ab-angle->ABCF C

Percentage Accurate: 80.1% → 80.2%

Time: 20.6s

Alternatives: 8

Speedup: 1.3×

• 62.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* PI (/ angle 180))))
  (+ (pow (* a (cos t_0)) 2) (pow (* b (sin t_0)) 2))))

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), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Initial Program: 80.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (let* ((t_0 (* PI (/ angle 180))))
  (+ (pow (* a (cos t_0)) 2) (pow (* b (sin t_0)) 2))))

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), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 80.2% accurate, 1.2× speedup

Math

\[\left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\frac{1}{90} \cdot \left(\pi \cdot \left|angle\right|\right) - \frac{-1}{2} \cdot \pi\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot \left|angle\right|\right)\right)}^{2} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (*
  (+ 1/2 (* 1/2 (sin (- (* 1/90 (* PI (fabs angle))) (* -1/2 PI)))))
  (* a a))
 (pow (* b (sin (* (* 1/180 PI) (fabs angle)))) 2)))

C

double code(double a, double b, double angle) {
	return ((0.5 + (0.5 * sin(((0.011111111111111112 * (((double) M_PI) * fabs(angle))) - (-0.5 * ((double) M_PI)))))) * (a * a)) + pow((b * sin(((0.005555555555555556 * ((double) M_PI)) * fabs(angle)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return ((0.5 + (0.5 * Math.sin(((0.011111111111111112 * (Math.PI * Math.abs(angle))) - (-0.5 * Math.PI))))) * (a * a)) + Math.pow((b * Math.sin(((0.005555555555555556 * Math.PI) * Math.abs(angle)))), 2.0);
}

Python

def code(a, b, angle):
	return ((0.5 + (0.5 * math.sin(((0.011111111111111112 * (math.pi * math.fabs(angle))) - (-0.5 * math.pi))))) * (a * a)) + math.pow((b * math.sin(((0.005555555555555556 * math.pi) * math.fabs(angle)))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(0.5 + Float64(0.5 * sin(Float64(Float64(0.011111111111111112 * Float64(pi * abs(angle))) - Float64(-0.5 * pi))))) * Float64(a * a)) + (Float64(b * sin(Float64(Float64(0.005555555555555556 * pi) * abs(angle)))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((0.5 + (0.5 * sin(((0.011111111111111112 * (pi * abs(angle))) - (-0.5 * pi))))) * (a * a)) + ((b * sin(((0.005555555555555556 * pi) * abs(angle)))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(1/2 + N[(1/2 * N[Sin[N[(N[(1/90 * N[(Pi * N[Abs[angle], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(1/180 * Pi), $MachinePrecision] * N[Abs[angle], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 80.1%

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

3. Applied rewrites 80.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 80.2%

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

5. Applied rewrites 80.2%

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

7. Applied rewrites 80.2%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\frac{-1}{90}}\right)\right) \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

   9. metadata-eval N/A

      \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\color{blue}{\frac{1}{90}} \cdot \left(angle \cdot \pi\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

   10. *-commutative N/A

       \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right) \cdot \frac{1}{90}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

   11. lift-*.f64 N/A

       \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right) \cdot \frac{1}{90}} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

   12. lower-sin.f64 N/A

       \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90} + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

   13. add-flip N/A

       \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90} - \left(\mathsf{neg}\left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right)}\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

   14. lift-PI.f64 N/A

       \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90} - \left(\mathsf{neg}\left(\frac{\color{blue}{\pi}}{2}\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

   15. mult-flip N/A

       \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90} - \left(\mathsf{neg}\left(\color{blue}{\pi \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

   16. metadata-eval N/A

       \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90} - \left(\mathsf{neg}\left(\pi \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

   17. distribute-lft-neg-out N/A

       \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \sin \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90} - \color{blue}{\left(\mathsf{neg}\left(\pi\right)\right) \cdot \frac{1}{2}}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

9. Applied rewrites 80.1%

   \[\leadsto \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\sin \left(\frac{1}{90} \cdot \left(\pi \cdot angle\right) - \frac{-1}{2} \cdot \pi\right)}\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 80.1% accurate, 1.3× speedup

Math

\[\left(1 - \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(\pi \cdot angle\right)\right) \cdot \frac{1}{2}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (* (- 1 (- 1/2 (* (cos (* 1/90 (* PI angle))) 1/2))) (* a a))
 (pow (* b (sin (* (* 1/180 PI) angle))) 2)))

