ab-angle->ABCF C

Percentage Accurate: 80.3% → 80.3%

Time: 28.9s

Alternatives: 17

Speedup: 1.0×

• 35.8% 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: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (* -0.005555555555555556 angle)))) 2.0)
  (pow
   (* b (log1p (expm1 (sin (* angle (* PI -0.005555555555555556))))))
   2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((((double) M_PI) * (-0.005555555555555556 * angle)))), 2.0) + pow((b * log1p(expm1(sin((angle * (((double) M_PI) * -0.005555555555555556)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.PI * (-0.005555555555555556 * angle)))), 2.0) + Math.pow((b * Math.log1p(Math.expm1(Math.sin((angle * (Math.PI * -0.005555555555555556)))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.cos((math.pi * (-0.005555555555555556 * angle)))), 2.0) + math.pow((b * math.log1p(math.expm1(math.sin((angle * (math.pi * -0.005555555555555556)))))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(pi * Float64(-0.005555555555555556 * angle)))) ^ 2.0) + (Float64(b * log1p(expm1(sin(Float64(angle * Float64(pi * -0.005555555555555556)))))) ^ 2.0))
end

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

   1. add-sqr-sqrt 35.6%

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

   2. sqrt-unprod 58.6%

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

   3. associate-*r/ 58.6%

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

   4. associate-*r/ 58.7%

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

   5. frac-times 58.5%

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

   6. *-commutative 58.5%

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

   7. *-commutative 58.5%

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

   8. metadata-eval 58.5%

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

   9. metadata-eval 58.5%

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

   10. frac-times 58.7%

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

   11. associate-*r/ 58.6%

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

   12. associate-*r/ 58.6%

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

   13. sqrt-unprod 38.7%

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

   14. add-sqr-sqrt 74.7%

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

   15. log1p-expm1-u 74.8%

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

6. Applied egg-rr 74.8%

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

7. Taylor expanded in a around 0 74.6%

   \[\leadsto {\color{blue}{\left(a \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\right)}^{2} \]
8. Step-by-step derivation

   1. associate-*r* 74.8%

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

   2. *-commutative 74.8%

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

9. Simplified 74.8%

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

10. Final simplification 74.8%

    \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right)}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\right)}^{2} \]
11. Add Preprocessing

Alternative 2: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (log1p (expm1 (sin (* angle (* PI -0.005555555555555556)))))) 2.0)
  (pow (* a (cos (* angle (/ PI -180.0)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * log1p(expm1(sin((angle * (((double) M_PI) * -0.005555555555555556)))))), 2.0) + pow((a * cos((angle * (((double) M_PI) / -180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.log1p(Math.expm1(Math.sin((angle * (Math.PI * -0.005555555555555556)))))), 2.0) + Math.pow((a * Math.cos((angle * (Math.PI / -180.0)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.log1p(math.expm1(math.sin((angle * (math.pi * -0.005555555555555556)))))), 2.0) + math.pow((a * math.cos((angle * (math.pi / -180.0)))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(b * log1p(expm1(sin(Float64(angle * Float64(pi * -0.005555555555555556)))))) ^ 2.0) + (Float64(a * cos(Float64(angle * Float64(pi / -180.0)))) ^ 2.0))
end

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

   1. add-sqr-sqrt 35.6%

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

   2. sqrt-unprod 58.6%

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

   3. associate-*r/ 58.6%

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

   4. associate-*r/ 58.7%

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

   5. frac-times 58.5%

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

   6. *-commutative 58.5%

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

   7. *-commutative 58.5%

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

   8. metadata-eval 58.5%

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

   9. metadata-eval 58.5%

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

   10. frac-times 58.7%

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

   11. associate-*r/ 58.6%

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

   12. associate-*r/ 58.6%

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

   13. sqrt-unprod 38.7%

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

   14. add-sqr-sqrt 74.7%

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

   15. log1p-expm1-u 74.8%

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

6. Applied egg-rr 74.8%

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

7. Final simplification 74.8%

   \[\leadsto {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 3: 80.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around inf 74.7%

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

6. Final simplification 74.7%

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

Alternative 4: 80.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (/ PI -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 = angle * (((double) M_PI) / -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 = angle * (Math.PI / -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 = angle * (math.pi / -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(angle * Float64(pi / -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 = angle * (pi / -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[(angle * 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 74.7%

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

3. Simplified 74.8%

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

5. Final simplification 74.8%

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

Alternative 5: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Taylor expanded in angle around inf 74.2%

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

7. Final simplification 74.2%

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

Alternative 6: 80.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Final simplification 74.3%

