ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.8%

Time: 6.2s

Alternatives: 11

Speedup: 0.5×

• 36.4% 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: 79.8% 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: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \left(1 - \left(0.5 - 0.5 \cdot \left(\left(\cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\pi, 0.5, 0.011111111111111112 \cdot \left(\pi \cdot angle\_m\right)\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\pi, 0.5, 0\right)}{2}\right)\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (*
   (-
    1.0
    (-
     0.5
     (*
      0.5
      (*
       (+
        (cos (* (* angle_m PI) -0.005555555555555556))
        (sin (* (* 0.005555555555555556 angle_m) PI)))
       (*
        2.0
        (*
         (cos (/ (fma PI 0.5 (* 0.011111111111111112 (* PI angle_m))) 2.0))
         (cos (/ (fma PI 0.5 0.0) 2.0))))))))
   (* 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 ((1.0 - (0.5 - (0.5 * ((cos(((angle_m * ((double) M_PI)) * -0.005555555555555556)) + sin(((0.005555555555555556 * angle_m) * ((double) M_PI)))) * (2.0 * (cos((fma(((double) M_PI), 0.5, (0.011111111111111112 * (((double) M_PI) * angle_m))) / 2.0)) * cos((fma(((double) M_PI), 0.5, 0.0) / 2.0)))))))) * (a * a)) + pow((b * sin((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(Float64(Float64(1.0 - Float64(0.5 - Float64(0.5 * Float64(Float64(cos(Float64(Float64(angle_m * pi) * -0.005555555555555556)) + sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))) * Float64(2.0 * Float64(cos(Float64(fma(pi, 0.5, Float64(0.011111111111111112 * Float64(pi * angle_m))) / 2.0)) * cos(Float64(fma(pi, 0.5, 0.0) / 2.0)))))))) * Float64(a * a)) + (Float64(b * sin(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(N[(1.0 - N[(0.5 - N[(0.5 * N[(N[(N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[Cos[N[(N[(Pi * 0.5 + N[(0.011111111111111112 * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(Pi * 0.5 + 0.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. mult-flip N/A

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

   11. lift-/.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. metadata-eval N/A

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

   19. mult-flip N/A

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

   20. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Applied rewrites 79.8%

   \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \color{blue}{\left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) - \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

9. Applied rewrites 79.8%

   \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \color{blue}{\left(2 \cdot \left(\cos \left(\frac{\mathsf{fma}\left(\pi, 0.5, 0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)}{2}\right) \cdot \cos \left(\frac{\mathsf{fma}\left(\pi, 0.5, 0\right)}{2}\right)\right)\right)}\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
10. Add Preprocessing

Alternative 2: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sin \left(\frac{angle\_m}{180} \cdot \pi\right)\\ t_1 := \cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right)\\ \left(1 - \left(0.5 - 0.5 \cdot \left(\left(t\_1 + t\_0\right) \cdot \left(t\_1 - t\_0\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (/ angle_m 180.0) PI)))
        (t_1 (cos (* (* angle_m PI) -0.005555555555555556))))
   (+
    (* (- 1.0 (- 0.5 (* 0.5 (* (+ t_1 t_0) (- t_1 t_0))))) (* 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) {
	double t_0 = sin(((angle_m / 180.0) * ((double) M_PI)));
	double t_1 = cos(((angle_m * ((double) M_PI)) * -0.005555555555555556));
	return ((1.0 - (0.5 - (0.5 * ((t_1 + t_0) * (t_1 - t_0))))) * (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) {
	double t_0 = Math.sin(((angle_m / 180.0) * Math.PI));
	double t_1 = Math.cos(((angle_m * Math.PI) * -0.005555555555555556));
	return ((1.0 - (0.5 - (0.5 * ((t_1 + t_0) * (t_1 - t_0))))) * (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):
	t_0 = math.sin(((angle_m / 180.0) * math.pi))
	t_1 = math.cos(((angle_m * math.pi) * -0.005555555555555556))
	return ((1.0 - (0.5 - (0.5 * ((t_1 + t_0) * (t_1 - t_0))))) * (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)
	t_0 = sin(Float64(Float64(angle_m / 180.0) * pi))
	t_1 = cos(Float64(Float64(angle_m * pi) * -0.005555555555555556))
	return Float64(Float64(Float64(1.0 - Float64(0.5 - Float64(0.5 * Float64(Float64(t_1 + t_0) * Float64(t_1 - t_0))))) * 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)
	t_0 = sin(((angle_m / 180.0) * pi));
	t_1 = cos(((angle_m * pi) * -0.005555555555555556));
	tmp = ((1.0 - (0.5 - (0.5 * ((t_1 + t_0) * (t_1 - t_0))))) * (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_] := Block[{t$95$0 = N[Sin[N[(N[(angle$95$m / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(1.0 - N[(0.5 - N[(0.5 * N[(N[(t$95$1 + t$95$0), $MachinePrecision] * N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. mult-flip N/A

