ab-angle->ABCF C

Percentage Accurate: 79.6% → 79.6%

Time: 17.4s

Alternatives: 21

Speedup: 1.9×

• 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: 79.6% 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: 79.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := {\left(\frac{180}{angle\_m}\right)}^{-0.5}\\ {\left(a \cdot \cos \left(\pi \cdot \left(t\_0 \cdot t\_0\right)\right)\right)}^{2} + {\left(b \cdot \sin \left({\pi}^{0.16666666666666666} \cdot \left(\left(angle\_m \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (pow (/ 180.0 angle_m) -0.5)))
   (+
    (pow (* a (cos (* PI (* t_0 t_0)))) 2.0)
    (pow
     (*
      b
      (sin
       (*
        (pow PI 0.16666666666666666)
        (*
         (* angle_m (* 0.005555555555555556 (pow PI 0.6666666666666666)))
         (pow PI 0.16666666666666666)))))
     2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = pow((180.0 / angle_m), -0.5);
	return pow((a * cos((((double) M_PI) * (t_0 * t_0)))), 2.0) + pow((b * sin((pow(((double) M_PI), 0.16666666666666666) * ((angle_m * (0.005555555555555556 * pow(((double) M_PI), 0.6666666666666666))) * pow(((double) M_PI), 0.16666666666666666))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.pow((180.0 / angle_m), -0.5);
	return Math.pow((a * Math.cos((Math.PI * (t_0 * t_0)))), 2.0) + Math.pow((b * Math.sin((Math.pow(Math.PI, 0.16666666666666666) * ((angle_m * (0.005555555555555556 * Math.pow(Math.PI, 0.6666666666666666))) * Math.pow(Math.PI, 0.16666666666666666))))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pow((180.0 / angle_m), -0.5)
	return math.pow((a * math.cos((math.pi * (t_0 * t_0)))), 2.0) + math.pow((b * math.sin((math.pow(math.pi, 0.16666666666666666) * ((angle_m * (0.005555555555555556 * math.pow(math.pi, 0.6666666666666666))) * math.pow(math.pi, 0.16666666666666666))))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(180.0 / angle_m) ^ -0.5
	return Float64((Float64(a * cos(Float64(pi * Float64(t_0 * t_0)))) ^ 2.0) + (Float64(b * sin(Float64((pi ^ 0.16666666666666666) * Float64(Float64(angle_m * Float64(0.005555555555555556 * (pi ^ 0.6666666666666666))) * (pi ^ 0.16666666666666666))))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	t_0 = (180.0 / angle_m) ^ -0.5;
	tmp = ((a * cos((pi * (t_0 * t_0)))) ^ 2.0) + ((b * sin(((pi ^ 0.16666666666666666) * ((angle_m * (0.005555555555555556 * (pi ^ 0.6666666666666666))) * (pi ^ 0.16666666666666666))))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[Power[N[(180.0 / angle$95$m), $MachinePrecision], -0.5], $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(Pi * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[Power[Pi, 0.16666666666666666], $MachinePrecision] * N[(N[(angle$95$m * N[(0.005555555555555556 * N[Power[Pi, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.9%

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

   1. *-commutative N/A

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

   2. add-cube-cbrt N/A

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

   3. associate-*r* N/A

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

   4. add-sqr-sqrt N/A

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

   5. cbrt-prod N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

4. Applied egg-rr 82.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   2. inv-pow N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{{\left(\frac{180}{angle}\right)}^{-1}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   3. sqr-pow N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({\left(\frac{180}{angle}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{180}{angle}\right)}^{\left(\frac{-1}{2}\right)}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({\left(\frac{180}{angle}\right)}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{180}{angle}\right)}^{\left(\frac{-1}{2}\right)}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{{\left(\frac{180}{angle}\right)}^{\left(\frac{-1}{2}\right)}} \cdot {\left(\frac{180}{angle}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left({\color{blue}{\left(\frac{180}{angle}\right)}}^{\left(\frac{-1}{2}\right)} \cdot {\left(\frac{180}{angle}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   7. metadata-eval N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left({\left(\frac{180}{angle}\right)}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\frac{180}{angle}\right)}^{\left(\frac{-1}{2}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   8. pow-lowering-pow.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left({\left(\frac{180}{angle}\right)}^{\frac{-1}{2}} \cdot \color{blue}{{\left(\frac{180}{angle}\right)}^{\left(\frac{-1}{2}\right)}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   9. /-lowering-/.f64 N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left({\left(\frac{180}{angle}\right)}^{\frac{-1}{2}} \cdot {\color{blue}{\left(\frac{180}{angle}\right)}}^{\left(\frac{-1}{2}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right) \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}^{2} \]