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

Alternative 7: 75.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 2.6 \cdot 10^{-12}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \pi \cdot \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(b \cdot \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 2.6e-12)
   (+ (pow a 2.0) (pow (* b (* PI (* -0.005555555555555556 angle))) 2.0))
   (+
    (pow (* a (cos (* angle (/ PI -180.0)))) 2.0)
    (*
     PI
     (*
      (* -0.005555555555555556 angle)
      (* b (* (* -0.005555555555555556 angle) (* PI b))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.6e-12) {
		tmp = pow(a, 2.0) + pow((b * (((double) M_PI) * (-0.005555555555555556 * angle))), 2.0);
	} else {
		tmp = pow((a * cos((angle * (((double) M_PI) / -180.0)))), 2.0) + (((double) M_PI) * ((-0.005555555555555556 * angle) * (b * ((-0.005555555555555556 * angle) * (((double) M_PI) * b)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 2.6e-12) {
		tmp = Math.pow(a, 2.0) + Math.pow((b * (Math.PI * (-0.005555555555555556 * angle))), 2.0);
	} else {
		tmp = Math.pow((a * Math.cos((angle * (Math.PI / -180.0)))), 2.0) + (Math.PI * ((-0.005555555555555556 * angle) * (b * ((-0.005555555555555556 * angle) * (Math.PI * b)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if angle <= 2.6e-12:
		tmp = math.pow(a, 2.0) + math.pow((b * (math.pi * (-0.005555555555555556 * angle))), 2.0)
	else:
		tmp = math.pow((a * math.cos((angle * (math.pi / -180.0)))), 2.0) + (math.pi * ((-0.005555555555555556 * angle) * (b * ((-0.005555555555555556 * angle) * (math.pi * b)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 2.6e-12)
		tmp = Float64((a ^ 2.0) + (Float64(b * Float64(pi * Float64(-0.005555555555555556 * angle))) ^ 2.0));
	else
		tmp = Float64((Float64(a * cos(Float64(angle * Float64(pi / -180.0)))) ^ 2.0) + Float64(pi * Float64(Float64(-0.005555555555555556 * angle) * Float64(b * Float64(Float64(-0.005555555555555556 * angle) * Float64(pi * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (angle <= 2.6e-12)
		tmp = (a ^ 2.0) + ((b * (pi * (-0.005555555555555556 * angle))) ^ 2.0);
	else
		tmp = ((a * cos((angle * (pi / -180.0)))) ^ 2.0) + (pi * ((-0.005555555555555556 * angle) * (b * ((-0.005555555555555556 * angle) * (pi * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 2.6e-12], N[(N[Power[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[(Pi * N[(-0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(a * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(Pi * N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * N[(b * N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 2.59999999999999983e-12

   1. Initial program 85.1%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in angle around 0 84.9%

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

   6. Taylor expanded in angle around 0 80.5%

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

      1. *-commutative 80.5%

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

   8. Simplified 80.5%

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

      1. unpow2 80.5%

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

      2. associate-*r* 80.5%

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

      3. associate-*l* 78.9%

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

      4. *-commutative 78.9%

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

      5. associate-*l* 78.9%

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

   10. Applied egg-rr 78.9%

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

   11. Taylor expanded in angle around 0 65.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   12. Step-by-step derivation

       1. associate-*r* 65.6%

          \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\pi}^{2}\right)} \]

       2. metadata-eval 65.6%

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

       3. unpow2 65.6%

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

       4. swap-sqr 65.6%

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

       5. *-commutative 65.6%

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

       6. unpow2 65.6%

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

       7. unpow2 65.6%

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

       8. swap-sqr 65.6%

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

       9. swap-sqr 80.4%

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

       10. unpow2 80.4%

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

       11. associate-*r* 80.5%

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

       12. associate-*r* 80.5%

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

       13. *-commutative 80.5%

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

       14. associate-*r* 80.5%

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

       15. *-commutative 80.5%

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

       16. *-commutative 80.5%

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

   13. Simplified 80.5%

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

   if 2.59999999999999983e-12 < angle

   1. Initial program 49.1%

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

   3. Simplified 49.4%

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

   5. Taylor expanded in angle around 0 40.6%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 40.7%

         \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\pi}^{2}\right)} \]

      2. *-commutative 40.7%

         \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right)} \]

      3. associate-*r* 40.6%

         \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{\left(\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot {\pi}^{2}\right) \cdot {b}^{2}} \]

      4. unpow2 40.6%

         \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(angle \cdot angle\right)}\right) \cdot {\pi}^{2}\right) \cdot {b}^{2} \]