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

   11. lift-/.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. metadata-eval N/A

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

   19. mult-flip N/A

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

   20. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Applied rewrites 79.8%

   \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \color{blue}{\left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) - \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. mult-flip N/A

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

   5. lift-/.f64 79.8

      \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\color{blue}{\frac{angle}{180}} \cdot \pi\right)\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) - \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

9. Applied rewrites 79.8%

   \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\color{blue}{\frac{angle}{180}} \cdot \pi\right)\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) - \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. metadata-eval N/A

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

    4. mult-flip N/A

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

    5. lift-/.f64 79.8

       \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) - \sin \left(\color{blue}{\frac{angle}{180}} \cdot \pi\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

11. Applied rewrites 79.8%

    \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) - \sin \left(\color{blue}{\frac{angle}{180}} \cdot \pi\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 3: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \sin \left(\left(0.005555555555555556 \cdot angle\_m\right) \cdot \pi\right)\\ t_1 := \cos \left(\left(angle\_m \cdot \pi\right) \cdot -0.005555555555555556\right)\\ \left(1 - \left(0.5 - 0.5 \cdot \left(\left(t\_1 + t\_0\right) \cdot \left(t\_1 - t\_0\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (sin (* (* 0.005555555555555556 angle_m) PI)))
        (t_1 (cos (* (* angle_m PI) -0.005555555555555556))))
   (+
    (* (- 1.0 (- 0.5 (* 0.5 (* (+ t_1 t_0) (- t_1 t_0))))) (* 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) {
	double t_0 = sin(((0.005555555555555556 * angle_m) * ((double) M_PI)));
	double t_1 = cos(((angle_m * ((double) M_PI)) * -0.005555555555555556));
	return ((1.0 - (0.5 - (0.5 * ((t_1 + t_0) * (t_1 - t_0))))) * (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) {
	double t_0 = Math.sin(((0.005555555555555556 * angle_m) * Math.PI));
	double t_1 = Math.cos(((angle_m * Math.PI) * -0.005555555555555556));
	return ((1.0 - (0.5 - (0.5 * ((t_1 + t_0) * (t_1 - t_0))))) * (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):
	t_0 = math.sin(((0.005555555555555556 * angle_m) * math.pi))
	t_1 = math.cos(((angle_m * math.pi) * -0.005555555555555556))
	return ((1.0 - (0.5 - (0.5 * ((t_1 + t_0) * (t_1 - t_0))))) * (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)
	t_0 = sin(Float64(Float64(0.005555555555555556 * angle_m) * pi))
	t_1 = cos(Float64(Float64(angle_m * pi) * -0.005555555555555556))
	return Float64(Float64(Float64(1.0 - Float64(0.5 - Float64(0.5 * Float64(Float64(t_1 + t_0) * Float64(t_1 - t_0))))) * 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)
	t_0 = sin(((0.005555555555555556 * angle_m) * pi));
	t_1 = cos(((angle_m * pi) * -0.005555555555555556));
	tmp = ((1.0 - (0.5 - (0.5 * ((t_1 + t_0) * (t_1 - t_0))))) * (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_] := Block[{t$95$0 = N[Sin[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * -0.005555555555555556), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(1.0 - N[(0.5 - N[(0.5 * N[(N[(t$95$1 + t$95$0), $MachinePrecision] * N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. mult-flip N/A

