ab-angle->ABCF C

Percentage Accurate: 79.3% → 79.2%

Time: 14.3s

Alternatives: 8

Speedup: 2.0×

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

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.3% 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: 79.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)\right)\right) + {\left(b \cdot \sin t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))))
   (+
    (* a (* a (+ 0.5 (* 0.5 (cos (* 2.0 (expm1 (log1p t_0))))))))
    (pow (* b (sin t_0)) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	return (a * (a * (0.5 + (0.5 * cos((2.0 * expm1(log1p(t_0)))))))) + pow((b * sin(t_0)), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	return (a * (a * (0.5 + (0.5 * Math.cos((2.0 * Math.expm1(Math.log1p(t_0)))))))) + Math.pow((b * Math.sin(t_0)), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	return (a * (a * (0.5 + (0.5 * math.cos((2.0 * math.expm1(math.log1p(t_0)))))))) + math.pow((b * math.sin(t_0)), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	return Float64(Float64(a * Float64(a * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * expm1(log1p(t_0)))))))) + (Float64(b * sin(t_0)) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(a * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 82.2%

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

3. Simplified 82.2%

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

   1. expm1-log1p-u 66.5%

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

   2. associate-*r* 66.5%

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

   3. *-commutative 66.5%

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

6. Applied egg-rr 66.5%

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

   1. unpow-prod-down 66.5%

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

   2. unpow2 66.5%

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

   3. expm1-log1p-u 82.3%

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

   4. *-commutative 82.3%

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

   5. associate-*r* 82.3%

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

   6. associate-*l* 82.2%

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

   7. associate-*r* 82.3%

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

   8. *-commutative 82.3%

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

   9. expm1-log1p-u 66.5%

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

   10. expm1-log1p-u 82.3%

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

   11. *-commutative 82.3%

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

   12. associate-*r* 82.2%

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

   13. *-commutative 82.2%

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

8. Applied egg-rr 82.2%

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

   1. unpow2 82.2%

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

   2. *-commutative 82.2%

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

   3. associate-*r* 81.3%

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

   4. *-commutative 81.3%

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

   5. *-commutative 81.3%

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

   6. associate-*r* 82.3%

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

   7. *-commutative 82.3%

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

   8. sqr-cos-a 82.3%

      \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. *-commutative 82.3%

      \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. *-commutative 82.3%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.005555555555555556\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   11. associate-*l* 82.2%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

10. Applied egg-rr 82.2%

    \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
11. Step-by-step derivation

    1. associate-*r* 82.3%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    2. *-commutative 82.3%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

12. Simplified 82.3%

    \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
13. Step-by-step derivation

    1. expm1-log1p-u 66.5%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    2. expm1-undefine 66.5%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} - 1\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    3. *-commutative 66.5%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(angle \cdot \pi\right) \cdot 0.005555555555555556}\right)} - 1\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    4. associate-*r* 66.5%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{angle \cdot \left(\pi \cdot 0.005555555555555556\right)}\right)} - 1\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    5. *-commutative 66.5%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot 0.005555555555555556\right) \cdot angle}\right)} - 1\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    6. associate-*r* 66.5%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(0.005555555555555556 \cdot angle\right)}\right)} - 1\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

14. Applied egg-rr 66.5%

    \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} - 1\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
15. Step-by-step derivation

    1. expm1-define 66.5%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    2. *-commutative 66.5%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \color{blue}{\left(angle \cdot 0.005555555555555556\right)}\right)\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

16. Simplified 66.5%

    \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
17. Add Preprocessing

Alternative 2: 79.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} + a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)\right)\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)
  (*
   a
   (*
    a
    (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.005555555555555556 (* PI angle_m))))))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(a * N[(a * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