   10. metadata-eval 40.2

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left({\left(\frac{180}{angle}\right)}^{-0.5} \cdot {\left(\frac{180}{angle}\right)}^{\color{blue}{-0.5}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)\right)}^{2} \]

6. Applied egg-rr 40.2%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left({\left(\frac{180}{angle}\right)}^{-0.5} \cdot {\left(\frac{180}{angle}\right)}^{-0.5}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)\right)}^{2} \]

7. Final simplification 40.2%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left({\left(\frac{180}{angle}\right)}^{-0.5} \cdot {\left(\frac{180}{angle}\right)}^{-0.5}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left({\pi}^{0.16666666666666666} \cdot \left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right)\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 2: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ {\left(b \cdot \sin \left({\pi}^{0.16666666666666666} \cdot \left(\left(angle\_m \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (*
    b
    (sin
     (*
      (pow PI 0.16666666666666666)
      (*
       (* angle_m (* 0.005555555555555556 (pow PI 0.6666666666666666)))
       (pow PI 0.16666666666666666)))))
   2.0)
  (pow (* a (cos (* PI (/ angle_m 180.0)))) 2.0)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return pow((b * sin((pow(((double) M_PI), 0.16666666666666666) * ((angle_m * (0.005555555555555556 * pow(((double) M_PI), 0.6666666666666666))) * pow(((double) M_PI), 0.16666666666666666))))), 2.0) + pow((a * cos((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	return Math.pow((b * Math.sin((Math.pow(Math.PI, 0.16666666666666666) * ((angle_m * (0.005555555555555556 * Math.pow(Math.PI, 0.6666666666666666))) * Math.pow(Math.PI, 0.16666666666666666))))), 2.0) + Math.pow((a * Math.cos((Math.PI * (angle_m / 180.0)))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return math.pow((b * math.sin((math.pow(math.pi, 0.16666666666666666) * ((angle_m * (0.005555555555555556 * math.pow(math.pi, 0.6666666666666666))) * math.pow(math.pi, 0.16666666666666666))))), 2.0) + math.pow((a * math.cos((math.pi * (angle_m / 180.0)))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64((Float64(b * sin(Float64((pi ^ 0.16666666666666666) * Float64(Float64(angle_m * Float64(0.005555555555555556 * (pi ^ 0.6666666666666666))) * (pi ^ 0.16666666666666666))))) ^ 2.0) + (Float64(a * cos(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin(((pi ^ 0.16666666666666666) * ((angle_m * (0.005555555555555556 * (pi ^ 0.6666666666666666))) * (pi ^ 0.16666666666666666))))) ^ 2.0) + ((a * cos((pi * (angle_m / 180.0)))) ^ 2.0);
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(N[Power[Pi, 0.16666666666666666], $MachinePrecision] * N[(N[(angle$95$m * N[(0.005555555555555556 * N[Power[Pi, 0.6666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, 0.16666666666666666], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi * N[(angle$95$m / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

   1. *-commutative N/A

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

   2. add-cube-cbrt N/A

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

   3. associate-*r* N/A

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

   4. add-sqr-sqrt N/A

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

   5. cbrt-prod N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

4. Applied egg-rr 82.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} \]

5. Final simplification 82.1%

   \[\leadsto {\left(b \cdot \sin \left({\pi}^{0.16666666666666666} \cdot \left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right)\right)\right)}^{2} + {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 3: 79.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. *-commutative N/A

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

   2. add-cube-cbrt N/A

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

   3. associate-*r* N/A

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

   4. add-sqr-sqrt N/A

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

   5. cbrt-prod N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

4. Applied egg-rr 82.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}\right)}^{2} \]

   2. pow-prod-up N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}}\right)\right)}^{2} \]

   3. metadata-eval N/A

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

   4. pow1/3 N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. sqr-pow N/A

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

   9. pow2 N/A

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

   10. metadata-eval N/A

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

   11. pow1/3 N/A

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

   12. pow2 N/A

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

   13. associate-*r* N/A

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

   14. add-cube-cbrt N/A

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

   15. add-sqr-sqrt N/A

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

   16. associate-*r* N/A

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

6. Applied egg-rr 82.1%

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

7. Final simplification 82.1%

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

Alternative 4: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. associate-*r/ N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. PI-lowering-PI.f64 82.0

