ab-angle->ABCF C

Percentage Accurate: 80.0% → 78.6%

Time: 14.6s

Alternatives: 10

Speedup: N/A×

• 36.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a + \left(\left(angle\_m \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot angle\_m\right)\right) \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle\_m}}\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 3.4e-9)
   (+
    (* a a)
    (* (* (* angle_m 3.08641975308642e-5) (* b angle_m)) (* b (* PI PI))))
   (+
    (* a a)
    (* (* b b) (- 0.5 (* 0.5 (cos (* 2.0 (/ PI (/ 180.0 angle_m))))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.4e-9) {
		tmp = (a * a) + (((angle_m * 3.08641975308642e-5) * (b * angle_m)) * (b * (((double) M_PI) * ((double) M_PI))));
	} else {
		tmp = (a * a) + ((b * b) * (0.5 - (0.5 * cos((2.0 * (((double) M_PI) / (180.0 / angle_m)))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.4e-9) {
		tmp = (a * a) + (((angle_m * 3.08641975308642e-5) * (b * angle_m)) * (b * (Math.PI * Math.PI)));
	} else {
		tmp = (a * a) + ((b * b) * (0.5 - (0.5 * Math.cos((2.0 * (Math.PI / (180.0 / angle_m)))))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if angle_m <= 3.4e-9:
		tmp = (a * a) + (((angle_m * 3.08641975308642e-5) * (b * angle_m)) * (b * (math.pi * math.pi)))
	else:
		tmp = (a * a) + ((b * b) * (0.5 - (0.5 * math.cos((2.0 * (math.pi / (180.0 / angle_m)))))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 3.4e-9)
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(angle_m * 3.08641975308642e-5) * Float64(b * angle_m)) * Float64(b * Float64(pi * pi))));
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(b * b) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(pi / Float64(180.0 / angle_m))))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 3.4e-9)
		tmp = (a * a) + (((angle_m * 3.08641975308642e-5) * (b * angle_m)) * (b * (pi * pi)));
	else
		tmp = (a * a) + ((b * b) * (0.5 - (0.5 * cos((2.0 * (pi / (180.0 / angle_m)))))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[angle$95$m, 3.4e-9], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(angle$95$m * 3.08641975308642e-5), $MachinePrecision] * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(b * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.3999999999999998e-9

   1. Initial program 85.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 85.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 85.0%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{{b}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(b \cdot b\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(b \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. Simplified 73.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot angle\right) \cdot \color{blue}{\left(angle \cdot \left(b \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot angle\right) \cdot \left(\left(angle \cdot b\right) \cdot \color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\frac{1}{32400} \cdot angle\right) \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{32400} \cdot angle\right) \cdot \left(angle \cdot b\right)\right), \color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{32400} \cdot angle\right), \left(angle \cdot b\right)\right), \left(\color{blue}{b} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \frac{1}{32400}\right), \left(angle \cdot b\right)\right), \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \left(angle \cdot b\right)\right), \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \left(b \cdot angle\right)\right), \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 81.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 81.3%

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

   if 3.3999999999999998e-9 < angle

   1. Initial program 63.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 65.2%

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

      1. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right)\right), 2\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{180}\right), angle\right)\right)\right), 2\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 180\right), angle\right)\right)\right), 2\right)\right) \]

      5. PI-lowering-PI.f64 65.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 180\right), angle\right)\right)\right), 2\right)\right) \]

   7. Applied egg-rr 65.1%

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

      1. +-commutative N/A

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

      2. associate-/r/ N/A

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

      3. pow2 N/A

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

      4. associate-*l* N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(b \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot b\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left(a \cdot a\right)}\right) \]

   9. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{\left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right) + a \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a + \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \frac{\pi}{\frac{180}{angle}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 79.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 79.6%