3. Simplified 82.2%

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

   1. expm1-log1p-u 66.5%

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

   2. associate-*r* 66.5%

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

   3. *-commutative 66.5%

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

6. Applied egg-rr 66.5%

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

   1. unpow-prod-down 66.5%

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

   2. unpow2 66.5%

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

   3. expm1-log1p-u 82.3%

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

   4. *-commutative 82.3%

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

   5. associate-*r* 82.3%

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

   6. associate-*l* 82.2%

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

   7. associate-*r* 82.3%

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

   8. *-commutative 82.3%

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

   9. expm1-log1p-u 66.5%

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

   10. expm1-log1p-u 82.3%

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

   11. *-commutative 82.3%

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

   12. associate-*r* 82.2%

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

   13. *-commutative 82.2%

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

8. Applied egg-rr 82.2%

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

   1. unpow2 82.2%

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

   2. *-commutative 82.2%

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

   3. associate-*r* 81.3%

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

   4. *-commutative 81.3%

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

   5. *-commutative 81.3%

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

   6. associate-*r* 82.3%

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

   7. *-commutative 82.3%

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

   8. sqr-cos-a 82.3%

      \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   9. *-commutative 82.3%

      \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   10. *-commutative 82.3%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.005555555555555556\right)\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

   11. associate-*l* 82.2%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

10. Applied egg-rr 82.2%

    \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
11. Step-by-step derivation

    1. associate-*r* 82.3%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    2. *-commutative 82.3%

       \[\leadsto a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right)\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

12. Simplified 82.3%

    \[\leadsto a \cdot \left(a \cdot \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

13. Final simplification 82.3%

    \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + a \cdot \left(a \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\right) \]
14. Add Preprocessing

Alternative 3: 79.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)}^{2} + a \cdot \left(a \cdot \frac{1 + \cos \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{2}\right) \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow (* b (sin (* PI (* angle_m 0.005555555555555556)))) 2.0)
  (* a (* a (/ (+ 1.0 (cos (* angle_m (* PI 0.011111111111111112)))) 2.0)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * sin((((double) M_PI) * (angle_m * 0.005555555555555556)))), 2.0) + (a * (a * ((1.0 + cos((angle_m * (((double) M_PI) * 0.011111111111111112)))) / 2.0)));
}

Java

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

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((b * math.sin((math.pi * (angle_m * 0.005555555555555556)))), 2.0) + (a * (a * ((1.0 + math.cos((angle_m * (math.pi * 0.011111111111111112)))) / 2.0)))

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))) ^ 2.0) + Float64(a * Float64(a * Float64(Float64(1.0 + cos(Float64(angle_m * Float64(pi * 0.011111111111111112)))) / 2.0))))
end

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(a * N[(a * N[(N[(1.0 + N[Cos[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

3. Simplified 82.2%

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

   1. expm1-log1p-u 66.5%

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

   2. associate-*r* 66.5%

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

   3. *-commutative 66.5%

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

6. Applied egg-rr 66.5%

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

   1. unpow-prod-down 66.5%

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

   2. unpow2 66.5%

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

   3. expm1-log1p-u 82.3%

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

   4. *-commutative 82.3%

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

   5. associate-*r* 82.3%

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

   6. associate-*l* 82.2%

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

   7. associate-*r* 82.3%

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

   8. *-commutative 82.3%

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

   9. expm1-log1p-u 66.5%

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

   10. expm1-log1p-u 82.3%

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

   11. *-commutative 82.3%

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

   12. associate-*r* 82.2%

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

   13. *-commutative 82.2%

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

8. Applied egg-rr 82.2%

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

   1. unpow2 82.2%

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

   2. *-commutative 82.2%

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

   3. associate-*r* 81.3%

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

   4. *-commutative 81.3%

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

   5. *-commutative 81.3%

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

   6. associate-*r* 82.3%

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

   7. *-commutative 82.3%

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

   8. cos-mult 82.3%

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

10. Applied egg-rr 82.2%

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

    1. +-commutative 82.2%

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

    2. +-inverses 82.2%

       \[\leadsto a \cdot \left(a \cdot \frac{\cos \color{blue}{0} + \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) + angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}{2}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

