ab-angle->ABCF C

Percentage Accurate: 80.2% → 80.1%

Time: 40.6s

Alternatives: 10

Speedup: N/A×

• 36.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot \left(e^{\mathsf{log1p}\left(\sqrt[3]{0.005555555555555556}\right)} + -1\right)\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (cos
     (*
      (pow
       (* (cbrt angle) (+ (exp (log1p (cbrt 0.005555555555555556))) -1.0))
       2.0)
      (* (cbrt (* angle 0.005555555555555556)) PI))))
   2.0)
  (pow (* b (sin (/ PI (/ -180.0 angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((pow((cbrt(angle) * (exp(log1p(cbrt(0.005555555555555556))) + -1.0)), 2.0) * (cbrt((angle * 0.005555555555555556)) * ((double) M_PI))))), 2.0) + pow((b * sin((((double) M_PI) / (-180.0 / angle)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.pow((Math.cbrt(angle) * (Math.exp(Math.log1p(Math.cbrt(0.005555555555555556))) + -1.0)), 2.0) * (Math.cbrt((angle * 0.005555555555555556)) * Math.PI)))), 2.0) + Math.pow((b * Math.sin((Math.PI / (-180.0 / angle)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64((Float64(cbrt(angle) * Float64(exp(log1p(cbrt(0.005555555555555556))) + -1.0)) ^ 2.0) * Float64(cbrt(Float64(angle * 0.005555555555555556)) * pi)))) ^ 2.0) + (Float64(b * sin(Float64(pi / Float64(-180.0 / angle)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(N[Power[N[(N[Power[angle, 1/3], $MachinePrecision] * N[(N[Exp[N[Log[1 + N[Power[0.005555555555555556, 1/3], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[(angle * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

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

3. Simplified 77.9%

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

   1. associate-/r/ 77.9%

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

   2. *-commutative 77.9%

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

   3. add-sqr-sqrt 37.7%

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

   4. sqrt-unprod 64.3%

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

   5. associate-*r/ 64.3%

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

   6. associate-*r/ 64.3%

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

   7. frac-times 64.0%

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

   8. *-commutative 64.0%

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

   9. *-commutative 64.0%

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

   10. metadata-eval 64.0%

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

   11. metadata-eval 64.0%

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

   12. frac-times 64.3%

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

   13. associate-*r/ 64.3%

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

   14. associate-*r/ 64.3%

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

   15. sqrt-unprod 40.2%

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

   16. add-sqr-sqrt 77.8%

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

   17. *-commutative 77.8%

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

   18. add-cube-cbrt 77.9%

       \[\leadsto {\left(a \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\frac{angle}{180}} \cdot \sqrt[3]{\frac{angle}{180}}\right) \cdot \sqrt[3]{\frac{angle}{180}}\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

6. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. cbrt-prod 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

8. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \left({\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
9. Step-by-step derivation

   1. expm1-log1p-u 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{0.005555555555555556}\right)\right)}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   2. expm1-udef 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{0.005555555555555556}\right)} - 1\right)}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

10. Applied egg-rr 78.0%

    \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{0.005555555555555556}\right)} - 1\right)}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

11. Final simplification 78.0%

    \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot \left(e^{\mathsf{log1p}\left(\sqrt[3]{0.005555555555555556}\right)} + -1\right)\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 2: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(a \cdot \cos \left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right) \cdot {\left(\sqrt[3]{0.005555555555555556} \cdot {\left(\sqrt[3]{\sqrt[3]{angle}}\right)}^{3}\right)}^{2}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (/ PI (/ -180.0 angle)))) 2.0)
  (pow
   (*
    a
    (cos
     (*
      (* (cbrt (* angle 0.005555555555555556)) PI)
      (pow
       (* (cbrt 0.005555555555555556) (pow (cbrt (cbrt angle)) 3.0))
       2.0))))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) / (-180.0 / angle)))), 2.0) + pow((a * cos(((cbrt((angle * 0.005555555555555556)) * ((double) M_PI)) * pow((cbrt(0.005555555555555556) * pow(cbrt(cbrt(angle)), 3.0)), 2.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI / (-180.0 / angle)))), 2.0) + Math.pow((a * Math.cos(((Math.cbrt((angle * 0.005555555555555556)) * Math.PI) * Math.pow((Math.cbrt(0.005555555555555556) * Math.pow(Math.cbrt(Math.cbrt(angle)), 3.0)), 2.0)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi / Float64(-180.0 / angle)))) ^ 2.0) + (Float64(a * cos(Float64(Float64(cbrt(Float64(angle * 0.005555555555555556)) * pi) * (Float64(cbrt(0.005555555555555556) * (cbrt(cbrt(angle)) ^ 3.0)) ^ 2.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[(N[Power[N[(angle * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision] * Pi), $MachinePrecision] * N[Power[N[(N[Power[0.005555555555555556, 1/3], $MachinePrecision] * N[Power[N[Power[N[Power[angle, 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

