ab-angle->ABCF B

Average Error: 31.4 → 21.6

Time: 1.5min

Precision: binary64

Cost: 117640

Math

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

↓

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_1 := \sqrt[3]{e^{\cos t_0}}\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin t_0\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} \cdot \left(\log \left(t_1 \cdot t_1\right) + \log t_1\right)\\ \end{array} \]

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 (* 0.005555555555555556 PI)))
        (t_1 (cbrt (exp (cos t_0)))))
   (if (<= (/ angle 180.0) -2e-33)
     (*
      (log1p (expm1 (sin t_0)))
      (* (* -2.0 (fma a a (* b (- b)))) (cos (pow (cbrt t_0) 3.0))))
     (if (<= (/ angle 180.0) 1e-10)
       (* (* (* -0.011111111111111112 (* angle PI)) (- a b)) (+ a b))
       (*
        (pow
         (cbrt
          (*
           2.0
           (*
            (- (* b b) (* a a))
            (sin (* PI (* angle 0.005555555555555556))))))
         3.0)
        (+ (log (* t_1 t_1)) (log t_1)))))))

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 * (0.005555555555555556 * ((double) M_PI));
	double t_1 = cbrt(exp(cos(t_0)));
	double tmp;
	if ((angle / 180.0) <= -2e-33) {
		tmp = log1p(expm1(sin(t_0))) * ((-2.0 * fma(a, a, (b * -b))) * cos(pow(cbrt(t_0), 3.0)));
	} else if ((angle / 180.0) <= 1e-10) {
		tmp = ((-0.011111111111111112 * (angle * ((double) M_PI))) * (a - b)) * (a + b);
	} else {
		tmp = pow(cbrt((2.0 * (((b * b) - (a * a)) * sin((((double) M_PI) * (angle * 0.005555555555555556)))))), 3.0) * (log((t_1 * t_1)) + log(t_1));
	}
	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(0.005555555555555556 * pi))
	t_1 = cbrt(exp(cos(t_0)))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e-33)
		tmp = Float64(log1p(expm1(sin(t_0))) * Float64(Float64(-2.0 * fma(a, a, Float64(b * Float64(-b)))) * cos((cbrt(t_0) ^ 3.0))));
	elseif (Float64(angle / 180.0) <= 1e-10)
		tmp = Float64(Float64(Float64(-0.011111111111111112 * Float64(angle * pi)) * Float64(a - b)) * Float64(a + b));
	else
		tmp = Float64((cbrt(Float64(2.0 * Float64(Float64(Float64(b * b) - Float64(a * a)) * sin(Float64(pi * Float64(angle * 0.005555555555555556)))))) ^ 3.0) * Float64(log(Float64(t_1 * t_1)) + log(t_1)));
	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[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[N[Cos[t$95$0], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e-33], N[(N[Log[1 + N[(Exp[N[Sin[t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[(N[(-2.0 * N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-10], N[(N[(N[(-0.011111111111111112 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision] * N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Power[N[(2.0 * N[(N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Log[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision] + N[Log[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -2.0000000000000001e-33

   1. Initial program 45.8

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

   2. Simplified 45.8

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 1 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))))): 7 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 46.0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)} \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   4. Applied egg-rr 46.3

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \color{blue}{\left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{3}\right)}\right) \]

   if -2.0000000000000001e-33 < (/.f64 angle 180) < 1.00000000000000004e-10

   1. Initial program 18.6

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

   2. Simplified 18.6

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 1 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))))): 7 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 18.5

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

   4. Simplified 18.5

      \[\leadsto \color{blue}{\pi \cdot \left(\left(a \cdot a - b \cdot b\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]
      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)))): 21 points increase in error, 26 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, 32 points decrease in error

   5. Taylor expanded in a around 0 18.5

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

   6. Simplified 0.3

      \[\leadsto \color{blue}{\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)} \]
      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)): 37 points increase in error, 11 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)))): 64 points increase in error, 30 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)))): 1 points increase in error, 1 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)))): 1 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))))): 21 points increase in error, 24 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, 0 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))))): 1 points increase in error, 1 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)))): 7 points increase in error, 7 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)))))): 15 points increase in error, 26 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

   if 1.00000000000000004e-10 < (/.f64 angle 180)

   1. Initial program 48.9

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

   2. Applied egg-rr 49.0

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

   3. Applied egg-rr 49.0

      \[\leadsto {\left(\sqrt[3]{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{angle \cdot \left(0.005555555555555556 \cdot \pi\right)}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} \cdot \left(\log \left(\sqrt[3]{e^{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	21.6
Cost	59076

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin t_0\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} \cdot {\left(\sqrt[3]{\cos t_0}\right)}^{3}\\ \end{array} \]

Alternative 2
Error	21.5
Cost	53064

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} \cdot {\left(\sqrt[3]{\cos t_0}\right)}^{3}\\ \end{array} \]