    3. cos-0 82.2%

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

    4. distribute-lft-out 82.2%

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

    5. distribute-lft-out 82.2%

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

    6. metadata-eval 82.2%

       \[\leadsto a \cdot \left(a \cdot \frac{1 + \cos \left(angle \cdot \left(\pi \cdot \color{blue}{0.011111111111111112}\right)\right)}{2}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

12. Simplified 82.2%

    \[\leadsto a \cdot \left(a \cdot \color{blue}{\frac{1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{2}}\right) + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

13. Final simplification 82.2%

    \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + a \cdot \left(a \cdot \frac{1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{2}\right) \]
14. Add Preprocessing

Alternative 4: 62.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\\ \mathbf{if}\;b \leq 3 \cdot 10^{+141}:\\ \;\;\;\;{\left(a \cdot \cos t\_0\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \sin t\_0\right)}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI angle_m))))
   (if (<= b 3e+141) (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * angle_m);
	double tmp;
	if (b <= 3e+141) {
		tmp = pow((a * cos(t_0)), 2.0);
	} else {
		tmp = pow((b * sin(t_0)), 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (Math.PI * angle_m);
	double tmp;
	if (b <= 3e+141) {
		tmp = Math.pow((a * Math.cos(t_0)), 2.0);
	} else {
		tmp = Math.pow((b * Math.sin(t_0)), 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = 0.005555555555555556 * (math.pi * angle_m)
	tmp = 0
	if b <= 3e+141:
		tmp = math.pow((a * math.cos(t_0)), 2.0)
	else:
		tmp = math.pow((b * math.sin(t_0)), 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(0.005555555555555556 * Float64(pi * angle_m))
	tmp = 0.0
	if (b <= 3e+141)
		tmp = Float64(a * cos(t_0)) ^ 2.0;
	else
		tmp = Float64(b * sin(t_0)) ^ 2.0;
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	t_0 = 0.005555555555555556 * (pi * angle_m);
	tmp = 0.0;
	if (b <= 3e+141)
		tmp = (a * cos(t_0)) ^ 2.0;
	else
		tmp = (b * sin(t_0)) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 3e+141], 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]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.9999999999999999e141

   1. Initial program 79.7%

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

   3. Simplified 79.8%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 62.6%

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

      1. unpow2 62.6%

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

      2. *-commutative 62.6%

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

      3. unpow2 62.6%

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

      4. swap-sqr 62.6%

         \[\leadsto \color{blue}{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

      5. unpow2 62.6%

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

      6. *-commutative 62.6%

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

   7. Simplified 62.6%

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

   if 2.9999999999999999e141 < b

   1. Initial program 93.9%

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

   3. Simplified 93.9%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 57.1%

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

      1. *-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

         \[\leadsto \color{blue}{\left(b \cdot b\right)} \cdot \left(\sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \]

      4. swap-sqr 76.7%

         \[\leadsto \color{blue}{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

      5. unpow2 76.7%

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

      6. *-commutative 76.7%

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

   7. Simplified 76.7%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{+141}:\\ \;\;\;\;{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{\left(b \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{+159}:\\ \;\;\;\;{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{a}^{6}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 4.4e+159)
   (pow (* a (cos (* 0.005555555555555556 (* PI angle_m)))) 2.0)
   (cbrt (pow a 6.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 4.4e+159) {
		tmp = pow((a * cos((0.005555555555555556 * (((double) M_PI) * angle_m)))), 2.0);
	} else {
		tmp = cbrt(pow(a, 6.0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 4.4e+159) {
		tmp = Math.pow((a * Math.cos((0.005555555555555556 * (Math.PI * angle_m)))), 2.0);
	} else {
		tmp = Math.cbrt(Math.pow(a, 6.0));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 4.4e+159)
		tmp = Float64(a * cos(Float64(0.005555555555555556 * Float64(pi * angle_m)))) ^ 2.0;
	else
		tmp = cbrt((a ^ 6.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 4.4e+159], N[Power[N[(a * N[Cos[N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], N[Power[N[Power[a, 6.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.3999999999999998e159