C

double code(double a, double b, double angle) {
	return ((1.0 - (0.5 - (cos((0.011111111111111112 * (((double) M_PI) * angle))) * 0.5))) * (a * a)) + pow((b * sin(((0.005555555555555556 * ((double) M_PI)) * angle))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return ((1.0 - (0.5 - (Math.cos((0.011111111111111112 * (Math.PI * angle))) * 0.5))) * (a * a)) + Math.pow((b * Math.sin(((0.005555555555555556 * Math.PI) * angle))), 2.0);
}

Python

def code(a, b, angle):
	return ((1.0 - (0.5 - (math.cos((0.011111111111111112 * (math.pi * angle))) * 0.5))) * (a * a)) + math.pow((b * math.sin(((0.005555555555555556 * math.pi) * angle))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(1.0 - Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(pi * angle))) * 0.5))) * Float64(a * a)) + (Float64(b * sin(Float64(Float64(0.005555555555555556 * pi) * angle))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((1.0 - (0.5 - (cos((0.011111111111111112 * (pi * angle))) * 0.5))) * (a * a)) + ((b * sin(((0.005555555555555556 * pi) * angle))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(1 - N[(1/2 - N[(N[Cos[N[(1/90 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(1/180 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 80.1%

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

3. Applied rewrites 80.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 80.2%

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

5. Applied rewrites 80.2%

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

7. Applied rewrites 80.2%

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

9. Applied rewrites 80.2%

   \[\leadsto \color{blue}{\left(1 - \left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(\pi \cdot angle\right)\right) \cdot \frac{1}{2}\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\left(\frac{1}{180} \cdot \pi\right) \cdot angle\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 3: 80.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (a b angle)
  :precision binary64
  (+ (* (+ 1/2 1/2) (* a a)) (pow (* b (sin (* (* 1/180 PI) angle))) 2)))

C

double code(double a, double b, double angle) {
	return ((0.5 + 0.5) * (a * a)) + pow((b * sin(((0.005555555555555556 * ((double) M_PI)) * angle))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return ((0.5 + 0.5) * (a * a)) + Math.pow((b * Math.sin(((0.005555555555555556 * Math.PI) * angle))), 2.0);
}

Python

def code(a, b, angle):
	return ((0.5 + 0.5) * (a * a)) + math.pow((b * math.sin(((0.005555555555555556 * math.pi) * angle))), 2.0)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(0.5 + 0.5) * Float64(a * a)) + (Float64(b * sin(Float64(Float64(0.005555555555555556 * pi) * angle))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((0.5 + 0.5) * (a * a)) + ((b * sin(((0.005555555555555556 * pi) * angle))) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(1/2 + 1/2), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(1/180 * Pi), $MachinePrecision] * angle), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 80.1%

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

3. Applied rewrites 80.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 80.2%

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

5. Applied rewrites 80.2%

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

7. Applied rewrites 80.2%

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

8. Taylor expanded in angle around 0

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

10. Applied rewrites 80.1%

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

Alternative 4: 71.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|a\right| \leq \frac{3950954765291961}{253266331108459042877954581524118722595974501479640924072000569439126758509088631982403994686712878069348015540240526683495797795130113239006767262824338603946605334680267915264}:\\ \;\;\;\;\left(b \cdot b\right) \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \pi\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot \left|a\right|\right) \cdot \left|a\right|\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<=
     (fabs a)
     3950954765291961/253266331108459042877954581524118722595974501479640924072000569439126758509088631982403994686712878069348015540240526683495797795130113239006767262824338603946605334680267915264)
  (* (* b b) (pow (sin (* 1/180 (* angle PI))) 2))
  (+
   (* (* (- 1/2 (* 1/2 (cos (* 1/90 (* angle PI))))) b) b)
   (* (* (+ 1/2 1/2) (fabs a)) (fabs a)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(a) <= 1.56e-161) {
		tmp = (b * b) * pow(sin((0.005555555555555556 * (angle * ((double) M_PI)))), 2.0);
	} else {
		tmp = (((0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) * b) * b) + (((0.5 + 0.5) * fabs(a)) * fabs(a));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(a) <= 1.56e-161) {
		tmp = (b * b) * Math.pow(Math.sin((0.005555555555555556 * (angle * Math.PI))), 2.0);
	} else {
		tmp = (((0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) * b) * b) + (((0.5 + 0.5) * Math.abs(a)) * Math.abs(a));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(a) <= 1.56e-161:
		tmp = (b * b) * math.pow(math.sin((0.005555555555555556 * (angle * math.pi))), 2.0)
	else:
		tmp = (((0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) * b) * b) + (((0.5 + 0.5) * math.fabs(a)) * math.fabs(a))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(a) <= 1.56e-161)
		tmp = Float64(Float64(b * b) * (sin(Float64(0.005555555555555556 * Float64(angle * pi))) ^ 2.0));
	else
		tmp = Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) * b) * b) + Float64(Float64(Float64(0.5 + 0.5) * abs(a)) * abs(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(a) <= 1.56e-161)
		tmp = (b * b) * (sin((0.005555555555555556 * (angle * pi))) ^ 2.0);
	else
		tmp = (((0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))) * b) * b) + (((0.5 + 0.5) * abs(a)) * abs(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[a], $MachinePrecision], 3950954765291961/253266331108459042877954581524118722595974501479640924072000569439126758509088631982403994686712878069348015540240526683495797795130113239006767262824338603946605334680267915264], N[(N[(b * b), $MachinePrecision] * N[Power[N[Sin[N[(1/180 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1/2 - N[(1/2 * N[Cos[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision] * N[Abs[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.56e-161

