ab-angle->ABCF A

Percentage Accurate: 79.6% → 79.6%

Time: 19.6s

Alternatives: 13

Speedup: 2.0×

• Could not converge on a ground truth

  1. a = 1.054493674778967
  2. b = 4.530023022339602e+302
  3. angle = 1.3517940007339026e+268

• 35.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(angle\_m \cdot angle\_m, -2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right), 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot angle\_m, \left(\pi \cdot t\_0\right) \cdot \left(t\_0 \cdot \left(angle\_m \cdot \left(a \cdot \pi\right)\right)\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle\_m \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot -0.5\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0
         (fma
          (* angle_m angle_m)
          (* -2.8577960676726107e-8 (* PI PI))
          0.005555555555555556)))
   (if (<= (/ angle_m 180.0) 100000000000.0)
     (fma
      (* a angle_m)
      (* (* PI t_0) (* t_0 (* angle_m (* a PI))))
      (*
       b
       (*
        b
        (+
         0.5
         (* 0.5 (cos (* 2.0 (* angle_m (* 0.005555555555555556 PI)))))))))
     (/
      1.0
      (/
       1.0
       (fma
        a
        (fma
         a
         0.5
         (* a (* (cos (* PI (* angle_m 0.011111111111111112))) -0.5)))
        (* (* b b) (+ 0.5 0.5))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = fma((angle_m * angle_m), (-2.8577960676726107e-8 * (((double) M_PI) * ((double) M_PI))), 0.005555555555555556);
	double tmp;
	if ((angle_m / 180.0) <= 100000000000.0) {
		tmp = fma((a * angle_m), ((((double) M_PI) * t_0) * (t_0 * (angle_m * (a * ((double) M_PI))))), (b * (b * (0.5 + (0.5 * cos((2.0 * (angle_m * (0.005555555555555556 * ((double) M_PI))))))))));
	} else {
		tmp = 1.0 / (1.0 / fma(a, fma(a, 0.5, (a * (cos((((double) M_PI) * (angle_m * 0.011111111111111112))) * -0.5))), ((b * b) * (0.5 + 0.5))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = fma(Float64(angle_m * angle_m), Float64(-2.8577960676726107e-8 * Float64(pi * pi)), 0.005555555555555556)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 100000000000.0)
		tmp = fma(Float64(a * angle_m), Float64(Float64(pi * t_0) * Float64(t_0 * Float64(angle_m * Float64(a * pi)))), Float64(b * Float64(b * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(angle_m * Float64(0.005555555555555556 * pi)))))))));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(a, fma(a, 0.5, Float64(a * Float64(cos(Float64(pi * Float64(angle_m * 0.011111111111111112))) * -0.5))), Float64(Float64(b * b) * Float64(0.5 + 0.5)))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(-2.8577960676726107e-8 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 100000000000.0], N[(N[(a * angle$95$m), $MachinePrecision] * N[(N[(Pi * t$95$0), $MachinePrecision] * N[(t$95$0 * N[(angle$95$m * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * N[(b * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(angle$95$m * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(a * N[(a * 0.5 + N[(a * N[(N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e11

   1. Initial program 87.5%

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

   3. Taylor expanded in angle around 0

      \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + \frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto {\left(angle \cdot \color{blue}{\left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      2. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \left(a \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      3. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \frac{-1}{34992000} \cdot \color{blue}{\left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      4. associate-*l* N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{\left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

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

         \[\leadsto {\color{blue}{\left(angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \mathsf{PI}\left(\right)\right) + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto {\left(angle \cdot \left(\frac{1}{180} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot a\right)} + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      7. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\color{blue}{\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a} + \left(\frac{-1}{34992000} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot {angle}^{2}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      8. associate-*l* N/A

         \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \color{blue}{\frac{-1}{34992000} \cdot \left(\left(a \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot {angle}^{2}\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      9. associate-*r* N/A