      5. metadata-eval 40.6%

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

      6. swap-sqr 40.6%

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

      7. unpow2 40.6%

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

      8. swap-sqr 40.6%

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

      9. associate-*r* 40.6%

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

      10. *-commutative 40.6%

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

      11. associate-*r* 40.6%

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

      12. associate-*r* 40.6%

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

      13. *-commutative 40.6%

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

      14. associate-*r* 40.6%

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

      15. unpow2 40.6%

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

   7. Simplified 40.6%

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

      1. pow-prod-down 42.3%

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

      2. associate-*l* 42.3%

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

   9. Applied egg-rr 42.3%

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

       1. *-commutative 42.3%

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

       2. associate-*r* 42.4%

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

       3. *-commutative 42.4%

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

       4. associate-*r* 42.4%

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

       5. pow2 42.4%

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

       6. *-commutative 42.4%

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

       7. associate-*r* 42.4%

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

       8. associate-*l* 42.4%

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

       9. *-commutative 42.4%

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

       10. *-commutative 42.4%

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

       11. associate-*r* 43.3%

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

       12. *-commutative 43.3%

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

       13. associate-*l* 43.3%

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

       14. associate-*l* 43.3%

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

   11. Applied egg-rr 43.3%

       \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{\pi \cdot \left(\left(b \cdot \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot b\right)\right)\right) \cdot \left(-0.005555555555555556 \cdot angle\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 2.6 \cdot 10^{-12}:\\ \;\;\;\;{a}^{2} + {\left(b \cdot \left(\pi \cdot \left(-0.005555555555555556 \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \pi \cdot \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(b \cdot \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+172}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(\pi \cdot b\right) \cdot \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= b 1.45e+172)
   (+
    (pow (* a (cos (* angle (/ PI -180.0)))) 2.0)
    (*
     (* -0.005555555555555556 (* PI angle))
     (* b (* (* -0.005555555555555556 angle) (* PI b)))))
   (+
    (pow a 2.0)
    (*
     (* PI b)
     (*
      (* -0.005555555555555556 angle)
      (* -0.005555555555555556 (* PI (* angle b))))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e+172) {
		tmp = pow((a * cos((angle * (((double) M_PI) / -180.0)))), 2.0) + ((-0.005555555555555556 * (((double) M_PI) * angle)) * (b * ((-0.005555555555555556 * angle) * (((double) M_PI) * b))));
	} else {
		tmp = pow(a, 2.0) + ((((double) M_PI) * b) * ((-0.005555555555555556 * angle) * (-0.005555555555555556 * (((double) M_PI) * (angle * b)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (b <= 1.45e+172) {
		tmp = Math.pow((a * Math.cos((angle * (Math.PI / -180.0)))), 2.0) + ((-0.005555555555555556 * (Math.PI * angle)) * (b * ((-0.005555555555555556 * angle) * (Math.PI * b))));
	} else {
		tmp = Math.pow(a, 2.0) + ((Math.PI * b) * ((-0.005555555555555556 * angle) * (-0.005555555555555556 * (Math.PI * (angle * b)))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if b <= 1.45e+172:
		tmp = math.pow((a * math.cos((angle * (math.pi / -180.0)))), 2.0) + ((-0.005555555555555556 * (math.pi * angle)) * (b * ((-0.005555555555555556 * angle) * (math.pi * b))))
	else:
		tmp = math.pow(a, 2.0) + ((math.pi * b) * ((-0.005555555555555556 * angle) * (-0.005555555555555556 * (math.pi * (angle * b)))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (b <= 1.45e+172)
		tmp = Float64((Float64(a * cos(Float64(angle * Float64(pi / -180.0)))) ^ 2.0) + Float64(Float64(-0.005555555555555556 * Float64(pi * angle)) * Float64(b * Float64(Float64(-0.005555555555555556 * angle) * Float64(pi * b)))));
	else
		tmp = Float64((a ^ 2.0) + Float64(Float64(pi * b) * Float64(Float64(-0.005555555555555556 * angle) * Float64(-0.005555555555555556 * Float64(pi * Float64(angle * b))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (b <= 1.45e+172)
		tmp = ((a * cos((angle * (pi / -180.0)))) ^ 2.0) + ((-0.005555555555555556 * (pi * angle)) * (b * ((-0.005555555555555556 * angle) * (pi * b))));
	else
		tmp = (a ^ 2.0) + ((pi * b) * ((-0.005555555555555556 * angle) * (-0.005555555555555556 * (pi * (angle * b)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[b, 1.45e+172], N[(N[Power[N[(a * N[Cos[N[(angle * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(-0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision] * N[(b * N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(Pi * b), $MachinePrecision] * N[(N[(-0.005555555555555556 * angle), $MachinePrecision] * N[(-0.005555555555555556 * N[(Pi * N[(angle * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.45e172