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

Alternative 3: 78.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;angle\_m \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a + \left(\left(angle\_m \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot angle\_m\right)\right) \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\left(\pi \cdot angle\_m\right) \cdot 0.011111111111111112\right) \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= angle_m 3.4e-9)
   (+
    (* a a)
    (* (* (* angle_m 3.08641975308642e-5) (* b angle_m)) (* b (* PI PI))))
   (+
    (* a a)
    (*
     (* b b)
     (+ 0.5 (* (cos (* (* PI angle_m) 0.011111111111111112)) -0.5))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.4e-9) {
		tmp = (a * a) + (((angle_m * 3.08641975308642e-5) * (b * angle_m)) * (b * (((double) M_PI) * ((double) M_PI))));
	} else {
		tmp = (a * a) + ((b * b) * (0.5 + (cos(((((double) M_PI) * angle_m) * 0.011111111111111112)) * -0.5)));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (angle_m <= 3.4e-9) {
		tmp = (a * a) + (((angle_m * 3.08641975308642e-5) * (b * angle_m)) * (b * (Math.PI * Math.PI)));
	} else {
		tmp = (a * a) + ((b * b) * (0.5 + (Math.cos(((Math.PI * angle_m) * 0.011111111111111112)) * -0.5)));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if angle_m <= 3.4e-9:
		tmp = (a * a) + (((angle_m * 3.08641975308642e-5) * (b * angle_m)) * (b * (math.pi * math.pi)))
	else:
		tmp = (a * a) + ((b * b) * (0.5 + (math.cos(((math.pi * angle_m) * 0.011111111111111112)) * -0.5)))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (angle_m <= 3.4e-9)
		tmp = Float64(Float64(a * a) + Float64(Float64(Float64(angle_m * 3.08641975308642e-5) * Float64(b * angle_m)) * Float64(b * Float64(pi * pi))));
	else
		tmp = Float64(Float64(a * a) + Float64(Float64(b * b) * Float64(0.5 + Float64(cos(Float64(Float64(pi * angle_m) * 0.011111111111111112)) * -0.5))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (angle_m <= 3.4e-9)
		tmp = (a * a) + (((angle_m * 3.08641975308642e-5) * (b * angle_m)) * (b * (pi * pi)));
	else
		tmp = (a * a) + ((b * b) * (0.5 + (cos(((pi * angle_m) * 0.011111111111111112)) * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[angle$95$m, 3.4e-9], N[(N[(a * a), $MachinePrecision] + N[(N[(N[(angle$95$m * 3.08641975308642e-5), $MachinePrecision] * N[(b * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(b * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(N[(b * b), $MachinePrecision] * N[(0.5 + N[(N[Cos[N[(N[(Pi * angle$95$m), $MachinePrecision] * 0.011111111111111112), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 3.3999999999999998e-9

   1. Initial program 85.0%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 85.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 85.0%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{{b}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(b \cdot b\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(b \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. Simplified 73.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot angle\right) \cdot \color{blue}{\left(angle \cdot \left(b \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{32400} \cdot angle\right) \cdot \left(\left(angle \cdot b\right) \cdot \color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\left(\frac{1}{32400} \cdot angle\right) \cdot \left(angle \cdot b\right)\right) \cdot \color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\left(\frac{1}{32400} \cdot angle\right) \cdot \left(angle \cdot b\right)\right), \color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{32400} \cdot angle\right), \left(angle \cdot b\right)\right), \left(\color{blue}{b} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot \frac{1}{32400}\right), \left(angle \cdot b\right)\right), \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \left(angle \cdot b\right)\right), \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \left(b \cdot angle\right)\right), \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      13. PI-lowering-PI.f64 81.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \frac{1}{32400}\right), \mathsf{*.f64}\left(b, angle\right)\right), \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 81.3%

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

   if 3.3999999999999998e-9 < angle

   1. Initial program 63.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 65.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 65.2%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({\left(\sin \left(\mathsf{PI}\left(\right) \cdot \frac{angle}{180}\right) \cdot b\right)}^{2}\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot b\right)}^{2}\right)\right) \]