         \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \frac{-1}{34992000} \cdot \color{blue}{\left(a \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot {angle}^{2}\right)\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      10. *-commutative N/A

          \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \frac{-1}{34992000} \cdot \left(a \cdot \color{blue}{\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      11. *-commutative N/A

          \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \frac{-1}{34992000} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot a\right)}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

      12. associate-*r* N/A

          \[\leadsto {\left(angle \cdot \left(\left(\frac{1}{180} \cdot \mathsf{PI}\left(\right)\right) \cdot a + \color{blue}{\left(\frac{-1}{34992000} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot a}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)}^{2} \]

   5. Simplified 81.8%

      \[\leadsto {\color{blue}{\left(angle \cdot \left(a \cdot \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right)\right)\right)}}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]

   7. Applied egg-rr 81.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(angle \cdot a, \left(\pi \cdot \mathsf{fma}\left(angle \cdot angle, -2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right), 0.005555555555555556\right)\right) \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \mathsf{fma}\left(angle \cdot angle, -2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right), 0.005555555555555556\right)\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\right)} \]

   if 1e11 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 56.1%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}}} \]
   5. Step-by-step derivation

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

   6. Applied egg-rr 56.3%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot -0.5\right)\right)}, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}} \]

   7. Taylor expanded in angle around 0

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

   9. Simplified 56.5%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot -0.5\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \color{blue}{1}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 100000000000:\\ \;\;\;\;\mathsf{fma}\left(a \cdot angle, \left(\pi \cdot \mathsf{fma}\left(angle \cdot angle, -2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right), 0.005555555555555556\right)\right) \cdot \left(\mathsf{fma}\left(angle \cdot angle, -2.8577960676726107 \cdot 10^{-8} \cdot \left(\pi \cdot \pi\right), 0.005555555555555556\right) \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right), b \cdot \left(b \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot -0.5\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. add-sqr-sqrt N/A

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

   2. associate-*r* N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

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

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

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

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 79.4%

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

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

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

6. Applied egg-rr 79.4%

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

7. Final simplification 79.4%

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

Alternative 3: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. *-commutative N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

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

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

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

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

   6. /-lowering-/.f64 79.3

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

4. Applied egg-rr 79.3%

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

5. Final simplification 79.3%

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

Alternative 4: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-*l/ N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. div-inv N/A

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

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

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

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

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

   8. metadata-eval 79.3

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

4. Applied egg-rr 79.3%

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

5. Final simplification 79.3%

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

Alternative 5: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. associate-*l/ N/A

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

   2. div-inv N/A

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

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

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

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

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

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

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

   6. metadata-eval 79.3

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

4. Applied egg-rr 79.3%

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

5. Final simplification 79.3%

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

Alternative 6: 79.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

   1. add-sqr-sqrt N/A

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

   2. associate-*r* N/A

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

   3. rem-square-sqrt N/A

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

   4. associate-*r* N/A

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

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

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

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

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

   7. div-inv N/A

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

   8. associate-*l* N/A

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

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

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

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

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

   11. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 79.4%

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

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

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

6. Applied egg-rr 79.4%

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

7. Taylor expanded in angle around 0

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

9. Simplified 79.1%

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

10. Final simplification 79.1%

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

Alternative 7: 79.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in angle around 0

