ab-angle->ABCF A

Average Error: 20.3 → 21.0

Time: 19.7s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := 1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;angle \leq -1900:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot t_0\\ \mathbf{elif}\;angle \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + t_0 \cdot \frac{a}{\frac{2}{a}}\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos (* angle (* PI 0.011111111111111112))))))
   (if (<= angle -1900.0)
     (+ (pow b 2.0) (* (/ (* a a) 2.0) t_0))
     (if (<= angle 1.6e-26)
       (+
        (pow b 2.0)
        (* 3.08641975308642e-5 (* (* PI (* angle (* a PI))) (* a angle))))
       (+ (pow b 2.0) (* t_0 (/ a (/ 2.0 a))))))))

C

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

↓

double code(double a, double b, double angle) {
	double t_0 = 1.0 - cos((angle * (((double) M_PI) * 0.011111111111111112)));
	double tmp;
	if (angle <= -1900.0) {
		tmp = pow(b, 2.0) + (((a * a) / 2.0) * t_0);
	} else if (angle <= 1.6e-26) {
		tmp = pow(b, 2.0) + (3.08641975308642e-5 * ((((double) M_PI) * (angle * (a * ((double) M_PI)))) * (a * angle)));
	} else {
		tmp = pow(b, 2.0) + (t_0 * (a / (2.0 / a)));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle) {
	double t_0 = 1.0 - Math.cos((angle * (Math.PI * 0.011111111111111112)));
	double tmp;
	if (angle <= -1900.0) {
		tmp = Math.pow(b, 2.0) + (((a * a) / 2.0) * t_0);
	} else if (angle <= 1.6e-26) {
		tmp = Math.pow(b, 2.0) + (3.08641975308642e-5 * ((Math.PI * (angle * (a * Math.PI))) * (a * angle)));
	} else {
		tmp = Math.pow(b, 2.0) + (t_0 * (a / (2.0 / a)));
	}
	return tmp;
}

Python

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

↓

def code(a, b, angle):
	t_0 = 1.0 - math.cos((angle * (math.pi * 0.011111111111111112)))
	tmp = 0
	if angle <= -1900.0:
		tmp = math.pow(b, 2.0) + (((a * a) / 2.0) * t_0)
	elif angle <= 1.6e-26:
		tmp = math.pow(b, 2.0) + (3.08641975308642e-5 * ((math.pi * (angle * (a * math.pi))) * (a * angle)))
	else:
		tmp = math.pow(b, 2.0) + (t_0 * (a / (2.0 / a)))
	return tmp

Julia

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

↓

function code(a, b, angle)
	t_0 = Float64(1.0 - cos(Float64(angle * Float64(pi * 0.011111111111111112))))
	tmp = 0.0
	if (angle <= -1900.0)
		tmp = Float64((b ^ 2.0) + Float64(Float64(Float64(a * a) / 2.0) * t_0));
	elseif (angle <= 1.6e-26)
		tmp = Float64((b ^ 2.0) + Float64(3.08641975308642e-5 * Float64(Float64(pi * Float64(angle * Float64(a * pi))) * Float64(a * angle))));
	else
		tmp = Float64((b ^ 2.0) + Float64(t_0 * Float64(a / Float64(2.0 / a))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(a, b, angle)
	t_0 = 1.0 - cos((angle * (pi * 0.011111111111111112)));
	tmp = 0.0;
	if (angle <= -1900.0)
		tmp = (b ^ 2.0) + (((a * a) / 2.0) * t_0);
	elseif (angle <= 1.6e-26)
		tmp = (b ^ 2.0) + (3.08641975308642e-5 * ((pi * (angle * (a * pi))) * (a * angle)));
	else
		tmp = (b ^ 2.0) + (t_0 * (a / (2.0 / a)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[a_, b_, angle_] := Block[{t$95$0 = N[(1.0 - N[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[angle, -1900.0], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(N[(a * a), $MachinePrecision] / 2.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 1.6e-26], N[(N[Power[b, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[(N[(Pi * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(t$95$0 * N[(a / N[(2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if angle < -1900

   1. Initial program 45.7

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

   2. Taylor expanded in angle around 0 45.5

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

   3. Applied egg-rr 45.4

      \[\leadsto \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]

   4. Simplified 45.4

      \[\leadsto \color{blue}{\frac{a \cdot a}{2} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      Proof

      [Start] 45.4	\[ \frac{\left(a \cdot a\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [<=] 45.4	\[ \frac{\color{blue}{{a}^{2}} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-*l/ [<=] 45.4	\[ \color{blue}{\frac{{a}^{2}}{2} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 45.4	\[ \frac{\color{blue}{a \cdot a}}{2} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      mul0-rgt [=>] 45.4	\[ \frac{a \cdot a}{2} \cdot \left(\cos \color{blue}{0} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      cos-0 [=>] 45.4	\[ \frac{a \cdot a}{2} \cdot \left(\color{blue}{1} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 45.4	\[ \frac{a \cdot a}{2} \cdot \left(1 - \cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

   if -1900 < angle < 1.6000000000000001e-26

   1. Initial program 0.3

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

   2. Taylor expanded in angle around 0 0.4

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

   3. Taylor expanded in angle around 0 0.6

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

   4. Simplified 0.6

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

      [Start] 0.6	\[ {\left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 0.6	\[ {\left(0.005555555555555556 \cdot \color{blue}{\left(\left(a \cdot \pi\right) \cdot angle\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 0.6	\[ {\left(0.005555555555555556 \cdot \color{blue}{\left(a \cdot \left(\pi \cdot angle\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [<=] 0.6	\[ {\left(0.005555555555555556 \cdot \left(a \cdot \color{blue}{\left(angle \cdot \pi\right)}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   5. Taylor expanded in a around 0 15.5