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

   11. lift-/.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. metadata-eval N/A

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

   19. mult-flip N/A

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

   20. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Applied rewrites 79.8%

   \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \color{blue}{\left(\left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) + \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot -0.005555555555555556\right) - \sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right)\right)\right)}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 4: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\_m\right)\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (*
   (-
    1.0
    (-
     0.5
     (*
      0.5
      (cos
       (*
        2.0
        (*
         (cbrt (* PI PI))
         (* (cbrt PI) (* 0.005555555555555556 angle_m))))))))
   (* 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 ((1.0 - (0.5 - (0.5 * cos((2.0 * (cbrt((((double) M_PI) * ((double) M_PI))) * (cbrt(((double) M_PI)) * (0.005555555555555556 * angle_m)))))))) * (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 ((1.0 - (0.5 - (0.5 * Math.cos((2.0 * (Math.cbrt((Math.PI * Math.PI)) * (Math.cbrt(Math.PI) * (0.005555555555555556 * angle_m)))))))) * (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(Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(cbrt(Float64(pi * pi)) * Float64(cbrt(pi) * Float64(0.005555555555555556 * angle_m)))))))) * Float64(a * a)) + (Float64(b * sin(Float64(pi * Float64(angle_m / 180.0)))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(N[(1.0 - N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[Power[N[(Pi * Pi), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[Pi, 1/3], $MachinePrecision] * N[(0.005555555555555556 * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. mult-flip N/A

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

   11. lift-/.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. metadata-eval N/A

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

   19. mult-flip N/A

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

   20. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. add-cube-cbrt N/A

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

   4. lift-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. mult-flip N/A

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

   7. lift-/.f64 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. cbrt-unprod N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-cbrt.f64 N/A

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

   16. lift-PI.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. lift-PI.f64 N/A

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

   21. lower-cbrt.f64 79.8

       \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \frac{angle}{180}\right)\right)\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   22. lift-/.f64 N/A

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

   23. mult-flip N/A

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

   24. metadata-eval N/A

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

   25. *-commutative N/A

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

7. Applied rewrites 79.8%

   \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \color{blue}{\left(\sqrt[3]{\pi \cdot \pi} \cdot \left(\sqrt[3]{\pi} \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right)}\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 5: 79.8% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (* (pow (- (cos (* (* 0.005555555555555556 angle_m) PI))) 2.0) (* 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 (pow(-cos(((0.005555555555555556 * angle_m) * ((double) M_PI))), 2.0) * (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 (Math.pow(-Math.cos(((0.005555555555555556 * angle_m) * Math.PI)), 2.0) * (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 (math.pow(-math.cos(((0.005555555555555556 * angle_m) * math.pi)), 2.0) * (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((Float64(-cos(Float64(Float64(0.005555555555555556 * angle_m) * pi))) ^ 2.0) * 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 = ((-cos(((0.005555555555555556 * angle_m) * pi)) ^ 2.0) * (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[(N[Power[(-N[Cos[N[(N[(0.005555555555555556 * angle$95$m), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), 2.0], $MachinePrecision] * N[(a * a), $MachinePrecision]), $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 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 79.8%

