ab-angle->ABCF A

Percentage Accurate: 79.4% → 79.4%

Time: 17.0s

Alternatives: 12

Speedup: 2.0×

• 37.1% 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: 79.4% 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: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot angle\_m\right) \cdot \left(\left(\pi \cdot \left(a \cdot angle\_m\right)\right) \cdot t\_0\right), \pi \cdot t\_0, \mathsf{fma}\left(0.5, \cos \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

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) 5.0)
     (fma
      (* (* a angle_m) (* (* PI (* a angle_m)) t_0))
      (* PI t_0)
      (* (fma 0.5 (cos (* (* angle_m PI) 0.011111111111111112)) 0.5) (* b b)))
     (fma
      a
      (* a (+ 0.5 (* -0.5 (cos (* PI (* angle_m 0.011111111111111112))))))
      (* b b)))))

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) <= 5.0) {
		tmp = fma(((a * angle_m) * ((((double) M_PI) * (a * angle_m)) * t_0)), (((double) M_PI) * t_0), (fma(0.5, cos(((angle_m * ((double) M_PI)) * 0.011111111111111112)), 0.5) * (b * b)));
	} else {
		tmp = fma(a, (a * (0.5 + (-0.5 * cos((((double) M_PI) * (angle_m * 0.011111111111111112)))))), (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = fma(Float64(Float64(angle_m * angle_m) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5.0)
		tmp = fma(Float64(Float64(a * angle_m) * Float64(Float64(pi * Float64(a * angle_m)) * t_0)), Float64(pi * t_0), Float64(fma(0.5, cos(Float64(Float64(angle_m * pi) * 0.011111111111111112)), 0.5) * Float64(b * b)));
	else
		tmp = fma(a, Float64(a * Float64(0.5 + Float64(-0.5 * cos(Float64(pi * Float64(angle_m * 0.011111111111111112)))))), Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5.0], N[(N[(N[(a * angle$95$m), $MachinePrecision] * N[(N[(Pi * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision] + N[(N[(0.5 * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(0.5 + N[(-0.5 * N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5

   1. Initial program 83.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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. lower-*.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 78.9%

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

   7. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right) \cdot \left(a \cdot angle\right), \pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right), \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(b \cdot b\right)\right)} \]

   if 5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 60.1%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 60.3%

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

   7. Applied egg-rr 60.3%

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

4. Final simplification 74.8%

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

Alternative 2: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = ((a * sin(((angle_m * pi) * 0.005555555555555556))) ^ 2.0) + ((b * cos(((angle_m * pi) / 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[(angle$95$m * Pi), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.9%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 77.9

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

4. Applied egg-rr 77.9%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. lift-*.f64 N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f64 77.9

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

6. Applied egg-rr 77.9%

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

Alternative 3: 79.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.9%

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

   1. lift-PI.f64 N/A

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

   2. associate-*l/ N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 77.9

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

4. Applied egg-rr 77.9%

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

6. Applied egg-rr 77.9%

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

Alternative 4: 79.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
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 77.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. unpow1 N/A

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

   7. unpow1 N/A

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

   8. unpow2 N/A

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

   9. *-rgt-identity N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-fma.f64 77.7

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

7. Applied egg-rr 77.7%

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

Alternative 5: 79.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
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 77.7%

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

   1. *-rgt-identity N/A

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

   2. pow2 N/A

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

   3. lift-*.f64 77.7

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

7. Applied egg-rr 77.7%

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

8. Final simplification 77.7%

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

Alternative 6: 79.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left(angle\_m \cdot angle\_m\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 5:\\ \;\;\;\;\mathsf{fma}\left(a \cdot angle\_m, \left(\left(\pi \cdot \left(a \cdot angle\_m\right)\right) \cdot t\_0\right) \cdot \left(\pi \cdot t\_0\right), \mathsf{fma}\left(0.5, \cos \left(\left(angle\_m \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

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) 5.0)
     (fma
      (* a angle_m)
      (* (* (* PI (* a angle_m)) t_0) (* PI t_0))
      (* (fma 0.5 (cos (* (* angle_m PI) 0.011111111111111112)) 0.5) (* b b)))
     (fma
      a
      (* a (+ 0.5 (* -0.5 (cos (* PI (* angle_m 0.011111111111111112))))))
      (* b b)))))