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

4. Applied egg-rr 82.0%

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

   1. clear-num N/A

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

   2. inv-pow N/A

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

   3. pow-to-exp N/A

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

   4. exp-lowering-exp.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. log-lowering-log.f64 N/A

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

   7. /-lowering-/.f64 39.4

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

6. Applied egg-rr 39.4%

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

   1. exp-to-pow N/A

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

   2. inv-pow N/A

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

   3. div-inv N/A

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

   4. associate-/r* N/A

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

   5. metadata-eval N/A

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

   6. /-lowering-/.f64 N/A

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

   7. /-lowering-/.f64 82.0

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

8. Applied egg-rr 82.0%

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

Alternative 5: 79.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 82.0%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. *-commutative N/A

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

   4. add-sqr-sqrt N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\frac{angle}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)\right) \]

   6. div-inv N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(\left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot angle\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

   10. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \color{blue}{\left(\left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)}\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right) \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\frac{1}{180} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right) \]

   14. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\frac{1}{180} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right) \]

   15. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\left(angle \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)\right)\right)\right) \]

   16. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\frac{1}{180} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(angle \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)\right)\right)\right) \]

   17. PI-lowering-PI.f64 82.0

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

6. Applied egg-rr 82.0%

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\left(0.005555555555555556 \cdot \sqrt{\pi}\right) \cdot \left(angle \cdot \sqrt{\pi}\right)\right)}\right)\right)\right) \]
7. Add Preprocessing

Alternative 6: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := b \cdot \sin t\_0\\ \mathsf{fma}\left(t\_1, t\_1, \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right)\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))) (t_1 (* b (sin t_0))))
   (fma t_1 t_1 (* (* a a) (+ 0.5 (* 0.5 (cos (* 2.0 t_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);
	double t_1 = b * sin(t_0);
	return fma(t_1, t_1, ((a * a) * (0.5 + (0.5 * cos((2.0 * t_0))))));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(b * sin(t_0))
	return fma(t_1, t_1, Float64(Float64(a * a) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * t_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]}, Block[{t$95$1 = N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$1 * t$95$1 + N[(N[(a * a), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 81.9%

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 82.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := b \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathsf{fma}\left(t\_0, t\_0, \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right), 0.5\right)\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* b (sin (* angle_m (* PI 0.005555555555555556))))))
   (fma
    t_0
    t_0
    (* (* a a) (fma 0.5 (cos (* (* PI angle_m) 0.011111111111111112)) 0.5)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = b * sin((angle_m * (((double) M_PI) * 0.005555555555555556)));
	return fma(t_0, t_0, ((a * a) * fma(0.5, cos(((((double) M_PI) * angle_m) * 0.011111111111111112)), 0.5)));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(b * sin(Float64(angle_m * Float64(pi * 0.005555555555555556))))
	return fma(t_0, t_0, Float64(Float64(a * a) * fma(0.5, cos(Float64(Float64(pi * angle_m) * 0.011111111111111112)), 0.5)))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(b * N[Sin[N[(angle$95$m * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 * t$95$0 + N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 81.9%

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

   1. *-commutative N/A

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

   2. add-cube-cbrt N/A

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

   3. associate-*r* N/A

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

   4. add-sqr-sqrt N/A

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

   5. cbrt-prod N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

4. Applied egg-rr 82.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} \]

6. Applied egg-rr 82.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(a \cdot a\right)\right)} \]

7. Final simplification 82.0%

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\right) \]
8. Add Preprocessing

Alternative 8: 79.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((b * sin(((angle_m * sqrt(sqrt(pi))) * (0.005555555555555556 * sqrt((pi * sqrt(pi))))))) ^ 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[(N[(angle$95$m * N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.005555555555555556 * N[Sqrt[N[(Pi * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.9%

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

   1. *-commutative N/A

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

   2. add-cube-cbrt N/A

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

   3. associate-*r* N/A

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

   4. add-sqr-sqrt N/A

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

   5. cbrt-prod N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

4. Applied egg-rr 82.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{\frac{1}{6}} \cdot {\mathsf{PI}\left(\right)}^{\frac{1}{6}}\right)\right)}\right)}^{2} \]