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

Alternative 6: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \left(1 - \left(0.5 - 0.5 \cdot \cos \left(\left(\left(\pi + \pi\right) \cdot 0.005555555555555556\right) \cdot angle\_m\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (*
   (- 1.0 (- 0.5 (* 0.5 (cos (* (* (+ PI PI) 0.005555555555555556) angle_m)))))
   (* 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 ((1.0 - (0.5 - (0.5 * cos((((((double) M_PI) + ((double) M_PI)) * 0.005555555555555556) * angle_m))))) * (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 ((1.0 - (0.5 - (0.5 * Math.cos((((Math.PI + Math.PI) * 0.005555555555555556) * angle_m))))) * (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 ((1.0 - (0.5 - (0.5 * math.cos((((math.pi + math.pi) * 0.005555555555555556) * angle_m))))) * (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(Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(Float64(Float64(pi + pi) * 0.005555555555555556) * angle_m))))) * 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 = ((1.0 - (0.5 - (0.5 * cos((((pi + pi) * 0.005555555555555556) * angle_m))))) * (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[(N[(1.0 - N[(0.5 - N[(0.5 * N[Cos[N[(N[(N[(Pi + Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision] * angle$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. mult-flip N/A

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

   11. lift-/.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. metadata-eval N/A

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

   19. mult-flip N/A

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

   20. lift-/.f64 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. mult-flip N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r* N/A

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

   8. lift-/.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f64 79.8

       \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \cos \left(\left(\color{blue}{\left(\pi + \pi\right)} \cdot 0.005555555555555556\right) \cdot angle\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Applied rewrites 79.8%

   \[\leadsto \left(1 - \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(\left(\left(\pi + \pi\right) \cdot 0.005555555555555556\right) \cdot angle\right)}\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 7: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(\pi, angle\_m \cdot 0.005555555555555556, \pi\right)\right)\right) \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle\_m}{180}\right)\right)}^{2} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (*
   (+ 0.5 (* 0.5 (cos (* 2.0 (fma PI (* angle_m 0.005555555555555556) PI)))))
   (* 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 ((0.5 + (0.5 * cos((2.0 * fma(((double) M_PI), (angle_m * 0.005555555555555556), ((double) M_PI)))))) * (a * a)) + pow((b * sin((((double) M_PI) * (angle_m / 180.0)))), 2.0);
}

Julia

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision] + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $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 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-neg.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. cos-+PI-rev N/A

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

   6. lift-neg.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. cos-+PI-rev N/A

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

   9. sqr-cos-a N/A

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

   10. lower-+.f64 N/A

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

   11. cos-2 N/A

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

   12. cos-sum N/A

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

   13. lower-*.f64 N/A

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

5. Applied rewrites 79.8%

   \[\leadsto \color{blue}{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, \pi\right)\right)\right)} \cdot \left(a \cdot a\right) + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 8: 79.7% accurate, 1.8× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (* 1.0 (* 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 (1.0 * (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 (1.0 * (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 (1.0 * (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(1.0 * 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 = (1.0 * (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[(1.0 * N[(a * a), $MachinePrecision]), $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 79.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. sqr-neg-rev N/A

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

   4. pow2 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. unpow-prod-down N/A

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

   9. lower-unsound-pow.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-unsound-*.f64 N/A