   1. Initial program 80.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 80.1%

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

   3. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 80.2%

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto {b}^{2} \cdot {\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}^{2} \]

      3. lower-pow.f64 N/A

         \[\leadsto {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{\color{blue}{2}} \]

      4. lower-sin.f64 N/A

         \[\leadsto {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. lower-PI.f64 34.6%

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

   8. Applied rewrites 34.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 34.6%

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

   10. Applied rewrites 34.6%

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

   if 1.56e-161 < a

   1. Initial program 80.1%

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

   3. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]
   5. Step-by-step derivation

   6. Applied rewrites 57.3%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a \]
   8. Step-by-step derivation

   9. Applied rewrites 57.4%

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

   10. Taylor expanded in angle around inf

       \[\leadsto \left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

       4. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

       6. lower-PI.f64 68.6%

          \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

   12. Applied rewrites 68.6%

       \[\leadsto \left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 68.6% accurate, 3.1× speedup

Math

\[\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+
 (* (* (- 1/2 (* 1/2 (cos (* 1/90 (* angle PI))))) b) b)
 (* (* (+ 1/2 1/2) a) a)))

C

double code(double a, double b, double angle) {
	return (((0.5 - (0.5 * cos((0.011111111111111112 * (angle * ((double) M_PI)))))) * b) * b) + (((0.5 + 0.5) * a) * a);
}

Java

public static double code(double a, double b, double angle) {
	return (((0.5 - (0.5 * Math.cos((0.011111111111111112 * (angle * Math.PI))))) * b) * b) + (((0.5 + 0.5) * a) * a);
}

Python

def code(a, b, angle):
	return (((0.5 - (0.5 * math.cos((0.011111111111111112 * (angle * math.pi))))) * b) * b) + (((0.5 + 0.5) * a) * a)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(0.011111111111111112 * Float64(angle * pi))))) * b) * b) + Float64(Float64(Float64(0.5 + 0.5) * a) * a))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (((0.5 - (0.5 * cos((0.011111111111111112 * (angle * pi))))) * b) * b) + (((0.5 + 0.5) * a) * a);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(N[(1/2 - N[(1/2 * N[Cos[N[(1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

3. Applied rewrites 68.7%

   \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

4. Taylor expanded in angle around 0

   \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]
5. Step-by-step derivation

6. Applied rewrites 57.3%

   \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

7. Taylor expanded in angle around 0

   \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a \]
8. Step-by-step derivation

9. Applied rewrites 57.4%

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

10. Taylor expanded in angle around inf

    \[\leadsto \left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]
11. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

    3. lower-cos.f64 N/A

       \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

    4. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

    5. lower-*.f64 N/A

       \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

    6. lower-PI.f64 68.6%

       \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

12. Applied rewrites 68.6%

    \[\leadsto \left(\color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)} \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]
13. Add Preprocessing