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) <= 5.0) {
		tmp = fma((a * angle_m), (((((double) M_PI) * (a * angle_m)) * t_0) * (((double) M_PI) * t_0)), (fma(0.5, cos(((angle_m * ((double) M_PI)) * 0.011111111111111112)), 0.5) * (b * b)));
	} else {
		tmp = fma(a, (a * (0.5 + (-0.5 * cos((((double) M_PI) * (angle_m * 0.011111111111111112)))))), (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = fma(Float64(Float64(angle_m * angle_m) * -2.8577960676726107e-8), Float64(pi * pi), 0.005555555555555556)
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5.0)
		tmp = fma(Float64(a * angle_m), Float64(Float64(Float64(pi * Float64(a * angle_m)) * t_0) * Float64(pi * t_0)), Float64(fma(0.5, cos(Float64(Float64(angle_m * pi) * 0.011111111111111112)), 0.5) * Float64(b * b)));
	else
		tmp = fma(a, Float64(a * Float64(0.5 + Float64(-0.5 * cos(Float64(pi * Float64(angle_m * 0.011111111111111112)))))), Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * -2.8577960676726107e-8), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.005555555555555556), $MachinePrecision]}, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5.0], N[(N[(a * angle$95$m), $MachinePrecision] * N[(N[(N[(Pi * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(Pi * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 * N[Cos[N[(N[(angle$95$m * Pi), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(0.5 + N[(-0.5 * N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5

   1. Initial program 83.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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. lower-*.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 78.9%

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

   7. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot angle, \left(\pi \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right) \cdot \left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \mathsf{fma}\left(\left(angle \cdot angle\right) \cdot -2.8577960676726107 \cdot 10^{-8}, \pi \cdot \pi, 0.005555555555555556\right)\right), \mathsf{fma}\left(0.5, \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right), 0.5\right) \cdot \left(b \cdot b\right)\right)} \]

   if 5 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 60.1%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 60.3%

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

   7. Applied egg-rr 60.3%

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

4. Final simplification 74.4%

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

Alternative 7: 79.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\_m\right)\right)\\ \mathbf{if}\;\frac{angle\_m}{180} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI (* a angle_m)))))
   (if (<= (/ angle_m 180.0) 5e-6)
     (fma t_0 t_0 (* b b))
     (fma
      a
      (* a (+ 0.5 (* -0.5 (cos (* PI (* angle_m 0.011111111111111112))))))
      (* b b)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * (a * angle_m));
	double tmp;
	if ((angle_m / 180.0) <= 5e-6) {
		tmp = fma(t_0, t_0, (b * b));
	} else {
		tmp = fma(a, (a * (0.5 + (-0.5 * cos((((double) M_PI) * (angle_m * 0.011111111111111112)))))), (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(0.005555555555555556 * Float64(pi * Float64(a * angle_m)))
	tmp = 0.0
	if (Float64(angle_m / 180.0) <= 5e-6)
		tmp = fma(t_0, t_0, Float64(b * b));
	else
		tmp = fma(a, Float64(a * Float64(0.5 + Float64(-0.5 * cos(Float64(pi * Float64(angle_m * 0.011111111111111112)))))), Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(angle$95$m / 180.0), $MachinePrecision], 5e-6], N[(t$95$0 * t$95$0 + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(0.5 + N[(-0.5 * N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 angle #s(literal 180 binary64)) < 5.00000000000000041e-6

   1. Initial program 83.3%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 83.3

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

   4. Applied egg-rr 83.3%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. *-commutative N/A

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

      13. pow-exp N/A

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

   6. Applied egg-rr 49.4%

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

   7. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. exp-sum N/A

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

      3. exp-to-pow N/A

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

      4. unpow2 N/A

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

      5. exp-to-pow N/A

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

      6. unpow2 N/A

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

      7. swap-sqr N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   9. Simplified 80.1%

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

   if 5.00000000000000041e-6 < (/.f64 angle #s(literal 180 binary64))