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

   6. Simplified 0.8

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]
      Proof

      [Start] 15.5	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot \left({a}^{2} \cdot {\pi}^{2}\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [=>] 15.5	\[ 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {a}^{2}\right) \cdot {\pi}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [=>] 15.5	\[ \color{blue}{\left(3.08641975308642 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {a}^{2}\right)\right) \cdot {\pi}^{2}} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 15.5	\[ \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot {a}^{2}\right)\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 15.5	\[ \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(a \cdot a\right)}\right)\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 15.5	\[ \left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(a \cdot a\right) \cdot \left(angle \cdot angle\right)\right)}\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]
      swap-sqr [<=] 0.8	\[ \left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)}\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [<=] 0.8	\[ \left(3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(a \cdot angle\right)}^{2}}\right) \cdot {\pi}^{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 0.8	\[ \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({\left(a \cdot angle\right)}^{2} \cdot {\pi}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 0.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)} \cdot {\pi}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 0.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) + {\left(b \cdot 1\right)}^{2} \]
      swap-sqr [<=] 0.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [<=] 0.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [<=] 0.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(a \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [<=] 0.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [=>] 0.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 0.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot {\left(\color{blue}{\left(angle \cdot a\right)} \cdot \pi\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [<=] 0.8	\[ 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(angle \cdot \left(a \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Applied egg-rr 0.8

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(angle \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

   if 1.6000000000000001e-26 < angle

   1. Initial program 42.3

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

   2. Taylor expanded in angle around 0 42.5

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

   3. Applied egg-rr 44.2

      \[\leadsto \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]

   4. Simplified 44.2

      \[\leadsto \color{blue}{\frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      Proof

      [Start] 44.2	\[ \frac{\left(a \cdot a\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [<=] 44.2	\[ \frac{\color{blue}{{a}^{2}} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-*l/ [<=] 44.2	\[ \color{blue}{\frac{{a}^{2}}{2} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 44.2	\[ \frac{\color{blue}{a \cdot a}}{2} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-/l* [=>] 44.2	\[ \color{blue}{\frac{a}{\frac{2}{a}}} \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      mul0-rgt [=>] 44.2	\[ \frac{a}{\frac{2}{a}} \cdot \left(\cos \color{blue}{0} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      cos-0 [=>] 44.2	\[ \frac{a}{\frac{2}{a}} \cdot \left(\color{blue}{1} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 44.2	\[ \frac{a}{\frac{2}{a}} \cdot \left(1 - \cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

3. Recombined 3 regimes into one program.

4. Final simplification 21.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -1900:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{2} \cdot \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot \frac{a}{\frac{2}{a}}\\ \end{array} \]

Alternatives

Alternative 1
Error	20.4
Cost	45440

\[{\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left|angle \cdot \frac{\pi}{180}\right|\right)\right)\right)\right)}^{2} + {b}^{2} \]

Alternative 2
Error	20.4
Cost	26368

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

Alternative 3
Error	20.4
Cost	26240

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

Alternative 4
Error	20.4
Cost	26240

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

Alternative 5
Error	20.4
Cost	26240

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

Alternative 6
Error	22.8
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.000165 \lor \neg \left(angle \leq 1.6 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{a}{\frac{2}{a}} \cdot 0 + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]

Alternative 7
Error	20.6
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -1900 \lor \neg \left(angle \leq 0.0185\right):\\ \;\;\;\;{b}^{2} + \left(1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right) \cdot \frac{a}{\frac{2}{a}}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right)\\ \end{array} \]

Alternative 8
Error	23.6
Cost	20360

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+82}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2}\\ \mathbf{elif}\;a \leq 3 \cdot 10^{-68}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \left(a \cdot {\pi}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \]

Alternative 9
Error	23.6
Cost	20360

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+82}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \cdot \left(a \cdot angle\right)\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-68}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(angle \cdot \left(a \cdot \left(a \cdot {\pi}^{2}\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \]

Alternative 10
Error	23.6
Cost	20360

\[\begin{array}{l} t_0 := angle \cdot \left(a \cdot \pi\right)\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{+153}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot t_0\right) \cdot \left(a \cdot angle\right)\right)\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+23}:\\ \;\;\;\;{b}^{2} + angle \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\left(a \cdot a\right) \cdot {\pi}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(t_0 \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	25.9
Cost	19840

\[{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \]

Alternative 12
Error	25.8
Cost	19840

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

Alternative 13
Error	25.8
Cost	19840

\[{b}^{2} + {\left(a \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2023031 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))