   1. Initial program 79.7%

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

   3. Simplified 79.8%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in a around inf 60.7%

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

      1. unpow2 60.7%

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

      2. *-commutative 60.7%

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

      3. unpow2 60.7%

         \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

      4. swap-sqr 60.7%

         \[\leadsto \color{blue}{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot \left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)} \]

      5. unpow2 60.7%

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

      6. *-commutative 60.7%

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

   7. Simplified 60.7%

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

   if 4.3999999999999998e159 < b

   1. Initial program 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 39.7%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 39.7%

         \[\leadsto \color{blue}{\sqrt{{a}^{2}} \cdot \sqrt{{a}^{2}}} \]

      2. sqrt-unprod 45.7%

         \[\leadsto \color{blue}{\sqrt{{a}^{2} \cdot {a}^{2}}} \]

      3. pow-prod-up 45.7%

         \[\leadsto \sqrt{\color{blue}{{a}^{\left(2 + 2\right)}}} \]

      4. metadata-eval 45.7%

         \[\leadsto \sqrt{{a}^{\color{blue}{4}}} \]

   7. Applied egg-rr 45.7%

      \[\leadsto \color{blue}{\sqrt{{a}^{4}}} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 48.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{{a}^{4}} \cdot \sqrt{{a}^{4}}\right) \cdot \sqrt{{a}^{4}}}} \]

      2. pow1/3 48.7%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{{a}^{4}} \cdot \sqrt{{a}^{4}}\right) \cdot \sqrt{{a}^{4}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 48.7%

         \[\leadsto {\left(\color{blue}{{a}^{4}} \cdot \sqrt{{a}^{4}}\right)}^{0.3333333333333333} \]

      4. sqrt-pow1 48.7%

         \[\leadsto {\left({a}^{4} \cdot \color{blue}{{a}^{\left(\frac{4}{2}\right)}}\right)}^{0.3333333333333333} \]

      5. metadata-eval 48.7%

         \[\leadsto {\left({a}^{4} \cdot {a}^{\color{blue}{2}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 48.7%

         \[\leadsto {\color{blue}{\left({a}^{\left(4 + 2\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 48.7%

         \[\leadsto {\left({a}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 48.7%

      \[\leadsto \color{blue}{{\left({a}^{6}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 48.7%

          \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]

   11. Simplified 48.7%

       \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 4.4 \cdot 10^{+159}:\\ \;\;\;\;{\left(a \cdot \cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{a}^{6}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Simplified 82.2%

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

   1. expm1-log1p-u 66.5%

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

   2. associate-*r* 66.5%

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

   3. *-commutative 66.5%

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

6. Applied egg-rr 66.5%

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

   1. unpow-prod-down 66.5%

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

   2. unpow2 66.5%

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

   3. expm1-log1p-u 82.3%

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

   4. *-commutative 82.3%

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

   5. associate-*r* 82.3%

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

   6. associate-*l* 82.2%

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

   7. associate-*r* 82.3%

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

   8. *-commutative 82.3%

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

   9. expm1-log1p-u 66.5%

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

   10. expm1-log1p-u 82.3%

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

   11. *-commutative 82.3%

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

   12. associate-*r* 82.2%

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

   13. *-commutative 82.2%

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

8. Applied egg-rr 82.2%

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

   1. add-sqr-sqrt 39.7%

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

   2. pow2 39.7%

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

   3. *-commutative 39.7%

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

   4. sqrt-prod 39.7%

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

   5. unpow2 39.7%

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

   6. sqrt-prod 30.7%

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

   7. add-sqr-sqrt 39.7%

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

   8. associate-*r* 39.7%

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

   9. *-commutative 39.7%

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

10. Applied egg-rr 39.7%

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

11. Taylor expanded in angle around 0 82.1%

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

12. Final simplification 82.1%

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

Alternative 7: 57.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 4.7 \cdot 10^{+154}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{a}^{6}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 4.7e+154) (* a a) (cbrt (pow a 6.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 4.7e+154) {
		tmp = a * a;
	} else {
		tmp = cbrt(pow(a, 6.0));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 4.7e+154) {
		tmp = a * a;
	} else {
		tmp = Math.cbrt(Math.pow(a, 6.0));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 4.7e+154)
		tmp = Float64(a * a);
	else
		tmp = cbrt((a ^ 6.0));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 4.7e+154], N[(a * a), $MachinePrecision], N[Power[N[Power[a, 6.0], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.69999999999999983e154