   1. Initial program 61.8%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 60.0%

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

   7. Applied egg-rr 59.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right), 0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\_m\right)\right)\\ \mathbf{if}\;a \leq 7 \cdot 10^{-168}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle\_m \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI (* a angle_m)))))
   (if (<= a 7e-168)
     (* (* b b) (fma 0.5 (cos (* PI (* angle_m 0.011111111111111112))) 0.5))
     (fma t_0 t_0 (* b b)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * (a * angle_m));
	double tmp;
	if (a <= 7e-168) {
		tmp = (b * b) * fma(0.5, cos((((double) M_PI) * (angle_m * 0.011111111111111112))), 0.5);
	} else {
		tmp = fma(t_0, t_0, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(0.005555555555555556 * Float64(pi * Float64(a * angle_m)))
	tmp = 0.0
	if (a <= 7e-168)
		tmp = Float64(Float64(b * b) * fma(0.5, cos(Float64(pi * Float64(angle_m * 0.011111111111111112))), 0.5));
	else
		tmp = fma(t_0, t_0, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 7e-168], N[(N[(b * b), $MachinePrecision] * N[(0.5 * N[Cos[N[(Pi * N[(angle$95$m * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 6.99999999999999964e-168

   1. Initial program 78.7%

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

   4. Applied egg-rr 32.1%

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

   5. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 59.3

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

   7. Simplified 59.3%

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

   if 6.99999999999999964e-168 < a

   1. Initial program 76.8%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 76.9

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

   4. Applied egg-rr 76.9%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. *-commutative N/A

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

      13. pow-exp N/A

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

   6. Applied egg-rr 44.3%

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

   7. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. exp-sum N/A

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

      3. exp-to-pow N/A

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

      4. unpow2 N/A

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

      5. exp-to-pow N/A

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

      6. unpow2 N/A

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

      7. swap-sqr N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   9. Simplified 71.6%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 7 \cdot 10^{-168}:\\ \;\;\;\;\left(b \cdot b\right) \cdot \mathsf{fma}\left(0.5, \cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right), 0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 9.3× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\_m\right)\right)\\ \mathbf{if}\;a \leq 8 \cdot 10^{-107}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* 0.005555555555555556 (* PI (* a angle_m)))))
   (if (<= a 8e-107) (* b b) (fma t_0 t_0 (* b b)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = 0.005555555555555556 * (((double) M_PI) * (a * angle_m));
	double tmp;
	if (a <= 8e-107) {
		tmp = b * b;
	} else {
		tmp = fma(t_0, t_0, (b * b));
	}
	return tmp;
}

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(0.005555555555555556 * Float64(pi * Float64(a * angle_m)))
	tmp = 0.0
	if (a <= 8e-107)
		tmp = Float64(b * b);
	else
		tmp = fma(t_0, t_0, Float64(b * b));
	end
	return tmp
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(0.005555555555555556 * N[(Pi * N[(a * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, 8e-107], N[(b * b), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < 8e-107

   1. Initial program 77.1%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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. lower-*.f64 59.5

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

   5. Simplified 59.5%

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

   if 8e-107 < a

   1. Initial program 79.1%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 79.1

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

   4. Applied egg-rr 79.1%

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

      1. lift-PI.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow-to-exp N/A

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

      7. lift-/.f64 N/A

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. lift-*.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. *-commutative N/A