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

5. Simplified 79.0%

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

6. Final simplification 79.0%

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

Alternative 8: 78.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;\frac{angle\_m}{180} \leq 100000000000:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(angle\_m, angle\_m \cdot \left(a \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), 0\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle\_m \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot -0.5\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= (/ angle_m 180.0) 100000000000.0)
   (/
    1.0
    (/
     1.0
     (fma
      a
      (fma angle_m (* angle_m (* a (* (* PI PI) 3.08641975308642e-5))) 0.0)
      (*
       (* b b)
       (+
        0.5
        (* 0.5 (cos (* 2.0 (* (* angle_m 0.005555555555555556) PI)))))))))
   (/
    1.0
    (/
     1.0
     (fma
      a
      (fma a 0.5 (* a (* (cos (* PI (* angle_m 0.011111111111111112))) -0.5)))
      (* (* b b) (+ 0.5 0.5)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if ((angle_m / 180.0) <= 100000000000.0) {
		tmp = 1.0 / (1.0 / fma(a, fma(angle_m, (angle_m * (a * ((((double) M_PI) * ((double) M_PI)) * 3.08641975308642e-5))), 0.0), ((b * b) * (0.5 + (0.5 * cos((2.0 * ((angle_m * 0.005555555555555556) * ((double) M_PI)))))))));
	} else {
		tmp = 1.0 / (1.0 / fma(a, fma(a, 0.5, (a * (cos((((double) M_PI) * (angle_m * 0.011111111111111112))) * -0.5))), ((b * b) * (0.5 + 0.5))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 100000000000.0)
		tmp = Float64(1.0 / Float64(1.0 / fma(a, fma(angle_m, Float64(angle_m * Float64(a * Float64(Float64(pi * pi) * 3.08641975308642e-5))), 0.0), Float64(Float64(b * b) * Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(angle_m * 0.005555555555555556) * pi)))))))));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(a, fma(a, 0.5, Float64(a * Float64(cos(Float64(pi * Float64(angle_m * 0.011111111111111112))) * -0.5))), Float64(Float64(b * b) * Float64(0.5 + 0.5)))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 100000000000.0], N[(1.0 / N[(1.0 / N[(a * N[(angle$95$m * N[(angle$95$m * N[(a * N[(N[(Pi * Pi), $MachinePrecision] * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(angle$95$m * 0.005555555555555556), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(a * N[(a * 0.5 + N[(a * N[(N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 1e11

   1. Initial program 87.5%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 71.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}}} \]
   5. Step-by-step derivation

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

   6. Applied egg-rr 71.8%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot -0.5\right)\right)}, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}} \]

   7. Taylor expanded in angle around 0

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\frac{-1}{2} \cdot a + \left(\frac{1}{32400} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \frac{1}{2} \cdot a\right)}, \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \frac{1}{2} \cdot a\right) + \frac{-1}{2} \cdot a}, \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\frac{1}{32400} \cdot \left(a \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \left(\frac{1}{2} \cdot a + \frac{-1}{2} \cdot a\right)}, \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \frac{1}{32400} \cdot \left(a \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + \left(\frac{1}{2} \cdot a + \frac{-1}{2} \cdot a\right), \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \frac{1}{32400} \cdot \color{blue}{\left(\left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + \left(\frac{1}{2} \cdot a + \frac{-1}{2} \cdot a\right), \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}} + \left(\frac{1}{2} \cdot a + \frac{-1}{2} \cdot a\right), \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{{angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + \left(\frac{1}{2} \cdot a + \frac{-1}{2} \cdot a\right), \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + \left(\frac{1}{2} \cdot a + \frac{-1}{2} \cdot a\right), \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} + \left(\frac{1}{2} \cdot a + \frac{-1}{2} \cdot a\right), \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      9. distribute-rgt-out N/A

         \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) + \color{blue}{a \cdot \left(\frac{1}{2} + \frac{-1}{2}\right)}, \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) + a \cdot \color{blue}{0}, \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

      11. mul0-rgt N/A

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) + \color{blue}{0}, \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

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

          \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(angle, angle \cdot \left(\frac{1}{32400} \cdot \left(a \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), 0\right)}, \left(b \cdot b\right) \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}} \]

   9. Simplified 80.5%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(angle, angle \cdot \left(a \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), 0\right)}, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}} \]

   if 1e11 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 56.1%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}}} \]
   5. Step-by-step derivation

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

   6. Applied egg-rr 56.3%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot -0.5\right)\right)}, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}} \]