      3. unpow-prod-down N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)}^{2} \cdot \color{blue}{{b}^{2}}\right)\right) \]

      4. pow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot {\color{blue}{b}}^{2}\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot {b}^{2}\right)\right) \]

      6. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right) \cdot angle}{180}\right)\right) \cdot {b}^{2}\right)\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{180} \cdot angle\right)\right) \cdot {b}^{2}\right)\right) \]

      8. associate-/r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot {b}^{2}\right)\right) \]

      9. sqr-sin-a N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right) \cdot {\color{blue}{b}}^{2}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{\mathsf{PI}\left(\right)}{\frac{180}{angle}}\right)\right), \color{blue}{\left({b}^{2}\right)}\right)\right) \]

   7. Applied egg-rr 65.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 3.4 \cdot 10^{-9}:\\ \;\;\;\;a \cdot a + \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(b \cdot angle\right)\right) \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + \left(b \cdot b\right) \cdot \left(0.5 + \cos \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.4% accurate, 18.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+49}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\_m\right) \cdot \left(angle\_m \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.1e+49)
   (* a a)
   (+
    (* a a)
    (* 3.08641975308642e-5 (* (* b angle_m) (* angle_m (* b (* PI PI))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.1e+49) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (3.08641975308642e-5 * ((b * angle_m) * (angle_m * (b * (((double) M_PI) * ((double) M_PI))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.1e+49) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (3.08641975308642e-5 * ((b * angle_m) * (angle_m * (b * (Math.PI * Math.PI)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 2.1e+49:
		tmp = a * a
	else:
		tmp = (a * a) + (3.08641975308642e-5 * ((b * angle_m) * (angle_m * (b * (math.pi * math.pi)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.1e+49)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(3.08641975308642e-5 * Float64(Float64(b * angle_m) * Float64(angle_m * Float64(b * Float64(pi * pi))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.1e+49)
		tmp = a * a;
	else
		tmp = (a * a) + (3.08641975308642e-5 * ((b * angle_m) * (angle_m * (b * (pi * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.1e+49], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[(b * angle$95$m), $MachinePrecision] * N[(angle$95$m * N[(b * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.10000000000000011e49

   1. Initial program 75.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 62.7%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 62.7%

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

   if 2.10000000000000011e49 < b

   1. Initial program 92.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 92.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 92.6%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{{b}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(b \cdot b\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(b \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. Simplified 74.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot \left(b \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot \color{blue}{angle}\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(\left(angle \cdot b\right) \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot angle\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot b\right) \cdot \color{blue}{\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\left(angle \cdot b\right), \color{blue}{\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right)}\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\left(b \cdot angle\right), \left(\color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot angle\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \left(\color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot angle\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{angle}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), angle\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right), angle\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right), angle\right)\right)\right)\right) \]

      11. PI-lowering-PI.f64 91.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right), angle\right)\right)\right)\right) \]

   10. Applied egg-rr 91.6%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.1 \cdot 10^{+49}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(b \cdot angle\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.4% accurate, 18.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.46 \cdot 10^{+49}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot \left(b \cdot \left(\left(b \cdot angle\_m\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.46e+49)
   (* a a)
   (+
    (* a a)
    (* 3.08641975308642e-5 (* angle_m (* b (* (* b angle_m) (* PI PI))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.46e+49) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (3.08641975308642e-5 * (angle_m * (b * ((b * angle_m) * (((double) M_PI) * ((double) M_PI))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.46e+49) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (3.08641975308642e-5 * (angle_m * (b * ((b * angle_m) * (Math.PI * Math.PI)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 1.46e+49:
		tmp = a * a
	else:
		tmp = (a * a) + (3.08641975308642e-5 * (angle_m * (b * ((b * angle_m) * (math.pi * math.pi)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.46e+49)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(3.08641975308642e-5 * Float64(angle_m * Float64(b * Float64(Float64(b * angle_m) * Float64(pi * pi))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.46e+49)
		tmp = a * a;
	else
		tmp = (a * a) + (3.08641975308642e-5 * (angle_m * (b * ((b * angle_m) * (pi * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.46e+49], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(3.08641975308642e-5 * N[(angle$95$m * N[(b * N[(N[(b * angle$95$m), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.46000000000000008e49