   1. Initial program 72.4%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in angle around 0 58.4%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 58.4%

         \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\pi}^{2}\right)} \]

      2. *-commutative 58.4%

         \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right)} \]

      3. associate-*r* 58.4%

         \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{\left(\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot {\pi}^{2}\right) \cdot {b}^{2}} \]

      4. unpow2 58.4%

         \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(angle \cdot angle\right)}\right) \cdot {\pi}^{2}\right) \cdot {b}^{2} \]

      5. metadata-eval 58.4%

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

      6. swap-sqr 58.4%

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

      7. unpow2 58.4%

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

      8. swap-sqr 58.4%

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

      9. associate-*r* 58.4%

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

      10. *-commutative 58.4%

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

      11. associate-*r* 58.4%

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

      12. associate-*r* 58.4%

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

      13. *-commutative 58.4%

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

      14. associate-*r* 58.4%

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

      15. unpow2 58.4%

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

   7. Simplified 58.4%

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

      1. pow-prod-down 66.8%

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

      2. associate-*l* 66.8%

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

   9. Applied egg-rr 66.8%

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

       1. unpow2 66.8%

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

       2. associate-*r* 66.8%

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

       3. associate-*l* 66.7%

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

       4. associate-*r* 66.7%

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

       5. *-commutative 66.7%

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

       6. *-commutative 66.7%

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

       7. associate-*r* 66.7%

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

       8. *-commutative 66.7%

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

       9. associate-*r* 66.7%

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

       10. *-commutative 66.7%

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

       11. associate-*r* 66.7%

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

       12. associate-*l* 66.7%

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

       13. *-commutative 66.7%

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

       14. *-commutative 66.7%

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

   11. Applied egg-rr 66.7%

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

   if 1.45e172 < b

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in angle around 0 99.7%

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

   6. Taylor expanded in angle around 0 99.7%

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

      1. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. unpow2 99.7%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   10. Applied egg-rr 100.0%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(\left(-0.005555555555555556 \cdot \left(\pi \cdot \left(b \cdot angle\right)\right)\right) \cdot \left(-0.005555555555555556 \cdot angle\right)\right) \cdot \left(\pi \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.45 \cdot 10^{+172}:\\ \;\;\;\;{\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \left(b \cdot \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{a}^{2} + \left(\pi \cdot b\right) \cdot \left(\left(-0.005555555555555556 \cdot angle\right) \cdot \left(-0.005555555555555556 \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 58.6%

   \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* 58.6%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\pi}^{2}\right)} \]

   2. *-commutative 58.6%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \color{blue}{\left({\pi}^{2} \cdot {b}^{2}\right)} \]

   3. associate-*r* 58.6%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \color{blue}{\left(\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot {\pi}^{2}\right) \cdot {b}^{2}} \]

   4. unpow2 58.6%

      \[\leadsto {\left(a \cdot \cos \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + \left(\left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(angle \cdot angle\right)}\right) \cdot {\pi}^{2}\right) \cdot {b}^{2} \]

   5. metadata-eval 58.6%

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

   6. swap-sqr 58.6%

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

   7. unpow2 58.6%

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

   8. swap-sqr 58.6%

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

   9. associate-*r* 58.6%

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

   10. *-commutative 58.6%

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

   11. associate-*r* 58.6%

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

   12. associate-*r* 58.6%

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

   13. *-commutative 58.6%

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

   14. associate-*r* 58.6%

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

   15. unpow2 58.6%

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

7. Simplified 58.6%

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

   1. pow-prod-down 69.6%

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

   2. associate-*l* 69.6%

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

9. Applied egg-rr 69.6%

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

    1. unpow2 69.6%

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

    2. *-commutative 69.6%

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

    3. associate-*r* 69.7%

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

    4. associate-*l* 69.7%

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

    5. *-commutative 69.7%

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

    6. *-commutative 69.7%

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

    7. *-commutative 69.7%

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

    8. associate-*l* 69.7%

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

    9. associate-*l* 69.7%

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

    10. *-commutative 69.7%

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

11. Applied egg-rr 69.7%

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

    1. associate-*r* 69.7%

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

    2. *-commutative 69.7%

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

    3. associate-*l* 69.7%

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

    4. associate-*r* 68.8%

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

    5. *-commutative 68.8%

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

    6. *-commutative 68.8%

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

    7. associate-*r* 68.8%

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

    8. *-commutative 68.8%

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

    9. *-commutative 68.8%

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

    10. associate-*r* 68.8%

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

    11. associate-*r* 68.8%

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

    12. *-commutative 68.8%

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

    13. associate-*r* 68.8%

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

    14. *-commutative 68.8%

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

    15. *-commutative 68.8%

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

13. Simplified 68.8%

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

14. Final simplification 68.8%

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

Alternative 10: 75.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Taylor expanded in angle around 0 68.7%