   7. Taylor expanded in angle around 0

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

   9. Simplified 56.5%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot -0.5\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \color{blue}{1}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 100000000000:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(angle, angle \cdot \left(a \cdot \left(\left(\pi \cdot \pi\right) \cdot 3.08641975308642 \cdot 10^{-5}\right)\right), 0\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot -0.5\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.05 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, b \cdot \left(b \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle\_m \cdot \left(\pi \cdot \pi\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right) \cdot -0.5\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5\right)\right)}}\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.05e+21)
   (fma
    (*
     angle_m
     (fma a (* a 3.08641975308642e-5) (* b (* b -3.08641975308642e-5))))
    (* angle_m (* PI PI))
    (* b b))
   (/
    1.0
    (/
     1.0
     (fma
      a
      (fma a 0.5 (* a (* (cos (* PI (* angle_m 0.011111111111111112))) -0.5)))
      (* (* b b) (+ 0.5 0.5)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.05e+21) {
		tmp = fma((angle_m * fma(a, (a * 3.08641975308642e-5), (b * (b * -3.08641975308642e-5)))), (angle_m * (((double) M_PI) * ((double) M_PI))), (b * b));
	} else {
		tmp = 1.0 / (1.0 / fma(a, fma(a, 0.5, (a * (cos((((double) M_PI) * (angle_m * 0.011111111111111112))) * -0.5))), ((b * b) * (0.5 + 0.5))));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.05e+21)
		tmp = fma(Float64(angle_m * fma(a, Float64(a * 3.08641975308642e-5), Float64(b * Float64(b * -3.08641975308642e-5)))), Float64(angle_m * Float64(pi * pi)), Float64(b * b));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(a, fma(a, 0.5, Float64(a * Float64(cos(Float64(pi * Float64(angle_m * 0.011111111111111112))) * -0.5))), Float64(Float64(b * b) * Float64(0.5 + 0.5)))));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.05e+21], N[(N[(angle$95$m * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision] + N[(b * N[(b * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(a * N[(a * 0.5 + N[(a * N[(N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.05e21

   1. Initial program 76.1%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Simplified 45.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

   7. Applied egg-rr 52.3%

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

   if 2.05e21 < b

   1. Initial program 89.5%

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

      1. flip-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(a, a \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}}} \]
   5. Step-by-step derivation

      1. sub-neg N/A

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

      2. distribute-lft-in N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. distribute-rgt-neg-in N/A

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

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

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

   6. Applied egg-rr 86.5%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot -0.5\right)\right)}, \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(angle \cdot 0.005555555555555556\right) \cdot \pi\right)\right)\right)\right)}} \]

   7. Taylor expanded in angle around 0

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

   9. Simplified 86.6%

      \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right) \cdot -0.5\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5 \cdot \color{blue}{1}\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.05 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, b \cdot \left(b \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle \cdot \left(\pi \cdot \pi\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.5, a \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot -0.5\right)\right), \left(b \cdot b\right) \cdot \left(0.5 + 0.5\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.3% accurate, 8.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, b \cdot \left(b \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle\_m \cdot \left(\pi \cdot \pi\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.7e+131)
   (fma
    (*
     angle_m
     (fma a (* a 3.08641975308642e-5) (* b (* b -3.08641975308642e-5))))
    (* angle_m (* PI PI))
    (* b b))
   (* b b)))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.7e+131) {
		tmp = fma((angle_m * fma(a, (a * 3.08641975308642e-5), (b * (b * -3.08641975308642e-5)))), (angle_m * (((double) M_PI) * ((double) M_PI))), (b * b));
	} else {
		tmp = b * b;
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.7e+131)
		tmp = fma(Float64(angle_m * fma(a, Float64(a * 3.08641975308642e-5), Float64(b * Float64(b * -3.08641975308642e-5)))), Float64(angle_m * Float64(pi * pi)), Float64(b * b));
	else
		tmp = Float64(b * b);
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.7e+131], N[(N[(angle$95$m * N[(a * N[(a * 3.08641975308642e-5), $MachinePrecision] + N[(b * N[(b * -3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(b * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.70000000000000004e131

