ab-angle->ABCF A

Average Accuracy: 69.1% → 68.3%

Time: 21.6s

Precision: binary64

Cost: 20424

Math

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

↓

\[\begin{array}{l} t_0 := \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;angle \leq -7 \cdot 10^{+22}:\\ \;\;\;\;{b}^{2} + a \cdot \left(a \cdot \frac{t_0 + -1}{-2}\right)\\ \mathbf{elif}\;angle \leq 3.9 \cdot 10^{-12}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{1 - t_0}{\frac{2}{a \cdot a}}\\ \end{array} \]

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 (cos (* angle (* PI 0.011111111111111112)))))
   (if (<= angle -7e+22)
     (+ (pow b 2.0) (* a (* a (/ (+ t_0 -1.0) -2.0))))
     (if (<= angle 3.9e-12)
       (+ (pow b 2.0) (pow (* a (* angle (/ PI 180.0))) 2.0))
       (+ (pow b 2.0) (/ (- 1.0 t_0) (/ 2.0 (* a 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 = cos((angle * (((double) M_PI) * 0.011111111111111112)));
	double tmp;
	if (angle <= -7e+22) {
		tmp = pow(b, 2.0) + (a * (a * ((t_0 + -1.0) / -2.0)));
	} else if (angle <= 3.9e-12) {
		tmp = pow(b, 2.0) + pow((a * (angle * (((double) M_PI) / 180.0))), 2.0);
	} else {
		tmp = pow(b, 2.0) + ((1.0 - t_0) / (2.0 / (a * 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 = Math.cos((angle * (Math.PI * 0.011111111111111112)));
	double tmp;
	if (angle <= -7e+22) {
		tmp = Math.pow(b, 2.0) + (a * (a * ((t_0 + -1.0) / -2.0)));
	} else if (angle <= 3.9e-12) {
		tmp = Math.pow(b, 2.0) + Math.pow((a * (angle * (Math.PI / 180.0))), 2.0);
	} else {
		tmp = Math.pow(b, 2.0) + ((1.0 - t_0) / (2.0 / (a * 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 = math.cos((angle * (math.pi * 0.011111111111111112)))
	tmp = 0
	if angle <= -7e+22:
		tmp = math.pow(b, 2.0) + (a * (a * ((t_0 + -1.0) / -2.0)))
	elif angle <= 3.9e-12:
		tmp = math.pow(b, 2.0) + math.pow((a * (angle * (math.pi / 180.0))), 2.0)
	else:
		tmp = math.pow(b, 2.0) + ((1.0 - t_0) / (2.0 / (a * 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 = cos(Float64(angle * Float64(pi * 0.011111111111111112)))
	tmp = 0.0
	if (angle <= -7e+22)
		tmp = Float64((b ^ 2.0) + Float64(a * Float64(a * Float64(Float64(t_0 + -1.0) / -2.0))));
	elseif (angle <= 3.9e-12)
		tmp = Float64((b ^ 2.0) + (Float64(a * Float64(angle * Float64(pi / 180.0))) ^ 2.0));
	else
		tmp = Float64((b ^ 2.0) + Float64(Float64(1.0 - t_0) / Float64(2.0 / Float64(a * 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 = cos((angle * (pi * 0.011111111111111112)));
	tmp = 0.0;
	if (angle <= -7e+22)
		tmp = (b ^ 2.0) + (a * (a * ((t_0 + -1.0) / -2.0)));
	elseif (angle <= 3.9e-12)
		tmp = (b ^ 2.0) + ((a * (angle * (pi / 180.0))) ^ 2.0);
	else
		tmp = (b ^ 2.0) + ((1.0 - t_0) / (2.0 / (a * 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[Cos[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[angle, -7e+22], N[(N[Power[b, 2.0], $MachinePrecision] + N[(a * N[(a * N[(N[(t$95$0 + -1.0), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[angle, 3.9e-12], N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[(angle * N[(Pi / 180.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(2.0 / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if angle < -7e22

   1. Initial program 26.6%

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

   2. Taylor expanded in angle around 0 27.4%

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

   3. Applied egg-rr 27.3%

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

      [Start] 27.4	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 27.4	\[ \color{blue}{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      swap-sqr [=>] 27.4	\[ \color{blue}{\left(a \cdot a\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      sin-mult [=>] 27.4	\[ \left(a \cdot a\right) \cdot \color{blue}{\frac{\cos \left(\frac{angle}{180} \cdot \pi - \frac{angle}{180} \cdot \pi\right) - \cos \left(\frac{angle}{180} \cdot \pi + \frac{angle}{180} \cdot \pi\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r/ [=>] 27.4	\[ \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi - \frac{angle}{180} \cdot \pi\right) - \cos \left(\frac{angle}{180} \cdot \pi + \frac{angle}{180} \cdot \pi\right)\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]

