ab-angle->ABCF B

Average Error: 31.4 → 21.5

Time: 51.7s

Precision: binary64

Cost: 59528

Math

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

↓

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := {b}^{2} - {a}^{2}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+306}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(\cos t_0 \cdot \sin t_0\right)\right) \cdot -2\right)\\ \mathbf{elif}\;t_1 \leq 10^{+307}:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* PI 0.005555555555555556)))
        (t_1 (- (pow b 2.0) (pow a 2.0))))
   (if (<= t_1 -5e+306)
     (* a (* (* a (* (cos t_0) (sin t_0))) -2.0))
     (if (<= t_1 1e+307)
       (*
        (sin (* PI (* angle 0.005555555555555556)))
        (*
         (* -2.0 (fma a a (* b (- b))))
         (cos (* 0.005555555555555556 (* angle PI)))))
       (* (* (* (* angle PI) -0.011111111111111112) (- a b)) (+ b a))))))

C

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

↓

double code(double a, double b, double angle) {
	double t_0 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_1 = pow(b, 2.0) - pow(a, 2.0);
	double tmp;
	if (t_1 <= -5e+306) {
		tmp = a * ((a * (cos(t_0) * sin(t_0))) * -2.0);
	} else if (t_1 <= 1e+307) {
		tmp = sin((((double) M_PI) * (angle * 0.005555555555555556))) * ((-2.0 * fma(a, a, (b * -b))) * cos((0.005555555555555556 * (angle * ((double) M_PI)))));
	} else {
		tmp = (((angle * ((double) M_PI)) * -0.011111111111111112) * (a - b)) * (b + a);
	}
	return tmp;
}

Julia

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

↓

function code(a, b, angle)
	t_0 = Float64(angle * Float64(pi * 0.005555555555555556))
	t_1 = Float64((b ^ 2.0) - (a ^ 2.0))
	tmp = 0.0
	if (t_1 <= -5e+306)
		tmp = Float64(a * Float64(Float64(a * Float64(cos(t_0) * sin(t_0))) * -2.0));
	elseif (t_1 <= 1e+307)
		tmp = Float64(sin(Float64(pi * Float64(angle * 0.005555555555555556))) * Float64(Float64(-2.0 * fma(a, a, Float64(b * Float64(-b)))) * cos(Float64(0.005555555555555556 * Float64(angle * pi)))));
	else
		tmp = Float64(Float64(Float64(Float64(angle * pi) * -0.011111111111111112) * Float64(a - b)) * Float64(b + a));
	end
	return tmp
end

Wolfram

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

↓

code[a_, b_, angle_] := Block[{t$95$0 = N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+306], N[(a * N[(N[(a * N[(N[Cos[t$95$0], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+307], N[(N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-2.0 * N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(angle * Pi), $MachinePrecision] * -0.011111111111111112), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < -4.99999999999999993e306

   1. Initial program 63.6

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

   2. Simplified 63.6

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      Proof
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 -2 (fma.f64 a a (*.f64 b (neg.f64 b)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) (fma.f64 a a (*.f64 b (neg.f64 b)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (fma.f64 a a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 b b))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (fma.f64 a a (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 a a) (pow.f64 b 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (pow.f64 b 2))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (-.f64 (pow.f64 a 2) (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 2 (neg.f64 (-.f64 (pow.f64 a 2) (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (pow.f64 a 2) (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (pow.f64 a 2)) (pow.f64 b 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (pow.f64 a 2))) (pow.f64 b 2))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 b 2) (neg.f64 (pow.f64 a 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 15 points increase in error, 17 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 64.0

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \color{blue}{{\left({\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}\right)}^{3}\right)}^{0.3333333333333333}}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   4. Applied egg-rr 63.6

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{b}\right)}^{2}, \sqrt[3]{b} \cdot b, a \cdot a\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   5. Taylor expanded in b around 0 63.5

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

   6. Simplified 3.1

      \[\leadsto \color{blue}{a \cdot \left(\left(a \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot -2\right)} \]
      Proof
      (*.f64 a (*.f64 (*.f64 a (*.f64 (cos.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))) (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))))) -2)): 0 points increase in error, 0 points decrease in error
      (*.f64 a (*.f64 (*.f64 a (*.f64 (cos.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/180))) (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))))) -2)): 8 points increase in error, 10 points decrease in error
      (*.f64 a (*.f64 (*.f64 a (*.f64 (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/180))))) -2)): 0 points increase in error, 0 points decrease in error
      (*.f64 a (*.f64 (*.f64 a (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/180))))) -2)): 16 points increase in error, 14 points decrease in error
      (*.f64 a (*.f64 (*.f64 a (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/180 (*.f64 angle (PI.f64))))))) -2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a (*.f64 a (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))) -2)): 0 points increase in error, 1 points decrease in error
      (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 a a) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))) -2): 50 points increase in error, 25 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))) -2): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 -2 (*.f64 (pow.f64 a 2) (*.f64 (cos.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))): 0 points increase in error, 0 points decrease in error

   if -4.99999999999999993e306 < (-.f64 (pow.f64 b 2) (pow.f64 a 2)) < 9.99999999999999986e306