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

3. Simplified 77.9%

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

   1. associate-/r/ 77.9%

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

   2. *-commutative 77.9%

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

   3. add-sqr-sqrt 37.7%

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

   4. sqrt-unprod 64.3%

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

   5. associate-*r/ 64.3%

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

   6. associate-*r/ 64.3%

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

   7. frac-times 64.0%

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

   8. *-commutative 64.0%

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

   9. *-commutative 64.0%

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

   10. metadata-eval 64.0%

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

   11. metadata-eval 64.0%

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

   12. frac-times 64.3%

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

   13. associate-*r/ 64.3%

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

   14. associate-*r/ 64.3%

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

   15. sqrt-unprod 40.2%

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

   16. add-sqr-sqrt 77.8%

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

   17. *-commutative 77.8%

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

   18. add-cube-cbrt 77.9%

       \[\leadsto {\left(a \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\frac{angle}{180}} \cdot \sqrt[3]{\frac{angle}{180}}\right) \cdot \sqrt[3]{\frac{angle}{180}}\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

6. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. cbrt-prod 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

8. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \left({\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
9. Step-by-step derivation

   1. add-cube-cbrt 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{angle}} \cdot \sqrt[3]{\sqrt[3]{angle}}\right) \cdot \sqrt[3]{\sqrt[3]{angle}}\right)} \cdot \sqrt[3]{0.005555555555555556}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   2. pow3 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{angle}}\right)}^{3}} \cdot \sqrt[3]{0.005555555555555556}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

10. Applied egg-rr 78.0%

    \[\leadsto {\left(a \cdot \cos \left({\left(\color{blue}{{\left(\sqrt[3]{\sqrt[3]{angle}}\right)}^{3}} \cdot \sqrt[3]{0.005555555555555556}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

11. Final simplification 78.0%

    \[\leadsto {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(a \cdot \cos \left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right) \cdot {\left(\sqrt[3]{0.005555555555555556} \cdot {\left(\sqrt[3]{\sqrt[3]{angle}}\right)}^{3}\right)}^{2}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 3: 80.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(a \cdot \cos \left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right) \cdot {\left(\sqrt[3]{angle} \cdot {\left({0.005555555555555556}^{0.16666666666666666}\right)}^{2}\right)}^{2}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (/ PI (/ -180.0 angle)))) 2.0)
  (pow
   (*
    a
    (cos
     (*
      (* (cbrt (* angle 0.005555555555555556)) PI)
      (pow
       (*
        (cbrt angle)
        (pow (pow 0.005555555555555556 0.16666666666666666) 2.0))
       2.0))))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((((double) M_PI) / (-180.0 / angle)))), 2.0) + pow((a * cos(((cbrt((angle * 0.005555555555555556)) * ((double) M_PI)) * pow((cbrt(angle) * pow(pow(0.005555555555555556, 0.16666666666666666), 2.0)), 2.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((Math.PI / (-180.0 / angle)))), 2.0) + Math.pow((a * Math.cos(((Math.cbrt((angle * 0.005555555555555556)) * Math.PI) * Math.pow((Math.cbrt(angle) * Math.pow(Math.pow(0.005555555555555556, 0.16666666666666666), 2.0)), 2.0)))), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(b * sin(Float64(pi / Float64(-180.0 / angle)))) ^ 2.0) + (Float64(a * cos(Float64(Float64(cbrt(Float64(angle * 0.005555555555555556)) * pi) * (Float64(cbrt(angle) * ((0.005555555555555556 ^ 0.16666666666666666) ^ 2.0)) ^ 2.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[Sin[N[(Pi / N[(-180.0 / angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(N[(N[Power[N[(angle * 0.005555555555555556), $MachinePrecision], 1/3], $MachinePrecision] * Pi), $MachinePrecision] * N[Power[N[(N[Power[angle, 1/3], $MachinePrecision] * N[Power[N[Power[0.005555555555555556, 0.16666666666666666], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.7%