   1. Initial program 75.7%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Simplified 45.8%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

   7. Applied egg-rr 52.2%

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

   if 2.70000000000000004e131 < b

   1. Initial program 100.0%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

      \[\leadsto \color{blue}{b \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.7 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \mathsf{fma}\left(a, a \cdot 3.08641975308642 \cdot 10^{-5}, b \cdot \left(b \cdot -3.08641975308642 \cdot 10^{-5}\right)\right), angle \cdot \left(\pi \cdot \pi\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.5% accurate, 10.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-55}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(angle\_m \cdot \left(angle\_m \cdot \left(\pi \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 1.3e-55)
   (* b b)
   (fma
    (* angle_m (* angle_m (* PI PI)))
    (* 3.08641975308642e-5 (* a a))
    (* b b))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 1.3e-55) {
		tmp = b * b;
	} else {
		tmp = fma((angle_m * (angle_m * (((double) M_PI) * ((double) M_PI)))), (3.08641975308642e-5 * (a * a)), (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 1.3e-55)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(angle_m * Float64(angle_m * Float64(pi * pi))), Float64(3.08641975308642e-5 * Float64(a * a)), Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 1.3e-55], N[(b * b), $MachinePrecision], N[(N[(angle$95$m * N[(angle$95$m * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.2999999999999999e-55

   1. Initial program 76.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 59.4

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

   5. Simplified 59.4%

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

   if 1.2999999999999999e-55 < a

   1. Initial program 85.2%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Simplified 35.4%

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

   6. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 63.6

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

   8. Simplified 63.6%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-55}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right), 3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{+104}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(angle\_m \cdot \pi\right) \cdot \left(angle\_m \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 3.7e+104)
   (* b b)
   (* (* 3.08641975308642e-5 (* a a)) (* (* angle_m PI) (* angle_m PI)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.7e+104) {
		tmp = b * b;
	} else {
		tmp = (3.08641975308642e-5 * (a * a)) * ((angle_m * ((double) M_PI)) * (angle_m * ((double) M_PI)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 3.7e+104) {
		tmp = b * b;
	} else {
		tmp = (3.08641975308642e-5 * (a * a)) * ((angle_m * Math.PI) * (angle_m * Math.PI));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 3.7e+104:
		tmp = b * b
	else:
		tmp = (3.08641975308642e-5 * (a * a)) * ((angle_m * math.pi) * (angle_m * math.pi))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 3.7e+104)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(3.08641975308642e-5 * Float64(a * a)) * Float64(Float64(angle_m * pi) * Float64(angle_m * pi)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 3.7e+104)
		tmp = b * b;
	else
		tmp = (3.08641975308642e-5 * (a * a)) * ((angle_m * pi) * (angle_m * pi));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 3.7e+104], N[(b * b), $MachinePrecision], N[(N[(3.08641975308642e-5 * N[(a * a), $MachinePrecision]), $MachinePrecision] * N[(N[(angle$95$m * Pi), $MachinePrecision] * N[(angle$95$m * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.6999999999999998e104

   1. Initial program 75.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 60.4

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

   5. Simplified 60.4%

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

   if 3.6999999999999998e104 < a

   1. Initial program 91.4%

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

   3. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-out N/A

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

   5. Simplified 39.0%

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

   6. Taylor expanded in b around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. unswap-sqr N/A

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

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

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

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

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

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

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

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

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

      14. PI-lowering-PI.f64 60.0

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

   8. Simplified 60.0%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.7 \cdot 10^{+104}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(3.08641975308642 \cdot 10^{-5} \cdot \left(a \cdot a\right)\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.4% accurate, 74.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 54.9

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

5. Simplified 54.9%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))