   1. Initial program 25.1

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

   2. Simplified 25.1

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      Proof
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 -2 (fma.f64 a a (*.f64 b (neg.f64 b)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) (fma.f64 a a (*.f64 b (neg.f64 b)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (fma.f64 a a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 b b))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (fma.f64 a a (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 a a) (pow.f64 b 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (pow.f64 b 2))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (-.f64 (pow.f64 a 2) (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 2 (neg.f64 (-.f64 (pow.f64 a 2) (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (pow.f64 a 2) (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (pow.f64 a 2)) (pow.f64 b 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (pow.f64 a 2))) (pow.f64 b 2))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 b 2) (neg.f64 (pow.f64 a 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 15 points increase in error, 17 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in angle around inf 25.1

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

   4. Taylor expanded in angle around inf 25.1

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

   5. Simplified 25.1

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \]
      Proof
      (sin.f64 (*.f64 (PI.f64) (*.f64 1/180 angle))): 0 points increase in error, 0 points decrease in error
      (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/180 angle) (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (sin.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/180 (*.f64 angle (PI.f64))))): 44 points increase in error, 44 points decrease in error

   if 9.99999999999999986e306 < (-.f64 (pow.f64 b 2) (pow.f64 a 2))

   1. Initial program 63.6

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

   2. Simplified 63.6

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      Proof
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 -2 (fma.f64 a a (*.f64 b (neg.f64 b)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) (fma.f64 a a (*.f64 b (neg.f64 b)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (fma.f64 a a (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 b b))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (fma.f64 a a (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 a a) (pow.f64 b 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 (neg.f64 2) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (pow.f64 b 2))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (-.f64 (pow.f64 a 2) (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (Rewrite<= distribute-rgt-neg-out_binary64 (*.f64 2 (neg.f64 (-.f64 (pow.f64 a 2) (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (pow.f64 a 2) (pow.f64 b 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (pow.f64 a 2)) (pow.f64 b 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (pow.f64 a 2))) (pow.f64 b 2))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 b 2) (neg.f64 (pow.f64 a 2))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 (*.f64 2 (Rewrite<= sub-neg_binary64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 15 points increase in error, 17 points decrease in error
      (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in angle around 0 63.6

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

   4. Simplified 63.5

      \[\leadsto \color{blue}{\pi \cdot \left(\left(a \cdot a - b \cdot b\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]
      Proof
      (*.f64 (PI.f64) (*.f64 (-.f64 (*.f64 a a) (*.f64 b b)) (*.f64 angle -1/90))): 0 points increase in error, 0 points decrease in error
      (*.f64 (PI.f64) (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 b b)) (*.f64 angle -1/90))): 0 points increase in error, 0 points decrease in error
      (*.f64 (PI.f64) (*.f64 (-.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2))) (*.f64 angle -1/90))): 0 points increase in error, 0 points decrease in error
      (*.f64 (PI.f64) (*.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (pow.f64 a 2) (neg.f64 (pow.f64 b 2)))) (*.f64 angle -1/90))): 0 points increase in error, 0 points decrease in error
      (*.f64 (PI.f64) (*.f64 (+.f64 (pow.f64 a 2) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 b 2)))) (*.f64 angle -1/90))): 0 points increase in error, 0 points decrease in error
      (*.f64 (PI.f64) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (pow.f64 b 2)) (pow.f64 a 2))) (*.f64 angle -1/90))): 0 points increase in error, 0 points decrease in error
      (*.f64 (PI.f64) (*.f64 (+.f64 (*.f64 -1 (pow.f64 b 2)) (pow.f64 a 2)) (Rewrite<= *-commutative_binary64 (*.f64 -1/90 angle)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (PI.f64) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1/90 angle) (+.f64 (*.f64 -1 (pow.f64 b 2)) (pow.f64 a 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 -1/90 angle) (+.f64 (*.f64 -1 (pow.f64 b 2)) (pow.f64 a 2))) (PI.f64))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 -1/90 angle) (*.f64 (+.f64 (*.f64 -1 (pow.f64 b 2)) (pow.f64 a 2)) (PI.f64)))): 22 points increase in error, 29 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1/90 (*.f64 angle (*.f64 (+.f64 (*.f64 -1 (pow.f64 b 2)) (pow.f64 a 2)) (PI.f64))))): 22 points increase in error, 21 points decrease in error