   1. Initial program 75.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 62.7%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 62.7%

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

   if 1.46000000000000008e49 < b

   1. Initial program 92.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 92.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 92.6%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{{b}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(b \cdot b\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(b \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. Simplified 74.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \left(\left(angle \cdot b\right) \cdot \color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \left(\left(angle \cdot b\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{b}\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \left(\left(\left(angle \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{b}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\left(\left(angle \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{b}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot b\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), b\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), b\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), b\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right), b\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right), b\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 88.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right), b\right)\right)\right)\right) \]

   10. Applied egg-rr 88.3%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.46 \cdot 10^{+49}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(b \cdot \left(\left(b \cdot angle\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.4% accurate, 18.9× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+49}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot \left(\pi \cdot \left(\left(b \cdot angle\_m\right) \cdot \left(b \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 1.8e+49)
   (* a a)
   (+
    (* a a)
    (* 3.08641975308642e-5 (* angle_m (* PI (* (* b angle_m) (* b PI))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.8e+49) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (3.08641975308642e-5 * (angle_m * (((double) M_PI) * ((b * angle_m) * (b * ((double) M_PI))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 1.8e+49) {
		tmp = a * a;
	} else {
		tmp = (a * a) + (3.08641975308642e-5 * (angle_m * (Math.PI * ((b * angle_m) * (b * Math.PI)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 1.8e+49:
		tmp = a * a
	else:
		tmp = (a * a) + (3.08641975308642e-5 * (angle_m * (math.pi * ((b * angle_m) * (b * math.pi)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 1.8e+49)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(a * a) + Float64(3.08641975308642e-5 * Float64(angle_m * Float64(pi * Float64(Float64(b * angle_m) * Float64(b * pi))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 1.8e+49)
		tmp = a * a;
	else
		tmp = (a * a) + (3.08641975308642e-5 * (angle_m * (pi * ((b * angle_m) * (b * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 1.8e+49], N[(a * a), $MachinePrecision], N[(N[(a * a), $MachinePrecision] + N[(3.08641975308642e-5 * N[(angle$95$m * N[(Pi * N[(N[(b * angle$95$m), $MachinePrecision] * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.79999999999999998e49

   1. Initial program 75.5%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 62.7%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 62.7%

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

   if 1.79999999999999998e49 < b

   1. Initial program 92.6%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 92.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 92.6%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{{b}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(b \cdot b\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(b \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. Simplified 74.9%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \left(\left(angle \cdot b\right) \cdot \color{blue}{\left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right)\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \left(\left(angle \cdot b\right) \cdot \left(\left(b \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \left(\left(\left(angle \cdot b\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\left(\left(angle \cdot b\right) \cdot \left(b \cdot \mathsf{PI}\left(\right)\right)\right), \color{blue}{\mathsf{PI}\left(\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(angle \cdot b\right), \left(b \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(b \cdot angle\right), \left(b \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \left(b \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(b, \mathsf{PI}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      10. PI-lowering-PI.f64 88.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(b, angle\right), \mathsf{*.f64}\left(b, \mathsf{PI.f64}\left(\right)\right)\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 88.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.8 \cdot 10^{+49}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot \left(\left(b \cdot angle\right) \cdot \left(b \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.6% accurate, 23.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+158}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle\_m \cdot \left(angle\_m \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 5e+158)
   (* a a)
   (* angle_m (* angle_m (* 3.08641975308642e-5 (* b (* b (* PI PI))))))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5e+158) {
		tmp = a * a;
	} else {
		tmp = angle_m * (angle_m * (3.08641975308642e-5 * (b * (b * (((double) M_PI) * ((double) M_PI))))));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 5e+158) {
		tmp = a * a;
	} else {
		tmp = angle_m * (angle_m * (3.08641975308642e-5 * (b * (b * (Math.PI * Math.PI)))));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 5e+158:
		tmp = a * a
	else:
		tmp = angle_m * (angle_m * (3.08641975308642e-5 * (b * (b * (math.pi * math.pi)))))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 5e+158)
		tmp = Float64(a * a);
	else
		tmp = Float64(angle_m * Float64(angle_m * Float64(3.08641975308642e-5 * Float64(b * Float64(b * Float64(pi * pi))))));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 5e+158)
		tmp = a * a;
	else
		tmp = angle_m * (angle_m * (3.08641975308642e-5 * (b * (b * (pi * pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 5e+158], N[(a * a), $MachinePrecision], N[(angle$95$m * N[(angle$95$m * N[(3.08641975308642e-5 * N[(b * N[(b * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.9999999999999996e158