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

3. Simplified 77.9%

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

   1. associate-/r/ 77.9%

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

   2. *-commutative 77.9%

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

   3. add-sqr-sqrt 37.7%

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

   4. sqrt-unprod 64.3%

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

   5. associate-*r/ 64.3%

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

   6. associate-*r/ 64.3%

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

   7. frac-times 64.0%

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

   8. *-commutative 64.0%

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

   9. *-commutative 64.0%

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

   10. metadata-eval 64.0%

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

   11. metadata-eval 64.0%

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

   12. frac-times 64.3%

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

   13. associate-*r/ 64.3%

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

   14. associate-*r/ 64.3%

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

   15. sqrt-unprod 40.2%

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

   16. add-sqr-sqrt 77.8%

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

   17. *-commutative 77.8%

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

   18. add-cube-cbrt 77.9%

       \[\leadsto {\left(a \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\frac{angle}{180}} \cdot \sqrt[3]{\frac{angle}{180}}\right) \cdot \sqrt[3]{\frac{angle}{180}}\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

6. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. cbrt-prod 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

8. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \left({\color{blue}{\left(\sqrt[3]{angle} \cdot \sqrt[3]{0.005555555555555556}\right)}}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot \color{blue}{\left(\sqrt{\sqrt[3]{0.005555555555555556}} \cdot \sqrt{\sqrt[3]{0.005555555555555556}}\right)}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   2. pow2 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot \color{blue}{{\left(\sqrt{\sqrt[3]{0.005555555555555556}}\right)}^{2}}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   3. pow1/3 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot {\left(\sqrt{\color{blue}{{0.005555555555555556}^{0.3333333333333333}}}\right)}^{2}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   4. sqrt-pow1 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot {\color{blue}{\left({0.005555555555555556}^{\left(\frac{0.3333333333333333}{2}\right)}\right)}}^{2}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

   5. metadata-eval 78.0%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot {\left({0.005555555555555556}^{\color{blue}{0.16666666666666666}}\right)}^{2}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

10. Applied egg-rr 78.0%

    \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{angle} \cdot \color{blue}{{\left({0.005555555555555556}^{0.16666666666666666}\right)}^{2}}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

11. Final simplification 78.0%

    \[\leadsto {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} + {\left(a \cdot \cos \left(\left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right) \cdot {\left(\sqrt[3]{angle} \cdot {\left({0.005555555555555556}^{0.16666666666666666}\right)}^{2}\right)}^{2}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 4: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 77.8%

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

5. Taylor expanded in angle around 0 77.9%

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

6. Taylor expanded in angle around inf 77.9%

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

7. Final simplification 77.9%

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

Alternative 5: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 77.8%

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

5. Taylor expanded in angle around 0 77.9%

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

   1. add-cube-cbrt 77.6%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\color{blue}{\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}}}{-180}\right)\right)}^{2} \]

   2. unpow2 77.6%

      \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \sqrt[3]{\pi}}{-180}\right)\right)}^{2} \]

7. Applied egg-rr 77.6%

   \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(angle \cdot \frac{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2} \cdot \sqrt[3]{\pi}}}{-180}\right)\right)}^{2} \]

8. Taylor expanded in angle around inf 77.9%

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

   1. metadata-eval 77.9%

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

   2. associate-/r/ 77.9%

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

   3. associate-/l* 77.9%

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

   4. *-commutative 77.9%

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

   5. associate-/l* 77.9%

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

   6. metadata-eval 77.9%

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

   7. associate-/l* 78.0%

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

10. Simplified 78.0%

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

11. Final simplification 78.0%

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

Alternative 6: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 77.9%

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

   1. associate-/r/ 77.9%

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

   2. *-commutative 77.9%

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

   3. add-sqr-sqrt 37.7%

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

   4. sqrt-unprod 64.3%

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

   5. associate-*r/ 64.3%

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

   6. associate-*r/ 64.3%

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

   7. frac-times 64.0%

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

   8. *-commutative 64.0%

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

   9. *-commutative 64.0%

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

   10. metadata-eval 64.0%

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

   11. metadata-eval 64.0%

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

   12. frac-times 64.3%

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

   13. associate-*r/ 64.3%

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

   14. associate-*r/ 64.3%

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

   15. sqrt-unprod 40.2%

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

   16. add-sqr-sqrt 77.8%

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

   17. *-commutative 77.8%

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

   18. add-cube-cbrt 77.9%

       \[\leadsto {\left(a \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\frac{angle}{180}} \cdot \sqrt[3]{\frac{angle}{180}}\right) \cdot \sqrt[3]{\frac{angle}{180}}\right)} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