   5. Taylor expanded in a around 0 63.6

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

   6. Simplified 3.4

      \[\leadsto \color{blue}{\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)} \]
      Proof
      (*.f64 (*.f64 (*.f64 -1/90 (*.f64 angle (PI.f64))) (-.f64 a b)) (+.f64 b a)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 -1/90 angle) (PI.f64))) (-.f64 a b)) (+.f64 b a)): 20 points increase in error, 16 points decrease in error
      (*.f64 (*.f64 (*.f64 (*.f64 -1/90 angle) (PI.f64)) (-.f64 a b)) (Rewrite<= +-commutative_binary64 (+.f64 a b))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 (*.f64 -1/90 angle) (PI.f64)) (*.f64 (-.f64 a b) (+.f64 a b)))): 63 points increase in error, 46 points decrease in error
      (*.f64 (*.f64 (*.f64 -1/90 angle) (PI.f64)) (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 a b) (-.f64 a b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 -1/90 angle) (PI.f64)) (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 a a) (*.f64 b b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 -1/90 angle) (PI.f64)) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 b b))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 -1/90 angle) (PI.f64)) (-.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 -1/90 angle) (*.f64 (PI.f64) (-.f64 (pow.f64 a 2) (pow.f64 b 2))))): 30 points increase in error, 19 points decrease in error
      (*.f64 (*.f64 -1/90 angle) (*.f64 (PI.f64) (Rewrite=> sub-neg_binary64 (+.f64 (pow.f64 a 2) (neg.f64 (pow.f64 b 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/90 angle) (*.f64 (PI.f64) (+.f64 (pow.f64 a 2) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 b 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/90 angle) (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (pow.f64 a 2) (PI.f64)) (*.f64 (*.f64 -1 (pow.f64 b 2)) (PI.f64))))): 0 points increase in error, 1 points decrease in error
      (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (*.f64 -1/90 angle) (*.f64 (pow.f64 a 2) (PI.f64))) (*.f64 (*.f64 -1/90 angle) (*.f64 (*.f64 -1 (pow.f64 b 2)) (PI.f64))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64))))) (*.f64 (*.f64 -1/90 angle) (*.f64 (*.f64 -1 (pow.f64 b 2)) (PI.f64)))): 14 points increase in error, 8 points decrease in error
      (+.f64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))) (*.f64 (*.f64 -1/90 angle) (*.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (pow.f64 b 2))) (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))) (*.f64 (*.f64 -1/90 angle) (Rewrite=> distribute-lft-neg-out_binary64 (neg.f64 (*.f64 (pow.f64 b 2) (PI.f64)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))) (Rewrite=> distribute-rgt-neg-out_binary64 (neg.f64 (*.f64 (*.f64 -1/90 angle) (*.f64 (pow.f64 b 2) (PI.f64)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> unsub-neg_binary64 (-.f64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))) (*.f64 (*.f64 -1/90 angle) (*.f64 (pow.f64 b 2) (PI.f64))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))) (*.f64 (neg.f64 (*.f64 -1/90 angle)) (*.f64 (pow.f64 b 2) (PI.f64))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))) (*.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 -1/90) angle)) (*.f64 (pow.f64 b 2) (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))) (*.f64 (*.f64 (Rewrite=> metadata-eval 1/90) angle) (*.f64 (pow.f64 b 2) (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))) (Rewrite<= associate-*r*_binary64 (*.f64 1/90 (*.f64 angle (*.f64 (pow.f64 b 2) (PI.f64)))))): 8 points increase in error, 13 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/90 (*.f64 angle (*.f64 (pow.f64 b 2) (PI.f64)))) (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 21.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -5 \cdot 10^{+306}:\\ \;\;\;\;a \cdot \left(\left(a \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right) \cdot -2\right)\\ \mathbf{elif}\;{b}^{2} - {a}^{2} \leq 10^{+307}:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.5
Cost	27336

\[\begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2000:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(t_0 \cdot \left(2 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 30000000:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(t_0 \cdot \left(-2 \cdot \left(a \cdot a + b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	21.5
Cost	27336