   1. Initial program 75.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 61.6%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 61.6%

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

   if 4.9999999999999996e158 < b

   1. Initial program 99.9%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 99.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{{b}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(b \cdot b\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(b \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. Simplified 78.7%

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

   9. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

       16. PI-lowering-PI.f64 72.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

   11. Simplified 72.2%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot \left(\left(b \cdot b\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right)\right), \color{blue}{angle}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(b \cdot b\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right)\right), angle\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\left(\left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{32400}\right)\right), angle\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \left(\frac{1}{32400} \cdot \left(\left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), angle\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(b \cdot b\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right), angle\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{32400}, \left(b \cdot \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), angle\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(b, \left(b \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), angle\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), angle\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), angle\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right)\right)\right), angle\right) \]

       13. PI-lowering-PI.f64 78.7%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right)\right)\right)\right)\right), angle\right) \]

   13. Applied egg-rr 78.7%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{+158}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.6% accurate, 23.1× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+154}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(angle\_m \cdot angle\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (if (<= b 2.4e+154)
   (* a a)
   (* b (* (* b (* 3.08641975308642e-5 (* PI PI))) (* angle_m angle_m)))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.4e+154) {
		tmp = a * a;
	} else {
		tmp = b * ((b * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI)))) * (angle_m * angle_m));
	}
	return tmp;
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double tmp;
	if (b <= 2.4e+154) {
		tmp = a * a;
	} else {
		tmp = b * ((b * (3.08641975308642e-5 * (Math.PI * Math.PI))) * (angle_m * angle_m));
	}
	return tmp;
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	tmp = 0
	if b <= 2.4e+154:
		tmp = a * a
	else:
		tmp = b * ((b * (3.08641975308642e-5 * (math.pi * math.pi))) * (angle_m * angle_m))
	return tmp

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	tmp = 0.0
	if (b <= 2.4e+154)
		tmp = Float64(a * a);
	else
		tmp = Float64(b * Float64(Float64(b * Float64(3.08641975308642e-5 * Float64(pi * pi))) * Float64(angle_m * angle_m)));
	end
	return tmp
end

MATLAB

angle_m = abs(angle);
function tmp_2 = code(a, b, angle_m)
	tmp = 0.0;
	if (b <= 2.4e+154)
		tmp = a * a;
	else
		tmp = b * ((b * (3.08641975308642e-5 * (pi * pi))) * (angle_m * angle_m));
	end
	tmp_2 = tmp;
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := If[LessEqual[b, 2.4e+154], N[(a * a), $MachinePrecision], N[(b * N[(N[(b * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle$95$m * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.40000000000000015e154

   1. Initial program 75.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 61.8%

         \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

   5. Simplified 61.8%

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

   if 2.40000000000000015e154 < b

   1. Initial program 99.8%

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

   3. Taylor expanded in angle around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

      2. *-lowering-*.f64 99.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in angle around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{{b}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

      13. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(b \cdot b\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right)\right) \]

      17. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(b \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