   2. pow-prod-up N/A

      \[\leadsto {\left(a \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\left(angle \cdot \left(\frac{1}{180} \cdot {\mathsf{PI}\left(\right)}^{\frac{2}{3}}\right)\right) \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{\left(\frac{1}{6} + \frac{1}{6}\right)}}\right)\right)}^{2} \]

   3. metadata-eval N/A

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

   4. pow1/3 N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. div-inv N/A

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

   8. sqr-pow N/A

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

   9. pow2 N/A

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

   10. metadata-eval N/A

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

   11. pow1/3 N/A

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

   12. pow2 N/A

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

   13. associate-*r* N/A

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

   14. add-cube-cbrt N/A

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

   15. add-sqr-sqrt N/A

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

   16. associate-*r* N/A

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

6. Applied egg-rr 82.1%

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

7. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 81.8

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

9. Simplified 81.8%

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

10. Final simplification 81.8%

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

Alternative 9: 79.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\\ \mathsf{fma}\left(t\_0, t\_0, a \cdot a\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* b (sin (* PI (* angle_m 0.005555555555555556))))))
   (fma t_0 t_0 (* a a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = b * sin((((double) M_PI) * (angle_m * 0.005555555555555556)));
	return fma(t_0, t_0, (a * a));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(b * sin(Float64(pi * Float64(angle_m * 0.005555555555555556))))
	return fma(t_0, t_0, Float64(a * a))
end

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 82.0%

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

5. Taylor expanded in angle around 0

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \color{blue}{1}\right) \]
6. Step-by-step derivation

7. Simplified 81.8%

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \left(a \cdot a\right) \cdot \color{blue}{1}\right) \]

8. Final simplification 81.8%

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot a\right) \]
9. Add Preprocessing

Alternative 10: 79.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := b \cdot \sin \left(angle\_m \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ \mathsf{fma}\left(t\_0, t\_0, a \cdot a\right) \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* b (sin (* angle_m (* PI 0.005555555555555556))))))
   (fma t_0 t_0 (* a a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = b * sin((angle_m * (((double) M_PI) * 0.005555555555555556)));
	return fma(t_0, t_0, (a * a));
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(b * sin(Float64(angle_m * Float64(pi * 0.005555555555555556))))
	return fma(t_0, t_0, Float64(a * a))
end

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. *-commutative N/A

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

   2. add-cube-cbrt N/A

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

   3. associate-*r* N/A

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

   4. add-sqr-sqrt N/A

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

   5. cbrt-prod N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

4. Applied egg-rr 82.1%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} \]

6. Applied egg-rr 82.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(a \cdot a\right)\right)} \]

7. Taylor expanded in angle around 0

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right), b \cdot \sin \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{180}\right)\right), \color{blue}{1} \cdot \left(a \cdot a\right)\right) \]
8. Step-by-step derivation

9. Simplified 81.7%

   \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), \color{blue}{1} \cdot \left(a \cdot a\right)\right) \]

10. Final simplification 81.7%

    \[\leadsto \mathsf{fma}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), a \cdot a\right) \]
11. Add Preprocessing

Alternative 11: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. /-lowering-/.f64 N/A

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

   4. PI-lowering-PI.f64 N/A

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

   5. /-lowering-/.f64 82.0

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

4. Applied egg-rr 82.0%

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

5. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 81.7

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

7. Simplified 81.7%

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

Alternative 12: 79.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 81.7

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

5. Simplified 81.7%

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

Alternative 13: 66.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right)\\ \mathbf{if}\;b \leq 2.5 \cdot 10^{-80}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* b (* 0.005555555555555556 (* PI angle_m)))))
   (if (<= b 2.5e-80)
     (* (* a a) (fma 0.5 (cos (* (* PI angle_m) 0.011111111111111112)) 0.5))
     (fma
      t_0
      t_0
      (*
       (* a a)
       (+
        0.5
        (* 0.5 (cos (* 2.0 (* PI (* angle_m 0.005555555555555556)))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = b * (0.005555555555555556 * (((double) M_PI) * angle_m));
	double tmp;
	if (b <= 2.5e-80) {
		tmp = (a * a) * fma(0.5, cos(((((double) M_PI) * angle_m) * 0.011111111111111112)), 0.5);
	} else {
		tmp = fma(t_0, t_0, ((a * a) * (0.5 + (0.5 * cos((2.0 * (((double) M_PI) * (angle_m * 0.005555555555555556))))))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(b * Float64(0.005555555555555556 * Float64(pi * angle_m)))
	tmp = 0.0
	if (b <= 2.5e-80)
		tmp = Float64(Float64(a * a) * fma(0.5, cos(Float64(Float64(pi * angle_m) * 0.011111111111111112)), 0.5));
	else
		tmp = fma(t_0, t_0, Float64(Float64(a * a) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(pi * Float64(angle_m * 0.005555555555555556))))))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(b * N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, 2.5e-80], N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(N[(a * a), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 2.5e-80