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \left(a \cdot a - b \cdot b\right) \cdot \left(\left(\cos t_0 \cdot \sin t_0\right) \cdot -2\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	21.4
Cost	27336

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_2 := a \cdot a - b \cdot b\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-9}:\\ \;\;\;\;t_2 \cdot \left(\left(\cos t_0 \cdot \sin t_0\right) \cdot -2\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_1 \cdot \left(-2 \cdot \left(t_2 \cdot \cos t_1\right)\right)\\ \end{array} \]

Alternative 4
Error	22.9
Cost	27080

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	22.9
Cost	27080

\[\begin{array}{l} t_0 := 2 \cdot \left(b \cdot b\right)\\ t_1 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2000:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot t_0\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_1 \cdot \left(\cos t_1 \cdot t_0\right)\\ \end{array} \]

Alternative 6
Error	22.8
Cost	27080

\[\begin{array}{l} t_0 := \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2000:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(t_0 \cdot \left(2 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 30000000:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(t_0 \cdot \left(-2 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	22.9
Cost	27080

\[\begin{array}{l} t_0 := 2 \cdot \left(b \cdot b\right)\\ t_1 := \pi \cdot \frac{angle}{180}\\ \mathbf{if}\;\frac{angle}{180} \leq -2000:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot t_0\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 \cdot \sin t_1\right) \cdot \cos t_1\\ \end{array} \]

Alternative 8
Error	22.5
Cost	20552

\[\begin{array}{l} t_0 := \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 50000000:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	23.4
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -680000:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;angle \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot {\left({\left(a \cdot \left(a \cdot \pi\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\\ \end{array} \]

Alternative 10
Error	23.4
Cost	20104

\[\begin{array}{l} \mathbf{if}\;angle \leq -680000:\\ \;\;\;\;2 \cdot \left({b}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 2.8 \cdot 10^{+53}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot {\left({\left(a \cdot \left(a \cdot \pi\right)\right)}^{3}\right)}^{0.3333333333333333}\right)\\ \end{array} \]

Alternative 11
Error	23.3
Cost	13572

\[\begin{array}{l} \mathbf{if}\;angle \leq -680000:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(2 \cdot \left(b \cdot b\right)\right)\\ \mathbf{elif}\;angle \leq 1.16 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(\left(1 + a \cdot \left(a \cdot \pi\right)\right) + -1\right)\right)\\ \end{array} \]

Alternative 12
Error	23.8
Cost	7432

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.0336733639279729 \cdot 10^{+28}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.16 \cdot 10^{+19}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(\left(1 + a \cdot \left(a \cdot \pi\right)\right) + -1\right)\right)\\ \end{array} \]

Alternative 13
Error	23.8
Cost	7432

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.0336733639279729 \cdot 10^{+28}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.16 \cdot 10^{+19}:\\ \;\;\;\;\left(\left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(\left(1 + a \cdot \left(a \cdot \pi\right)\right) + -1\right)\right)\\ \end{array} \]

Alternative 14
Error	31.4
Cost	7304

\[\begin{array}{l} \mathbf{if}\;a \leq -2.4246603172779114 \cdot 10^{-63}:\\ \;\;\;\;a \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right)\\ \mathbf{elif}\;a \leq 3.710556084562489 \cdot 10^{-32}:\\ \;\;\;\;b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 15
Error	24.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.0336733639279729 \cdot 10^{+28}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(a - b\right)\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(b + a\right)\right)\right)\\ \end{array} \]

Alternative 16
Error	24.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.0336733639279729 \cdot 10^{+28}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	38.4
Cost	7176

\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right)\\ \mathbf{if}\;a \leq -8.094074239291841 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 129921.89477107806:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	32.5
Cost	7176

\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right)\\ \mathbf{if}\;a \leq -2.4246603172779114 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.5349671239479256 \cdot 10^{-79}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 19
Error	32.6
Cost	7176

\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right)\\ \mathbf{if}\;a \leq -2.4246603172779114 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.5349671239479256 \cdot 10^{-79}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 20
Error	32.5
Cost	7176

\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right)\\ \mathbf{if}\;a \leq -2.4246603172779114 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 1.5349671239479256 \cdot 10^{-79}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 21
Error	31.4
Cost	7176

\[\begin{array}{l} t_0 := a \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot -0.011111111111111112\right)\right)\\ \mathbf{if}\;a \leq -2.4246603172779114 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3.710556084562489 \cdot 10^{-32}:\\ \;\;\;\;b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 22
Error	43.6
Cost	6912

\[-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right) \]

Reproduce

herbie shell --seed 2022317 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))