Alternative 6: 60.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 45000000000000001093790020844307659491309099240666167757791051529882576298935378714552136028009316752614990502760270500706746399264931364234169536023864079809685017457168411131102292168329892380638610266183315693802356736:\\ \;\;\;\;\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left|b\right|\right) \cdot \left|b\right| + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left|b\right| \cdot \left(\frac{-1}{2} \cdot \left(\left|b\right| \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + \frac{1}{2} \cdot \left|b\right|\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<=
     (fabs b)
     45000000000000001093790020844307659491309099240666167757791051529882576298935378714552136028009316752614990502760270500706746399264931364234169536023864079809685017457168411131102292168329892380638610266183315693802356736)
  (+ (* (* (- 1/2 1/2) (fabs b)) (fabs b)) (* (* (+ 1/2 1/2) a) a))
  (*
   (fabs b)
   (+
    (* -1/2 (* (fabs b) (cos (* -1/90 (* angle PI)))))
    (* 1/2 (fabs b))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 4.5e+220) {
		tmp = (((0.5 - 0.5) * fabs(b)) * fabs(b)) + (((0.5 + 0.5) * a) * a);
	} else {
		tmp = fabs(b) * ((-0.5 * (fabs(b) * cos((-0.011111111111111112 * (angle * ((double) M_PI)))))) + (0.5 * fabs(b)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 4.5e+220) {
		tmp = (((0.5 - 0.5) * Math.abs(b)) * Math.abs(b)) + (((0.5 + 0.5) * a) * a);
	} else {
		tmp = Math.abs(b) * ((-0.5 * (Math.abs(b) * Math.cos((-0.011111111111111112 * (angle * Math.PI))))) + (0.5 * Math.abs(b)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 4.5e+220:
		tmp = (((0.5 - 0.5) * math.fabs(b)) * math.fabs(b)) + (((0.5 + 0.5) * a) * a)
	else:
		tmp = math.fabs(b) * ((-0.5 * (math.fabs(b) * math.cos((-0.011111111111111112 * (angle * math.pi))))) + (0.5 * math.fabs(b)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 4.5e+220)
		tmp = Float64(Float64(Float64(Float64(0.5 - 0.5) * abs(b)) * abs(b)) + Float64(Float64(Float64(0.5 + 0.5) * a) * a));
	else
		tmp = Float64(abs(b) * Float64(Float64(-0.5 * Float64(abs(b) * cos(Float64(-0.011111111111111112 * Float64(angle * pi))))) + Float64(0.5 * abs(b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 4.5e+220)
		tmp = (((0.5 - 0.5) * abs(b)) * abs(b)) + (((0.5 + 0.5) * a) * a);
	else
		tmp = abs(b) * ((-0.5 * (abs(b) * cos((-0.011111111111111112 * (angle * pi))))) + (0.5 * abs(b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 45000000000000001093790020844307659491309099240666167757791051529882576298935378714552136028009316752614990502760270500706746399264931364234169536023864079809685017457168411131102292168329892380638610266183315693802356736], N[(N[(N[(N[(1/2 - 1/2), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[Abs[b], $MachinePrecision] * N[(N[(-1/2 * N[(N[Abs[b], $MachinePrecision] * N[Cos[N[(-1/90 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1/2 * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5000000000000001e220

   1. Initial program 80.1%

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

   3. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]
   5. Step-by-step derivation

   6. Applied rewrites 57.3%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a \]
   8. Step-by-step derivation

   9. Applied rewrites 57.4%

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

   if 4.5000000000000001e220 < b

   1. Initial program 80.1%

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

   3. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right)} \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(b \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)\right)} \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      3. lift--.f64 N/A

         \[\leadsto \left(b \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)}\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      4. lift-*.f64 N/A

         \[\leadsto \left(b \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)}\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(b \cdot \color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)}\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      6. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(b \cdot \frac{1}{2} + b \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)\right)} \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      7. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(b \cdot \frac{1}{2} + b \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)\right)} \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{b \cdot \frac{1}{2}} + b \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      9. lower-*.f64 N/A

         \[\leadsto \left(b \cdot \frac{1}{2} + \color{blue}{b \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)}\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      10. lower-*.f64 N/A

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)}\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      11. metadata-eval 68.6%

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \left(\color{blue}{\frac{-1}{2}} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \left(\frac{-1}{2} \cdot \color{blue}{\cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)}\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      13. cos-neg-rev N/A

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \left(\frac{-1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)}\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \left(\frac{-1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right)}\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      15. lift-*.f64 N/A