   1. Initial program 79.3%

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

   3. Simplified 79.4%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 60.6%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. unpow2 60.6%

         \[\leadsto \color{blue}{a \cdot a} \]

   7. Applied egg-rr 60.6%

      \[\leadsto \color{blue}{a \cdot a} \]

   if 4.69999999999999983e154 < b

   1. Initial program 99.5%

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

   3. Simplified 99.5%

      \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
   4. Add Preprocessing

   5. Taylor expanded in angle around 0 41.0%

      \[\leadsto \color{blue}{{a}^{2}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 41.0%

         \[\leadsto \color{blue}{\sqrt{{a}^{2}} \cdot \sqrt{{a}^{2}}} \]

      2. sqrt-unprod 46.4%

         \[\leadsto \color{blue}{\sqrt{{a}^{2} \cdot {a}^{2}}} \]

      3. pow-prod-up 46.4%

         \[\leadsto \sqrt{\color{blue}{{a}^{\left(2 + 2\right)}}} \]

      4. metadata-eval 46.4%

         \[\leadsto \sqrt{{a}^{\color{blue}{4}}} \]

   7. Applied egg-rr 46.4%

      \[\leadsto \color{blue}{\sqrt{{a}^{4}}} \]
   8. Step-by-step derivation

      1. add-cbrt-cube 49.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{{a}^{4}} \cdot \sqrt{{a}^{4}}\right) \cdot \sqrt{{a}^{4}}}} \]

      2. pow1/3 49.0%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{{a}^{4}} \cdot \sqrt{{a}^{4}}\right) \cdot \sqrt{{a}^{4}}\right)}^{0.3333333333333333}} \]

      3. add-sqr-sqrt 49.0%

         \[\leadsto {\left(\color{blue}{{a}^{4}} \cdot \sqrt{{a}^{4}}\right)}^{0.3333333333333333} \]

      4. sqrt-pow1 49.0%

         \[\leadsto {\left({a}^{4} \cdot \color{blue}{{a}^{\left(\frac{4}{2}\right)}}\right)}^{0.3333333333333333} \]

      5. metadata-eval 49.0%

         \[\leadsto {\left({a}^{4} \cdot {a}^{\color{blue}{2}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 49.0%

         \[\leadsto {\color{blue}{\left({a}^{\left(4 + 2\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 49.0%

         \[\leadsto {\left({a}^{\color{blue}{6}}\right)}^{0.3333333333333333} \]

   9. Applied egg-rr 49.0%

      \[\leadsto \color{blue}{{\left({a}^{6}\right)}^{0.3333333333333333}} \]
   10. Step-by-step derivation

       1. unpow1/3 49.0%

          \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]

   11. Simplified 49.0%

       \[\leadsto \color{blue}{\sqrt[3]{{a}^{6}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 56.8% accurate, 139.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m) :precision binary64 (* a a))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return a * a;
}

Fortran

angle_m = abs(angle)
real(8) function code(a, b, angle_m)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle_m
    code = a * a
end function

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return a * a;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return a * a

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(a * a)
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = a * a;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(a * a), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

3. Simplified 82.2%

   \[\leadsto \color{blue}{{\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 57.9%

   \[\leadsto \color{blue}{{a}^{2}} \]
6. Step-by-step derivation

   1. unpow2 57.9%

      \[\leadsto \color{blue}{a \cdot a} \]

7. Applied egg-rr 57.9%

   \[\leadsto \color{blue}{a \cdot a} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024129 
(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)))