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

      13. pow-exp N/A

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

   6. Applied egg-rr 43.3%

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

   7. Taylor expanded in angle around 0

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

      1. distribute-rgt-in N/A

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

      2. exp-sum N/A

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

      3. exp-to-pow N/A

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

      4. unpow2 N/A

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

      5. exp-to-pow N/A

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

      6. unpow2 N/A

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

      7. swap-sqr N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

   9. Simplified 73.7%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-107}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right), 0.005555555555555556 \cdot \left(\pi \cdot \left(a \cdot angle\right)\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+83}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(angle\_m \cdot \left(angle\_m \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 4.1e+83)
   (* b b)
   (* (* PI (* angle_m (* angle_m (* a a)))) (* PI 3.08641975308642e-5))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.1e+83) {
		tmp = b * b;
	} else {
		tmp = (((double) M_PI) * (angle_m * (angle_m * (a * a)))) * (((double) M_PI) * 3.08641975308642e-5);
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.1e+83) {
		tmp = b * b;
	} else {
		tmp = (Math.PI * (angle_m * (angle_m * (a * a)))) * (Math.PI * 3.08641975308642e-5);
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 4.1e+83:
		tmp = b * b
	else:
		tmp = (math.pi * (angle_m * (angle_m * (a * a)))) * (math.pi * 3.08641975308642e-5)
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 4.1e+83)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(pi * Float64(angle_m * Float64(angle_m * Float64(a * a)))) * Float64(pi * 3.08641975308642e-5));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 4.1e+83)
		tmp = b * b;
	else
		tmp = (pi * (angle_m * (angle_m * (a * a)))) * (pi * 3.08641975308642e-5);
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 4.1e+83], N[(b * b), $MachinePrecision], N[(N[(Pi * N[(angle$95$m * N[(angle$95$m * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.1000000000000001e83

   1. Initial program 76.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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. lower-*.f64 61.0

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

   5. Simplified 61.0%

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

   if 4.1000000000000001e83 < a

   1. Initial program 83.0%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 83.0

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 50.1%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-PI.f64 N/A

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

      19. lower-PI.f64 57.8

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

   10. Simplified 57.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. associate-*r* N/A

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

       8. lift-*.f64 N/A

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

       9. lift-*.f64 N/A

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

       10. associate-*l* N/A

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

       11. associate-*r* N/A

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

       12. lower-*.f64 N/A

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

       13. lower-*.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. associate-*l* N/A

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

       16. lower-*.f64 N/A

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

       17. lower-*.f64 N/A

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

       18. lower-*.f64 62.0

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

   12. Applied egg-rr 62.0%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+83}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.0% accurate, 12.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+83}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(angle\_m \cdot angle\_m\right) \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= a 4.1e+83)
   (* b b)
   (* a (* (* angle_m angle_m) (* a (* PI (* PI 3.08641975308642e-5)))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.1e+83) {
		tmp = b * b;
	} else {
		tmp = a * ((angle_m * angle_m) * (a * (((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (a <= 4.1e+83) {
		tmp = b * b;
	} else {
		tmp = a * ((angle_m * angle_m) * (a * (Math.PI * (Math.PI * 3.08641975308642e-5))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if a <= 4.1e+83:
		tmp = b * b
	else:
		tmp = a * ((angle_m * angle_m) * (a * (math.pi * (math.pi * 3.08641975308642e-5))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (a <= 4.1e+83)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * Float64(Float64(angle_m * angle_m) * Float64(a * Float64(pi * Float64(pi * 3.08641975308642e-5)))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (a <= 4.1e+83)
		tmp = b * b;
	else
		tmp = a * ((angle_m * angle_m) * (a * (pi * (pi * 3.08641975308642e-5))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[a, 4.1e+83], N[(b * b), $MachinePrecision], N[(a * N[(N[(angle$95$m * angle$95$m), $MachinePrecision] * N[(a * N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 4.1000000000000001e83

   1. Initial program 76.6%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   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. lower-*.f64 61.0

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

   5. Simplified 61.0%

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

   if 4.1000000000000001e83 < a

   1. Initial program 83.0%

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

      1. lift-PI.f64 N/A

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

      2. associate-*l/ N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 83.0

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in angle around 0

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

      1. lower-fma.f64 N/A

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

   7. Simplified 50.1%

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

   8. Taylor expanded in b around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-PI.f64 N/A

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

      19. lower-PI.f64 57.8

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

   10. Simplified 57.8%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-PI.f64 N/A

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

       4. lift-PI.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. associate-*l* N/A

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

       12. associate-*l* N/A

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

       13. lower-*.f64 N/A

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

       14. lower-*.f64 N/A

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

   12. Applied egg-rr 61.8%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.1 \cdot 10^{+83}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(angle \cdot angle\right) \cdot \left(a \cdot \left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.3% 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 77.9%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
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. lower-*.f64 52.7

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

5. Simplified 52.7%

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

Reproduce

herbie shell --seed 2024207 
(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)))