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \left(\frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(angle \cdot \pi\right) \cdot \frac{1}{90}}\right)\right)\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      16. *-commutative N/A

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \left(\frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{90} \cdot \left(angle \cdot \pi\right)}\right)\right)\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      17. distribute-lft-neg-in N/A

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \left(\frac{-1}{2} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{90}\right)\right) \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      18. lower-*.f64 N/A

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \left(\frac{-1}{2} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{90}\right)\right) \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

      19. metadata-eval 68.6%

          \[\leadsto \left(b \cdot \frac{1}{2} + b \cdot \left(\frac{-1}{2} \cdot \cos \left(\color{blue}{\frac{-1}{90}} \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

   5. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\left(b \cdot \frac{1}{2} + b \cdot \left(\frac{-1}{2} \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right)\right)} \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

   6. Taylor expanded in a around 0

      \[\leadsto \color{blue}{b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + \frac{1}{2} \cdot b\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto b \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot b\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{\frac{1}{2} \cdot b}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \color{blue}{\frac{1}{2}} \cdot b\right) \]

      4. lower-*.f64 N/A

         \[\leadsto b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot b\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot b\right) \]

      6. lower-*.f64 N/A

         \[\leadsto b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot b\right) \]

      7. lower-*.f64 N/A

         \[\leadsto b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{1}{2} \cdot b\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + \frac{1}{2} \cdot b\right) \]

      9. lower-*.f64 27.1%

         \[\leadsto b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + \frac{1}{2} \cdot \color{blue}{b}\right) \]

   8. Applied rewrites 27.1%

      \[\leadsto \color{blue}{b \cdot \left(\frac{-1}{2} \cdot \left(b \cdot \cos \left(\frac{-1}{90} \cdot \left(angle \cdot \pi\right)\right)\right) + \frac{1}{2} \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 60.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|b\right| \leq 45000000000000001093790020844307659491309099240666167757791051529882576298935378714552136028009316752614990502760270500706746399264931364234169536023864079809685017457168411131102292168329892380638610266183315693802356736:\\ \;\;\;\;\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot \left|b\right|\right) \cdot \left|b\right| + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(\pi \cdot angle\right)\right) \cdot \frac{1}{2}\right) \cdot \left(\left|b\right| \cdot \left|b\right|\right)\\ \end{array} \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (if (<=
     (fabs b)
     45000000000000001093790020844307659491309099240666167757791051529882576298935378714552136028009316752614990502760270500706746399264931364234169536023864079809685017457168411131102292168329892380638610266183315693802356736)
  (+ (* (* (- 1/2 1/2) (fabs b)) (fabs b)) (* (* (+ 1/2 1/2) a) a))
  (*
   (- 1/2 (* (cos (* 1/90 (* PI angle))) 1/2))
   (* (fabs b) (fabs b)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (fabs(b) <= 4.5e+220) {
		tmp = (((0.5 - 0.5) * fabs(b)) * fabs(b)) + (((0.5 + 0.5) * a) * a);
	} else {
		tmp = (0.5 - (cos((0.011111111111111112 * (((double) M_PI) * angle))) * 0.5)) * (fabs(b) * fabs(b));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (Math.abs(b) <= 4.5e+220) {
		tmp = (((0.5 - 0.5) * Math.abs(b)) * Math.abs(b)) + (((0.5 + 0.5) * a) * a);
	} else {
		tmp = (0.5 - (Math.cos((0.011111111111111112 * (Math.PI * angle))) * 0.5)) * (Math.abs(b) * Math.abs(b));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if math.fabs(b) <= 4.5e+220:
		tmp = (((0.5 - 0.5) * math.fabs(b)) * math.fabs(b)) + (((0.5 + 0.5) * a) * a)
	else:
		tmp = (0.5 - (math.cos((0.011111111111111112 * (math.pi * angle))) * 0.5)) * (math.fabs(b) * math.fabs(b))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (abs(b) <= 4.5e+220)
		tmp = Float64(Float64(Float64(Float64(0.5 - 0.5) * abs(b)) * abs(b)) + Float64(Float64(Float64(0.5 + 0.5) * a) * a));
	else
		tmp = Float64(Float64(0.5 - Float64(cos(Float64(0.011111111111111112 * Float64(pi * angle))) * 0.5)) * Float64(abs(b) * abs(b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (abs(b) <= 4.5e+220)
		tmp = (((0.5 - 0.5) * abs(b)) * abs(b)) + (((0.5 + 0.5) * a) * a);
	else
		tmp = (0.5 - (cos((0.011111111111111112 * (pi * angle))) * 0.5)) * (abs(b) * abs(b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[N[Abs[b], $MachinePrecision], 45000000000000001093790020844307659491309099240666167757791051529882576298935378714552136028009316752614990502760270500706746399264931364234169536023864079809685017457168411131102292168329892380638610266183315693802356736], N[(N[(N[(N[(1/2 - 1/2), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(N[(1/2 - N[(N[Cos[N[(1/90 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[b], $MachinePrecision] * N[Abs[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.5000000000000001e220