   1. Initial program 82.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 82.5%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. PI-lowering-PI.f64 75.4

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

   7. Simplified 75.4%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 63.0

          \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \color{blue}{\pi}\right) \cdot 0.011111111111111112\right), 0.5\right) \]

   10. Simplified 63.0%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)} \]

   if 2.5e-80 < b

   1. Initial program 81.0%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 81.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. PI-lowering-PI.f64 68.0

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

   7. Simplified 68.0%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)}, \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(angle \cdot \mathsf{PI}\left(\right)\right)}\right), \left(a \cdot a\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right)\right)\right)\right) \]

      3. PI-lowering-PI.f64 78.1

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

   10. Simplified 78.1%

       \[\leadsto \mathsf{fma}\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}, \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.5 \cdot 10^{-80}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), \left(a \cdot a\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\_m\right)\right), b \cdot \sin \left(\pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 2e+60)
   (fma
    (* b (* 0.005555555555555556 (* PI angle_m)))
    (* b (sin (* PI (* angle_m 0.005555555555555556))))
    (* a a))
   (fma
    (* angle_m angle_m)
    (* PI (* PI (* b (* b 3.08641975308642e-5))))
    (* a a))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 2e+60) {
		tmp = fma((b * (0.005555555555555556 * (((double) M_PI) * angle_m))), (b * sin((((double) M_PI) * (angle_m * 0.005555555555555556)))), (a * a));
	} else {
		tmp = fma((angle_m * angle_m), (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5)))), (a * a));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 2e+60)
		tmp = fma(Float64(b * Float64(0.005555555555555556 * Float64(pi * angle_m))), Float64(b * sin(Float64(pi * Float64(angle_m * 0.005555555555555556)))), Float64(a * a));
	else
		tmp = fma(Float64(angle_m * angle_m), Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))), Float64(a * a));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 2e+60], N[(N[(b * N[(0.005555555555555556 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(b * N[Sin[N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1.9999999999999999e60

   1. Initial program 84.9%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 85.0%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. PI-lowering-PI.f64 79.6

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

   7. Simplified 79.6%

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

   8. Taylor expanded in angle around 0

      \[\leadsto \mathsf{fma}\left(b \cdot \left(\frac{1}{180} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right), b \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(angle \cdot \frac{1}{180}\right)\right), \left(a \cdot a\right) \cdot \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Simplified 79.2%

       \[\leadsto \mathsf{fma}\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right), b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), \left(a \cdot a\right) \cdot \color{blue}{1}\right) \]

   if 1.9999999999999999e60 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 68.3%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 34.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {b}^{2}\right)}\right), a \cdot a\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} \cdot \frac{1}{32400}\right)}\right), a \cdot a\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{32400}\right)\right), a \cdot a\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      5. *-lowering-*.f64 56.8

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \color{blue}{\left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right)\right), a \cdot a\right) \]

   8. Simplified 56.8%

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \color{blue}{\left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)}\right), a \cdot a\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(b \cdot \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right), b \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-79}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(angle\_m \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.1e-79)
   (* (* a a) (fma 0.5 (cos (* (* PI angle_m) 0.011111111111111112)) 0.5))
   (if (<= b 9.6e+154)
     (fma
      (* angle_m angle_m)
      (* PI (* PI (* b (* b 3.08641975308642e-5))))
      (* a a))
     (* angle_m (* angle_m (* (* PI (* PI 3.08641975308642e-5)) (* b b)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.1e-79) {
		tmp = (a * a) * fma(0.5, cos(((((double) M_PI) * angle_m) * 0.011111111111111112)), 0.5);
	} else if (b <= 9.6e+154) {
		tmp = fma((angle_m * angle_m), (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5)))), (a * a));
	} else {
		tmp = angle_m * (angle_m * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (b * b)));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.1e-79)
		tmp = Float64(Float64(a * a) * fma(0.5, cos(Float64(Float64(pi * angle_m) * 0.011111111111111112)), 0.5));
	elseif (b <= 9.6e+154)
		tmp = fma(Float64(angle_m * angle_m), Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(angle_m * Float64(angle_m * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(b * b))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.1e-79], N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.6e+154], N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(angle$95$m * N[(angle$95$m * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.0999999999999999e-79

