutative 72.1%

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

8. Simplified 72.1%

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

   1. unpow2 72.1%

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

   2. associate-*r* 72.1%

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

   3. metadata-eval 72.1%

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

   4. div-inv 72.1%

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

   5. associate-*r* 72.1%

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

   6. *-commutative 72.1%

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

   7. associate-*r* 72.1%

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

   8. div-inv 72.1%

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

   9. metadata-eval 72.1%

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

   10. *-commutative 72.1%

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

10. Applied egg-rr 72.1%

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

11. Final simplification 72.1%

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

Alternative 13: 74.9% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   (* (* PI b) (* angle_m -0.005555555555555556))
   (* -0.005555555555555556 (* b (* PI angle_m))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(N[(Pi * b), $MachinePrecision] * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision] * N[(-0.005555555555555556 * N[(b * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in angle around 0 78.0%

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

6. Taylor expanded in angle around 0 72.1%

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

   1. associate-*r* 72.1%

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

   2. *-commutative 72.1%

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

   3. associate-*l* 72.1%

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

   4. *-commutative 72.1%

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

8. Simplified 72.1%

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

   1. unpow2 72.1%

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

   2. associate-*r* 72.1%

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

   3. metadata-eval 72.1%

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

   4. div-inv 72.1%

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

   5. associate-*l* 71.3%

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

   6. div-inv 71.3%

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

   7. metadata-eval 71.3%

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

   8. *-commutative 71.3%

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

   9. *-commutative 71.3%

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

   10. associate-*r* 71.4%

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

10. Applied egg-rr 71.4%

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

    1. pow1 71.4%

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

    2. associate-*r* 72.1%

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

    3. *-commutative 72.1%

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

    4. *-commutative 72.1%

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

    5. associate-*r* 72.1%

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

12. Applied egg-rr 72.1%

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

13. Final simplification 72.1%

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

Reproduce

herbie shell --seed 2024017 
(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)))