   1. Initial program 80.1%

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

   3. Applied rewrites 68.7%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

   4. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]
   5. Step-by-step derivation

   6. Applied rewrites 57.3%

      \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

   7. Taylor expanded in angle around 0

      \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a \]
   8. Step-by-step derivation

   9. Applied rewrites 57.4%

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

   if 4.5000000000000001e220 < b

   1. Initial program 80.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 80.1%

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

   3. Applied rewrites 80.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval 80.2%

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

   5. Applied rewrites 80.2%

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

   6. Taylor expanded in a around 0

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

      1. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot \color{blue}{{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2}} \]

      2. lower-pow.f64 N/A

         \[\leadsto {b}^{2} \cdot {\color{blue}{\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}}^{2} \]

      3. lower-pow.f64 N/A

         \[\leadsto {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{\color{blue}{2}} \]

      4. lower-sin.f64 N/A

         \[\leadsto {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      5. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. lower-*.f64 N/A

         \[\leadsto {b}^{2} \cdot {\sin \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. lower-PI.f64 34.6%

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

   8. Applied rewrites 34.6%

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

   10. Applied rewrites 26.8%

       \[\leadsto \color{blue}{\left(\frac{1}{2} - \cos \left(\frac{1}{90} \cdot \left(\pi \cdot angle\right)\right) \cdot \frac{1}{2}\right) \cdot \left(b \cdot b\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.4% accurate, 14.9× speedup

Math

\[\left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2}\right) \cdot a\right) \cdot a \]

FPCore

(FPCore (a b angle)
  :precision binary64
  (+ (* (* (- 1/2 1/2) b) b) (* (* (+ 1/2 1/2) a) a)))

C

double code(double a, double b, double angle) {
	return (((0.5 - 0.5) * b) * b) + (((0.5 + 0.5) * a) * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = (((0.5d0 - 0.5d0) * b) * b) + (((0.5d0 + 0.5d0) * a) * a)
end function

Java

public static double code(double a, double b, double angle) {
	return (((0.5 - 0.5) * b) * b) + (((0.5 + 0.5) * a) * a);
}

Python

def code(a, b, angle):
	return (((0.5 - 0.5) * b) * b) + (((0.5 + 0.5) * a) * a)

Julia

function code(a, b, angle)
	return Float64(Float64(Float64(Float64(0.5 - 0.5) * b) * b) + Float64(Float64(Float64(0.5 + 0.5) * a) * a))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (((0.5 - 0.5) * b) * b) + (((0.5 + 0.5) * a) * a);
end

Wolfram

code[a_, b_, angle_] := N[(N[(N[(N[(1/2 - 1/2), $MachinePrecision] * b), $MachinePrecision] * b), $MachinePrecision] + N[(N[(N[(1/2 + 1/2), $MachinePrecision] * a), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.1%

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

3. Applied rewrites 68.7%

   \[\leadsto \color{blue}{\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a} \]

4. Taylor expanded in angle around 0

   \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]
5. Step-by-step derivation

6. Applied rewrites 57.3%

   \[\leadsto \left(\left(\frac{1}{2} - \color{blue}{\frac{1}{2}}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\left(angle \cdot \pi\right) \cdot \frac{1}{90}\right)\right) \cdot a\right) \cdot a \]

7. Taylor expanded in angle around 0

   \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a \]
8. Step-by-step derivation

9. Applied rewrites 57.4%

   \[\leadsto \left(\left(\frac{1}{2} - \frac{1}{2}\right) \cdot b\right) \cdot b + \left(\left(\frac{1}{2} + \color{blue}{\frac{1}{2}}\right) \cdot a\right) \cdot a \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180)))) 2) (pow (* b (sin (* PI (/ angle 180)))) 2)))