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

   1. *-commutative 68.7%

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

8. Simplified 68.7%

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

9. Final simplification 68.7%

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

Alternative 11: 75.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Taylor expanded in angle around 0 68.7%

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

7. Final simplification 68.7%

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

Alternative 12: 75.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Taylor expanded in angle around 0 68.7%

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

   1. *-commutative 68.7%

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

8. Simplified 68.7%

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

   1. unpow2 68.7%

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

   2. associate-*r* 68.7%

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

   3. associate-*l* 67.9%

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

   4. *-commutative 67.9%

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

   5. associate-*l* 67.9%

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

10. Applied egg-rr 67.9%

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

11. Taylor expanded in angle around 0 57.6%

    \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\pi}^{2}\right)\right)} \]
12. Step-by-step derivation

    1. associate-*r* 57.6%

       \[\leadsto {\left(a \cdot 1\right)}^{2} + \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot {angle}^{2}\right) \cdot \left({b}^{2} \cdot {\pi}^{2}\right)} \]

    2. metadata-eval 57.6%

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

    3. unpow2 57.6%

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

    4. swap-sqr 57.6%

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

    5. *-commutative 57.6%

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

    6. unpow2 57.6%

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

    7. unpow2 57.6%

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

    8. swap-sqr 57.6%

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

    9. swap-sqr 68.7%

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

    10. unpow2 68.7%

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

    11. associate-*r* 68.7%

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

    12. associate-*r* 68.7%

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

    13. *-commutative 68.7%

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

    14. associate-*r* 68.7%

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

    15. *-commutative 68.7%

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

    16. *-commutative 68.7%

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

13. Simplified 68.7%

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

14. Final simplification 68.7%

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

Alternative 13: 73.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Taylor expanded in angle around 0 68.7%

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

   1. *-commutative 68.7%

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

8. Simplified 68.7%

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

   1. unpow2 68.7%

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

   2. associate-*r* 68.7%

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

   3. associate-*l* 67.9%

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

   4. *-commutative 67.9%

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

   5. associate-*l* 67.9%

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

10. Applied egg-rr 67.9%

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

11. Taylor expanded in b around 0 67.9%

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

12. Final simplification 67.9%

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

Alternative 14: 73.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Taylor expanded in angle around 0 68.7%

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

   1. *-commutative 68.7%

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

8. Simplified 68.7%

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

   1. unpow2 68.7%

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

   2. associate-*r* 68.7%

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

   3. associate-*l* 67.9%

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

   4. *-commutative 67.9%

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

   5. associate-*l* 67.9%

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

10. Applied egg-rr 67.9%

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

11. Final simplification 67.9%

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

Alternative 15: 73.5% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Taylor expanded in angle around 0 68.7%

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

   1. *-commutative 68.7%

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

8. Simplified 68.7%

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

   1. unpow2 68.7%

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

   2. associate-*r* 68.7%

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

   3. associate-*l* 67.9%

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

   4. *-commutative 67.9%

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

   5. associate-*l* 67.9%

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

10. Applied egg-rr 67.9%

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

11. Taylor expanded in b around 0 67.9%

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

    1. *-commutative 67.9%

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

    2. associate-*r* 67.9%

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

13. Simplified 67.9%

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

14. Final simplification 67.9%

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

Alternative 16: 75.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Taylor expanded in angle around 0 68.7%

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

   1. *-commutative 68.7%

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

8. Simplified 68.7%

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

   1. unpow2 68.7%

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

   2. *-commutative 68.7%

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

   3. associate-*l* 68.8%

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

   4. *-commutative 68.8%

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

   5. associate-*l* 68.7%

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

10. Applied egg-rr 68.7%

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

11. Final simplification 68.7%

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

Alternative 17: 75.1% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Simplified 74.8%

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

5. Taylor expanded in angle around 0 74.3%

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

6. Taylor expanded in angle around 0 68.7%

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

   1. *-commutative 68.7%

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

8. Simplified 68.7%

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

   1. unpow2 68.7%

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

   2. associate-*r* 68.7%

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

   3. *-commutative 68.7%

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

   4. associate-*l* 68.7%

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

   5. *-commutative 68.7%

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

   6. associate-*l* 68.7%

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

10. Applied egg-rr 68.7%

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

11. Final simplification 68.7%

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

Reproduce

herbie shell --seed 2024044 
(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)))