6. Applied egg-rr 78.0%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\sqrt[3]{angle \cdot 0.005555555555555556}\right)}^{2} \cdot \left(\sqrt[3]{angle \cdot 0.005555555555555556} \cdot \pi\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{\frac{-180}{angle}}\right)\right)}^{2} \]

7. Taylor expanded in angle around 0 78.0%

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

8. Final simplification 78.0%

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

Alternative 7: 73.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 77.8%

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

5. Taylor expanded in angle around 0 77.9%

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

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. unpow2 72.1%

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

   2. associate-*r* 72.1%

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

   3. associate-*l* 71.6%

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

   4. *-commutative 71.6%

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

   5. associate-*r* 71.6%

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

   6. *-commutative 71.6%

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

   7. associate-*r* 71.6%

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

   8. metadata-eval 71.6%

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

   9. associate-/r/ 71.6%

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

   10. clear-num 71.6%

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

   11. *-commutative 71.6%

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

   12. div-inv 71.6%

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

   13. metadata-eval 71.6%

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

8. Applied egg-rr 71.6%

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

9. Final simplification 71.6%

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

Alternative 8: 73.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 77.8%

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

5. Taylor expanded in angle around 0 77.9%

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

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. unpow2 72.1%

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

   2. associate-*r* 72.1%

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

   3. associate-*l* 71.6%

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

   4. *-commutative 71.6%

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

   5. associate-*r* 71.6%

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

   6. *-commutative 71.6%

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

   7. associate-*r* 71.6%

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

   8. metadata-eval 71.6%

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

   9. associate-/r/ 71.6%

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

   10. clear-num 71.6%

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

   11. *-commutative 71.6%

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

   12. div-inv 71.6%

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

   13. metadata-eval 71.6%

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

8. Applied egg-rr 71.6%

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

   1. metadata-eval 71.6%

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

   2. div-inv 71.6%

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

10. Applied egg-rr 71.6%

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

11. Final simplification 71.6%

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

Alternative 9: 75.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 77.8%

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

5. Taylor expanded in angle around 0 77.9%

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

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. unpow2 72.1%

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

   2. associate-*r* 72.1%

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

   3. *-commutative 72.1%

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

   4. associate-*r* 72.1%

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

   5. metadata-eval 72.1%

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

   6. associate-/r/ 72.1%

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

   7. clear-num 72.1%

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

   8. *-commutative 72.1%

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

   9. div-inv 72.1%

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

   10. metadata-eval 72.1%

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

   11. associate-*r* 72.1%

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

   12. *-commutative 72.1%

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

   13. associate-*r* 72.1%

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

   14. metadata-eval 72.1%

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

   15. associate-/r/ 72.1%

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

   16. clear-num 72.1%

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

   17. *-commutative 72.1%

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

   18. div-inv 72.1%

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

   19. metadata-eval 72.1%

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

8. Applied egg-rr 72.1%

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

9. Final simplification 72.1%

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

Alternative 10: 75.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Simplified 77.8%

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

5. Taylor expanded in angle around 0 77.9%

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

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. unpow2 72.1%

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

   2. associate-*r* 72.1%

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

   3. associate-*r* 72.1%

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

   4. *-commutative 72.1%

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

   5. associate-*r* 72.1%

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

   6. metadata-eval 72.1%

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

   7. associate-/r/ 72.1%

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

   8. clear-num 72.1%

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

   9. *-commutative 72.1%

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

   10. div-inv 72.1%

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

   11. metadata-eval 72.1%

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

   12. *-commutative 72.1%

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

   13. associate-*l* 72.1%

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

8. Applied egg-rr 72.1%

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

9. Final simplification 72.1%

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

Reproduce

herbie shell --seed 2024023 
(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)))