   1. Initial program 82.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 82.5%

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

   5. Taylor expanded in angle around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. PI-lowering-PI.f64 75.4

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

   7. Simplified 75.4%

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

   8. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 63.0

          \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \color{blue}{\pi}\right) \cdot 0.011111111111111112\right), 0.5\right) \]

   10. Simplified 63.0%

       \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right)} \]

   if 1.0999999999999999e-79 < b < 9.60000000000000059e154

   1. Initial program 64.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 28.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {b}^{2}\right)}\right), a \cdot a\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} \cdot \frac{1}{32400}\right)}\right), a \cdot a\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{32400}\right)\right), a \cdot a\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      5. *-lowering-*.f64 57.1

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \color{blue}{\left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right)\right), a \cdot a\right) \]

   8. Simplified 57.1%

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \color{blue}{\left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)}\right), a \cdot a\right) \]

   if 9.60000000000000059e154 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 44.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      14. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      18. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {b}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

      20. *-lowering-*.f64 55.0

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

   8. Simplified 55.0%

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

      1. associate-*l* N/A

         \[\leadsto \color{blue}{angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot angle} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot angle} \]

   10. Applied egg-rr 69.3%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right) \cdot angle} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-79}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right), 0.5\right)\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-79}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(angle\_m \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.2e-79)
   (* (* a a) (fma 0.5 (cos (* PI (* angle_m 0.011111111111111112))) 0.5))
   (if (<= b 9.6e+154)
     (fma
      (* angle_m angle_m)
      (* PI (* PI (* b (* b 3.08641975308642e-5))))
      (* a a))
     (* angle_m (* angle_m (* (* PI (* PI 3.08641975308642e-5)) (* b b)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.2e-79) {
		tmp = (a * a) * fma(0.5, cos((((double) M_PI) * (angle_m * 0.011111111111111112))), 0.5);
	} else if (b <= 9.6e+154) {
		tmp = fma((angle_m * angle_m), (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5)))), (a * a));
	} else {
		tmp = angle_m * (angle_m * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (b * b)));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.2e-79)
		tmp = Float64(Float64(a * a) * fma(0.5, cos(Float64(pi * Float64(angle_m * 0.011111111111111112))), 0.5));
	elseif (b <= 9.6e+154)
		tmp = fma(Float64(angle_m * angle_m), Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(angle_m * Float64(angle_m * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(b * b))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.2e-79], N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.6e+154], N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(angle$95$m * N[(angle$95$m * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.20000000000000003e-79

   1. Initial program 82.4%

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 82.5%

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

   5. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 62.9

          \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \color{blue}{\left(angle \cdot 0.011111111111111112\right)}\right), 0.5\right) \]

   7. Simplified 62.9%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)} \]

   if 1.20000000000000003e-79 < b < 9.60000000000000059e154

   1. Initial program 64.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 28.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {b}^{2}\right)}\right), a \cdot a\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} \cdot \frac{1}{32400}\right)}\right), a \cdot a\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{32400}\right)\right), a \cdot a\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      5. *-lowering-*.f64 57.1

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \color{blue}{\left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right)\right), a \cdot a\right) \]

   8. Simplified 57.1%

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \color{blue}{\left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)}\right), a \cdot a\right) \]

   if 9.60000000000000059e154 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 44.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      14. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      18. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {b}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

      20. *-lowering-*.f64 55.0

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

   8. Simplified 55.0%

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

      1. associate-*l* N/A

         \[\leadsto \color{blue}{angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot angle} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot angle} \]

   10. Applied egg-rr 69.3%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right) \cdot angle} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.2 \cdot 10^{-79}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle\_m \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(angle\_m \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.25e-80)
   (* (* a a) (fma 0.5 (cos (* angle_m (* PI 0.011111111111111112))) 0.5))
   (if (<= b 9.6e+154)
     (fma
      (* angle_m angle_m)
      (* PI (* PI (* b (* b 3.08641975308642e-5))))
      (* a a))
     (* angle_m (* angle_m (* (* PI (* PI 3.08641975308642e-5)) (* b b)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.25e-80) {
		tmp = (a * a) * fma(0.5, cos((angle_m * (((double) M_PI) * 0.011111111111111112))), 0.5);
	} else if (b <= 9.6e+154) {
		tmp = fma((angle_m * angle_m), (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5)))), (a * a));
	} else {
		tmp = angle_m * (angle_m * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (b * b)));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.25e-80)
		tmp = Float64(Float64(a * a) * fma(0.5, cos(Float64(angle_m * Float64(pi * 0.011111111111111112))), 0.5));
	elseif (b <= 9.6e+154)
		tmp = fma(Float64(angle_m * angle_m), Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(angle_m * Float64(angle_m * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(b * b))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.25e-80], N[(N[(a * a), $MachinePrecision] * N[(0.5 * N[Cos[N[(angle$95$m * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 9.6e+154], N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(angle$95$m * N[(angle$95$m * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < 1.25e-80