   4. Simplified 27.3%

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

      [Start] 27.3	\[ \frac{\left(a \cdot a\right) \cdot \left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 27.3	\[ \frac{\color{blue}{\left(\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot \left(a \cdot a\right)}}{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-/l* [=>] 27.3	\[ \color{blue}{\frac{\cos \left(\left(angle \cdot \pi\right) \cdot 0\right) - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{2}{a \cdot a}}} + {\left(b \cdot 1\right)}^{2} \]
      mul0-rgt [=>] 27.3	\[ \frac{\cos \color{blue}{0} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{2}{a \cdot a}} + {\left(b \cdot 1\right)}^{2} \]

   5. Applied egg-rr 27.4%

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

      [Start] 27.3	\[ \frac{\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{2}{a \cdot a}} + {\left(b \cdot 1\right)}^{2} \]
      associate-/r/ [=>] 27.3	\[ \color{blue}{\frac{\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{2} \cdot \left(a \cdot a\right)} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r* [=>] 27.3	\[ \color{blue}{\left(\frac{\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{2} \cdot a\right) \cdot a} + {\left(b \cdot 1\right)}^{2} \]
      frac-2neg [=>] 27.3	\[ \left(\color{blue}{\frac{-\left(\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{-2}} \cdot a\right) \cdot a + {\left(b \cdot 1\right)}^{2} \]
      neg-sub0 [=>] 27.3	\[ \left(\frac{\color{blue}{0 - \left(\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}}{-2} \cdot a\right) \cdot a + {\left(b \cdot 1\right)}^{2} \]
      cos-0 [=>] 27.3	\[ \left(\frac{0 - \left(\color{blue}{1} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{-2} \cdot a\right) \cdot a + {\left(b \cdot 1\right)}^{2} \]
      associate--r- [=>] 27.3	\[ \left(\frac{\color{blue}{\left(0 - 1\right) + \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}}{-2} \cdot a\right) \cdot a + {\left(b \cdot 1\right)}^{2} \]
      metadata-eval [=>] 27.3	\[ \left(\frac{\color{blue}{-1} + \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{-2} \cdot a\right) \cdot a + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 27.4	\[ \left(\frac{-1 + \cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}}{-2} \cdot a\right) \cdot a + {\left(b \cdot 1\right)}^{2} \]
      metadata-eval [=>] 27.4	\[ \left(\frac{-1 + \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{\color{blue}{-2}} \cdot a\right) \cdot a + {\left(b \cdot 1\right)}^{2} \]

   if -7e22 < angle < 3.89999999999999994e-12

   1. Initial program 98.4%

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

   2. Taylor expanded in angle around 0 98.0%

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

   3. Taylor expanded in angle around 0 97.3%

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

   4. Simplified 97.4%

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

      [Start] 97.3	\[ {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      metadata-eval [<=] 97.3	\[ {\left(a \cdot \left(\color{blue}{\frac{1}{180}} \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-/r/ [<=] 97.2	\[ {\left(a \cdot \color{blue}{\frac{1}{\frac{180}{angle \cdot \pi}}}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-/l* [<=] 97.3	\[ {\left(a \cdot \color{blue}{\frac{1 \cdot \left(angle \cdot \pi\right)}{180}}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [<=] 97.3	\[ {\left(a \cdot \frac{\color{blue}{\left(angle \cdot \pi\right) \cdot 1}}{180}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      *-rgt-identity [=>] 97.3	\[ {\left(a \cdot \frac{\color{blue}{angle \cdot \pi}}{180}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r/ [<=] 97.4	\[ {\left(a \cdot \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   if 3.89999999999999994e-12 < angle

   1. Initial program 32.8%

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

   2. Simplified 32.8%

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

      [Start] 32.8	\[ {\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 32.8	\[ {\left(a \cdot \sin \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*r/ [<=] 32.8	\[ {\left(a \cdot \sin \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
      associate-*l/ [=>] 32.8	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(\frac{angle \cdot \pi}{180}\right)}\right)}^{2} \]
      associate-*r/ [<=] 32.8	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \color{blue}{\left(angle \cdot \frac{\pi}{180}\right)}\right)}^{2} \]

   3. Taylor expanded in angle around 0 32.4%

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

   4. Applied egg-rr 21.8%

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

      [Start] 32.4	\[ {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
      add-log-exp [=>] 21.8	\[ \color{blue}{\log \left(e^{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}}\right)} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 21.8	\[ \log \left(e^{{\color{blue}{\left(\sin \left(angle \cdot \frac{\pi}{180}\right) \cdot a\right)}}^{2}}\right) + {\left(b \cdot 1\right)}^{2} \]
      div-inv [=>] 21.8	\[ \log \left(e^{{\left(\sin \left(angle \cdot \color{blue}{\left(\pi \cdot \frac{1}{180}\right)}\right) \cdot a\right)}^{2}}\right) + {\left(b \cdot 1\right)}^{2} \]
      metadata-eval [=>] 21.8	\[ \log \left(e^{{\left(\sin \left(angle \cdot \left(\pi \cdot \color{blue}{0.005555555555555556}\right)\right) \cdot a\right)}^{2}}\right) + {\left(b \cdot 1\right)}^{2} \]