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

3. Applied rewrites 79.8%

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

4. Taylor expanded in angle around 0

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

6. Applied rewrites 79.7%

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

Alternative 9: 66.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.55 \cdot 10^{-5}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;e^{2 \cdot \left(\log angle\_m + \left(\log b + \log \left(0.005555555555555556 \cdot \pi\right)\right)\right)} + {a}^{2}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.55e-5)
   (* a a)
   (+
    (exp
     (* 2.0 (+ (log angle_m) (+ (log b) (log (* 0.005555555555555556 PI))))))
    (pow a 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.55e-5) {
		tmp = a * a;
	} else {
		tmp = exp((2.0 * (log(angle_m) + (log(b) + log((0.005555555555555556 * ((double) M_PI))))))) + pow(a, 2.0);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.55e-5) {
		tmp = a * a;
	} else {
		tmp = Math.exp((2.0 * (Math.log(angle_m) + (Math.log(b) + Math.log((0.005555555555555556 * Math.PI)))))) + Math.pow(a, 2.0);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 1.55e-5:
		tmp = a * a
	else:
		tmp = math.exp((2.0 * (math.log(angle_m) + (math.log(b) + math.log((0.005555555555555556 * math.pi)))))) + math.pow(a, 2.0)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.55e-5)
		tmp = Float64(a * a);
	else
		tmp = Float64(exp(Float64(2.0 * Float64(log(angle_m) + Float64(log(b) + log(Float64(0.005555555555555556 * pi)))))) + (a ^ 2.0));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.55e-5)
		tmp = a * a;
	else
		tmp = exp((2.0 * (log(angle_m) + (log(b) + log((0.005555555555555556 * pi)))))) + (a ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.55e-5], N[(a * a), $MachinePrecision], N[(N[Exp[N[(2.0 * N[(N[Log[angle$95$m], $MachinePrecision] + N[(N[Log[b], $MachinePrecision] + N[Log[N[(0.005555555555555556 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.55000000000000007e-5

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

   if 1.55000000000000007e-5 < b

   1. Initial program 79.8%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. lower-unsound-exp.f64 N/A

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

      4. lower-unsound-*.f64 N/A

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

      5. lower-unsound-log.f64 45.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 45.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 45.0

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

      12. lift-/.f64 N/A

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

      13. mult-flip N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 44.8

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

   3. Applied rewrites 44.8%

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

      1. lift-log.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. log-prod N/A

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

      4. lower-unsound-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. metadata-eval N/A

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

      11. mult-flip N/A

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

      12. lift-/.f64 N/A

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

      13. lower-unsound-log.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. mult-flip N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower-unsound-log.f64 32.0

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

   5. Applied rewrites 32.0%

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

   6. Taylor expanded in angle around 0

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

      1. lower-+.f64 N/A

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

   8. Applied rewrites 37.1%

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

Alternative 10: 57.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 7.5 \cdot 10^{+107}:\\ \;\;\;\;\mathsf{fma}\left(a, a, \left(\left(\left(\pi \cdot \pi\right) \cdot \mathsf{fma}\left(-3.08641975308642 \cdot 10^{-5}, a \cdot a, \left(b \cdot b\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right) \cdot angle\_m\right) \cdot angle\_m\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 7.5e+107)
   (fma
    a
    a
    (*
     (*
      (*
       (* PI PI)
       (fma -3.08641975308642e-5 (* a a) (* (* b b) 3.08641975308642e-5)))
      angle_m)
     angle_m))
   (* a a)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 7.5e+107) {
		tmp = fma(a, a, ((((((double) M_PI) * ((double) M_PI)) * fma(-3.08641975308642e-5, (a * a), ((b * b) * 3.08641975308642e-5))) * angle_m) * angle_m));
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 7.5e+107)
		tmp = fma(a, a, Float64(Float64(Float64(Float64(pi * pi) * fma(-3.08641975308642e-5, Float64(a * a), Float64(Float64(b * b) * 3.08641975308642e-5))) * angle_m) * angle_m));
	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, 7.5e+107], N[(a * a + N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(-3.08641975308642e-5 * N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * angle$95$m), $MachinePrecision] * angle$95$m), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 7.4999999999999996e107

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 41.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   6. Applied rewrites 44.0%

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

   if 7.4999999999999996e107 < a

   1. Initial program 79.8%

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

   2. Taylor expanded in angle around 0

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

      1. lower-pow.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 57.1

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

   6. Applied rewrites 57.1%

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

Alternative 11: 56.7% accurate, 29.7× 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 =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(a, b, angle_m)
use fmin_fmax_functions
    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 79.8%

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

2. Taylor expanded in angle around 0

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

   1. lower-pow.f64 57.1

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

4. Applied rewrites 57.1%

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

   1. lift-pow.f64 N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 57.1

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

6. Applied rewrites 57.1%

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

Reproduce

herbie shell --seed 2025154 
(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)))