   1. Initial program 82.4%

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

      1. *-commutative N/A

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

      2. add-cube-cbrt N/A

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

      3. associate-*r* N/A

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

      4. add-sqr-sqrt N/A

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

      5. cbrt-prod N/A

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

      6. associate-*r* N/A

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

      7. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\left(\left(angle \cdot \left(0.005555555555555556 \cdot {\pi}^{0.6666666666666666}\right)\right) \cdot {\pi}^{0.16666666666666666}\right) \cdot {\pi}^{0.16666666666666666}\right)}\right)}^{2} \]

   6. Applied egg-rr 82.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right), \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(a \cdot a\right)\right)} \]

   7. Taylor expanded in b around 0

      \[\leadsto \color{blue}{{a}^{2} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\frac{1}{90} \cdot \left(angle \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. PI-lowering-PI.f64 62.9

          \[\leadsto \left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\color{blue}{\pi} \cdot 0.011111111111111112\right)\right), 0.5\right) \]

   9. Simplified 62.9%

      \[\leadsto \color{blue}{\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)} \]

   if 1.25e-80 < b < 9.60000000000000059e154

   1. Initial program 64.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 28.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {b}^{2}\right)}\right), a \cdot a\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} \cdot \frac{1}{32400}\right)}\right), a \cdot a\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{32400}\right)\right), a \cdot a\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      5. *-lowering-*.f64 57.1

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \color{blue}{\left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right)\right), a \cdot a\right) \]

   8. Simplified 57.1%

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \color{blue}{\left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)}\right), a \cdot a\right) \]

   if 9.60000000000000059e154 < 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. Add Preprocessing

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 44.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      14. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      18. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {b}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

      20. *-lowering-*.f64 55.0

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

   8. Simplified 55.0%

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

      1. associate-*l* N/A

         \[\leadsto \color{blue}{angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot angle} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot angle} \]

   10. Applied egg-rr 69.3%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right) \cdot angle} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.25 \cdot 10^{-80}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \mathsf{fma}\left(0.5, \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 65.4% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.05 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot angle\_m, \pi \cdot \left(\pi \cdot \left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.05e+153)
   (fma
    (* angle_m angle_m)
    (* PI (* PI (* b (* b 3.08641975308642e-5))))
    (* a a))
   (* a a)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.05e+153) {
		tmp = fma((angle_m * angle_m), (((double) M_PI) * (((double) M_PI) * (b * (b * 3.08641975308642e-5)))), (a * a));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.05e+153)
		tmp = fma(Float64(angle_m * angle_m), Float64(pi * Float64(pi * Float64(b * Float64(b * 3.08641975308642e-5)))), Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.05e+153], N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(Pi * N[(b * N[(b * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.05000000000000008e153

   1. Initial program 78.7%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 46.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot {b}^{2}\right)}\right), a \cdot a\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left({b}^{2} \cdot \frac{1}{32400}\right)}\right), a \cdot a\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \frac{1}{32400}\right)\right), a \cdot a\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \frac{1}{32400}\right)\right)}\right), a \cdot a\right) \]

      5. *-lowering-*.f64 59.9

         \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \left(b \cdot \color{blue}{\left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)}\right)\right), a \cdot a\right) \]