   5. Applied egg-rr 31.1%

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

      [Start] 21.8	\[ \log \left(e^{{\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right)}^{2}}\right) + {\left(b \cdot 1\right)}^{2} \]
      add-log-exp [<=] 32.4	\[ \color{blue}{{\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]
      unpow2 [=>] 32.4	\[ \color{blue}{\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right) \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right)} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 32.4	\[ \color{blue}{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot a\right) + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 32.4	\[ \left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right) \cdot \color{blue}{\left(a \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      swap-sqr [=>] 32.4	\[ \color{blue}{\left(a \cdot a\right) \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
      sin-mult [=>] 31.2	\[ \left(a \cdot a\right) \cdot \color{blue}{\frac{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) - angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) - \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) + angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]
      associate-*r/ [=>] 31.2	\[ \color{blue}{\frac{\left(a \cdot a\right) \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) - angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) - \cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right) + angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}{2}} + {\left(b \cdot 1\right)}^{2} \]

   6. Simplified 31.2%

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

      [Start] 31.1	\[ \frac{\left(a \cdot a\right) \cdot \left(\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)}{2} + {\left(b \cdot 1\right)}^{2} \]
      *-commutative [=>] 31.1	\[ \frac{\color{blue}{\left(\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right) \cdot \left(a \cdot a\right)}}{2} + {\left(b \cdot 1\right)}^{2} \]
      associate-/l* [=>] 31.1	\[ \color{blue}{\frac{\cos 0 - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{2}{a \cdot a}}} + {\left(b \cdot 1\right)}^{2} \]
      cos-0 [=>] 31.1	\[ \frac{\color{blue}{1} - \cos \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)}{\frac{2}{a \cdot a}} + {\left(b \cdot 1\right)}^{2} \]
      associate-*l* [=>] 31.2	\[ \frac{1 - \cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}}{\frac{2}{a \cdot a}} + {\left(b \cdot 1\right)}^{2} \]

3. Recombined 3 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq -7 \cdot 10^{+22}:\\ \;\;\;\;{b}^{2} + a \cdot \left(a \cdot \frac{\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) + -1}{-2}\right)\\ \mathbf{elif}\;angle \leq 3.9 \cdot 10^{-12}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{\frac{2}{a \cdot a}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	68.3%
Cost	26372

\[\begin{array}{l} \mathbf{if}\;angle \leq 3.9 \cdot 10^{-12}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \mathsf{expm1}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{1 - \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}{\frac{2}{a \cdot a}}\\ \end{array} \]

Alternative 2
Accuracy	68.9%
Cost	26240

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

Alternative 3
Accuracy	68.9%
Cost	26240

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

Alternative 4
Accuracy	68.3%
Cost	20425

\[\begin{array}{l} \mathbf{if}\;angle \leq -7 \cdot 10^{+22} \lor \neg \left(angle \leq 3.9 \cdot 10^{-12}\right):\\ \;\;\;\;{b}^{2} + a \cdot \left(a \cdot \frac{\cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) + -1}{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \end{array} \]

Alternative 5
Accuracy	68.3%
Cost	20424

\[\begin{array}{l} t_0 := \cos \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;angle \leq -7 \cdot 10^{+22}:\\ \;\;\;\;{b}^{2} + a \cdot \left(a \cdot \frac{t_0 + -1}{-2}\right)\\ \mathbf{elif}\;angle \leq 3.9 \cdot 10^{-12}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + \frac{a \cdot a}{\frac{2}{1 - t_0}}\\ \end{array} \]

Alternative 6
Accuracy	66.2%
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.0003:\\ \;\;\;\;{\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}\\ \mathbf{elif}\;angle \leq 2.1 \cdot 10^{+35}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]

Alternative 7
Accuracy	66.3%
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.062:\\ \;\;\;\;{\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}\\ \mathbf{elif}\;angle \leq 8.6 \cdot 10^{+34}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]

Alternative 8
Accuracy	66.3%
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.026:\\ \;\;\;\;{\left(b \cdot \cos \left(\frac{angle}{\frac{180}{\pi}}\right)\right)}^{2}\\ \mathbf{elif}\;angle \leq 1.25 \cdot 10^{+35}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;b \cdot b\\ \end{array} \]

Alternative 9
Accuracy	50.0%
Cost	192

\[b \cdot b \]

Reproduce

herbie shell --seed 2023136 
(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)))