   8. Simplified 76.6%

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

   9. Taylor expanded in a around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left({angle}^{2}\right), \color{blue}{\left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(angle \cdot angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\color{blue}{\frac{1}{32400}} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \left({b}^{2} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left({b}^{2}\right), \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\left(b \cdot b\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \left(\color{blue}{{\mathsf{PI}\left(\right)}^{2}} \cdot \frac{1}{32400}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\left({\mathsf{PI}\left(\right)}^{2}\right), \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

       16. PI-lowering-PI.f64 70.3%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(angle, angle\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, b\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right)\right) \]

   11. Simplified 70.3%

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

       1. *-commutative N/A

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

       2. associate-*l* N/A

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

       3. associate-*l* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(b, \color{blue}{\left(\left(b \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right) \cdot \left(angle \cdot angle\right)\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(b \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right), \color{blue}{\left(angle \cdot angle\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right)\right), \left(\color{blue}{angle} \cdot angle\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right), \left(angle \cdot angle\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right), \left(angle \cdot angle\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI}\left(\right)\right), \frac{1}{32400}\right)\right), \left(angle \cdot angle\right)\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right), \left(angle \cdot angle\right)\right)\right) \]

       11. *-lowering-*.f64 71.1%

           \[\leadsto \mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(b, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{PI.f64}\left(\right)\right), \frac{1}{32400}\right)\right), \mathsf{*.f64}\left(angle, \color{blue}{angle}\right)\right)\right) \]

   13. Applied egg-rr 71.1%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.4 \cdot 10^{+154}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(b \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \left(angle \cdot angle\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.6% accurate, 24.5× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle\_m \cdot \left(angle\_m \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \end{array} \]

FPCore

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

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	return (a * a) + (3.08641975308642e-5 * (angle_m * (angle_m * (b * (b * (((double) M_PI) * ((double) M_PI)))))));
}

Java

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

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	return (a * a) + (3.08641975308642e-5 * (angle_m * (angle_m * (b * (b * (math.pi * math.pi))))))

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	return Float64(Float64(a * a) + Float64(3.08641975308642e-5 * Float64(angle_m * Float64(angle_m * Float64(b * Float64(b * Float64(pi * pi)))))))
end

MATLAB

angle_m = abs(angle);
function tmp = code(a, b, angle_m)
	tmp = (a * a) + (3.08641975308642e-5 * (angle_m * (angle_m * (b * (b * (pi * pi))))));
end

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in angle around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left({a}^{2}\right)}, \mathsf{pow.f64}\left(\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right), 2\right)\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

   2. *-lowering-*.f64 79.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{*.f64}\left(b, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{/.f64}\left(angle, 180\right)\right)\right)\right)}, 2\right)\right) \]

5. Simplified 79.6%

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

6. Taylor expanded in angle around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

      \[\leadsto \mathsf{+.f64}\left(\left({a}^{2}\right), \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(a \cdot a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\color{blue}{{angle}^{2}} \cdot \left(\frac{1}{32400} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left({angle}^{2} \cdot \left(\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \color{blue}{\frac{1}{32400}}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \left(\frac{1}{32400} \cdot \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \color{blue}{\left({angle}^{2} \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(\left(angle \cdot angle\right) \cdot \left(\color{blue}{{b}^{2}} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \left(angle \cdot \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \color{blue}{\left(angle \cdot \left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \color{blue}{\left({b}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(\left(b \cdot b\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right)\right)\right)\right) \]

   17. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \left(b \cdot \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(a, a\right), \mathsf{*.f64}\left(\frac{1}{32400}, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(angle, \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right)\right)\right)\right) \]

8. Simplified 70.6%

   \[\leadsto \color{blue}{a \cdot a + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right)} \]
9. Add Preprocessing

Alternative 10: 56.8% accurate, 139.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.3%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 56.8%

      \[\leadsto \mathsf{*.f64}\left(a, \color{blue}{a}\right) \]

5. Simplified 56.8%

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

Reproduce

herbie shell --seed 2024160 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))