   8. Simplified 59.9%

      \[\leadsto \mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \color{blue}{\left(b \cdot \left(b \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)}\right), a \cdot a\right) \]

   if 1.05000000000000008e153 < a

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

Alternative 19: 61.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(angle\_m \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.8e+159)
   (* a a)
   (* angle_m (* angle_m (* (* PI (* PI 3.08641975308642e-5)) (* b b))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.8e+159) {
		tmp = a * a;
	} else {
		tmp = angle_m * (angle_m * ((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)) * (b * b)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.8e+159) {
		tmp = a * a;
	} else {
		tmp = angle_m * (angle_m * ((Math.PI * (Math.PI * 3.08641975308642e-5)) * (b * b)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 5.8e+159:
		tmp = a * a
	else:
		tmp = angle_m * (angle_m * ((math.pi * (math.pi * 3.08641975308642e-5)) * (b * b)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 5.8e+159)
		tmp = Float64(a * a);
	else
		tmp = Float64(angle_m * Float64(angle_m * Float64(Float64(pi * Float64(pi * 3.08641975308642e-5)) * Float64(b * b))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 5.8e+159)
		tmp = a * a;
	else
		tmp = angle_m * (angle_m * ((pi * (pi * 3.08641975308642e-5)) * (b * b)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 5.8e+159], N[(a * a), $MachinePrecision], N[(angle$95$m * N[(angle$95$m * N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.80000000000000029e159

   1. Initial program 79.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 60.0

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

   5. Simplified 60.0%

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

   if 5.80000000000000029e159 < b

   1. Initial program 99.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 48.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      14. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      18. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {b}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

      20. *-lowering-*.f64 56.9

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

   8. Simplified 56.9%

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

      1. associate-*l* N/A

         \[\leadsto \color{blue}{angle \cdot \left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot angle} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \left(b \cdot b\right)\right)\right)\right)\right) \cdot angle} \]

   10. Applied egg-rr 69.5%

       \[\leadsto \color{blue}{\left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right) \cdot angle} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left(\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 60.6% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5.8 \cdot 10^{+159}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(angle\_m \cdot angle\_m\right) \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5.8e+159)
   (* a a)
   (* (* angle_m angle_m) (* PI (* 3.08641975308642e-5 (* PI (* b b)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.8e+159) {
		tmp = a * a;
	} else {
		tmp = (angle_m * angle_m) * (((double) M_PI) * (3.08641975308642e-5 * (((double) M_PI) * (b * b))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5.8e+159) {
		tmp = a * a;
	} else {
		tmp = (angle_m * angle_m) * (Math.PI * (3.08641975308642e-5 * (Math.PI * (b * b))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 5.8e+159:
		tmp = a * a
	else:
		tmp = (angle_m * angle_m) * (math.pi * (3.08641975308642e-5 * (math.pi * (b * b))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 5.8e+159)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(angle_m * angle_m) * Float64(pi * Float64(3.08641975308642e-5 * Float64(pi * Float64(b * b)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 5.8e+159)
		tmp = a * a;
	else
		tmp = (angle_m * angle_m) * (pi * (3.08641975308642e-5 * (pi * (b * b))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 5.8e+159], N[(a * a), $MachinePrecision], N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(Pi * N[(3.08641975308642e-5 * N[(Pi * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.80000000000000029e159

   1. Initial program 79.2%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 60.0

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

   5. Simplified 60.0%

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

   if 5.80000000000000029e159 < b

   1. Initial program 99.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {a}^{2}} \]
   4. Step-by-step derivation

      1. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({angle}^{2}, \frac{-1}{32400} \cdot \left({a}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right), {a}^{2}\right)} \]

   5. Simplified 48.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot angle, \pi \cdot \left(\pi \cdot \mathsf{fma}\left(b \cdot b, 3.08641975308642 \cdot 10^{-5}, \left(a \cdot a\right) \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), a \cdot a\right)} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

      3. *-commutative N/A

         \[\leadsto {angle}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

      5. unpow2 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

      7. associate-*r* N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)} \]

      9. unpow2 N/A

         \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right) \]

      10. associate-*l* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot {b}^{2}\right)\right)\right)} \]

      11. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {b}^{2}\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      12. associate-*r* N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]

      14. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      16. *-commutative N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot {b}^{2}\right)}\right)\right) \]

      18. PI-lowering-PI.f64 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot {b}^{2}\right)\right)\right) \]

      19. unpow2 N/A

          \[\leadsto \left(angle \cdot angle\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(b \cdot b\right)}\right)\right)\right) \]

      20. *-lowering-*.f64 56.9

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

   8. Simplified 56.9%

      \[\leadsto \color{blue}{\left(angle \cdot angle\right) \cdot \left(\pi \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 56.9% accurate, 74.7× 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 81.9%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 56.0

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

5. Simplified 56.0%

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

Reproduce

herbie shell --seed 2024198 
(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)))