Alternative 3
Error	21.5
Cost	46276

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} \cdot \cos t_0\\ \end{array} \]

Alternative 4
Error	21.7
Cost	46212

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2000000:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \log \left(e^{\sin t_0}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} \cdot \cos t_0\\ \end{array} \]

Alternative 5
Error	21.5
Cost	46212

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-33}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\sin t_0\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{3} \cdot \cos t_0\\ \end{array} \]

Alternative 6
Error	21.4
Cost	40200

\[\begin{array}{l} t_0 := 2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-23}:\\ \;\;\;\;t_0 \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{t_0}\right)}^{3} \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 7
Error	21.6
Cost	33672

\[\begin{array}{l} t_0 := \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot t_0\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\\ \end{array} \]

Alternative 8
Error	21.6
Cost	33672

\[\begin{array}{l} t_0 := \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot t_0\right) \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\\ \end{array} \]

Alternative 9
Error	22.3
Cost	27336

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \cos t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -2000000:\\ \;\;\;\;b \cdot \left(\left(b \cdot t_1\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right) \cdot \left(t_1 \cdot \sin t_0\right)\\ \end{array} \]

Alternative 10
Error	21.4
Cost	27336

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_1 := \left(\sin t_0 \cdot \cos t_0\right) \cdot \left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	21.4
Cost	27336

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;\cos t_1 \cdot \left(\sin t_1 \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin t_0 \cdot \cos t_0\right) \cdot \left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right)\\ \end{array} \]

Alternative 12
Error	21.4
Cost	27336

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\left(2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin t_0 \cdot \cos t_0\right) \cdot \left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right)\\ \end{array} \]

Alternative 13
Error	22.6
Cost	26820

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2000000:\\ \;\;\;\;b \cdot \left(\left(b \cdot \cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot \left(2 \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \end{array} \]

Alternative 14
Error	22.4
Cost	20552

\[\begin{array}{l} t_0 := \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-10}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	23.2
Cost	13704

\[\begin{array}{l} t_0 := b \cdot \left(b \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{if}\;angle \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 1.42 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	23.1
Cost	13704

\[\begin{array}{l} \mathbf{if}\;angle \leq -3.3 \cdot 10^{-7}:\\ \;\;\;\;b \cdot \left(b \cdot \left(2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{elif}\;angle \leq 1.7 \cdot 10^{-8}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(2 \cdot \sin \left(\frac{angle}{\frac{180}{\pi}}\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	28.9
Cost	7696

\[\begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{if}\;b \leq -1.710298681464313 \cdot 10^{+143}:\\ \;\;\;\;b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq -1.974355042029666 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 7.256580097829589 \cdot 10^{-106}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.4337886292758712 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 18
Error	28.8
Cost	7696

\[\begin{array}{l} \mathbf{if}\;b \leq -1.710298681464313 \cdot 10^{+143}:\\ \;\;\;\;b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq -1.0601197293999757 \cdot 10^{-109}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(\pi \cdot \left(a \cdot a - b \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 7.256580097829589 \cdot 10^{-106}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.4337886292758712 \cdot 10^{+92}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 19
Error	32.8
Cost	7304

\[\begin{array}{l} \mathbf{if}\;a \leq -2.262712903415839 \cdot 10^{+40}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.6480168738985596 \cdot 10^{+98}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(angle \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 20
Error	32.8
Cost	7304

\[\begin{array}{l} \mathbf{if}\;a \leq -2.262712903415839 \cdot 10^{+40}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.6480168738985596 \cdot 10^{+98}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + b\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 21
Error	24.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.12 \cdot 10^{+33}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(1 + a \cdot \left(\pi \cdot a\right)\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \left(a - b\right)\right)\right)\right)\\ \end{array} \]

Alternative 22
Error	24.8
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.12 \cdot 10^{+33}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(1 + a \cdot \left(\pi \cdot a\right)\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(a + b\right)\\ \end{array} \]

Alternative 23
Error	38.8
Cost	7176

\[\begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;a \leq -5.659415514631142 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.6593844643277683 \cdot 10^{-165}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 24
Error	38.8
Cost	7176

\[\begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;a \leq -2.5924584532107125 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.6593844643277683 \cdot 10^{-165}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 25
Error	38.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5977364288848698 \cdot 10^{+66}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.6593844643277683 \cdot 10^{-165}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 26
Error	38.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -1.5977364288848698 \cdot 10^{+66}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.6593844643277683 \cdot 10^{-165}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 27
Error	38.8
Cost	7176

\[\begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;a \leq -2.5924584532107125 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 2.9244935162153585 \cdot 10^{+105}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 28
Error	35.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -8.551032365404731 \cdot 10^{+55}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 2.1885793581987048 \cdot 10^{+94}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 29
Error	33.0
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -8.551032365404731 \cdot 10^{+55}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.6480168738985596 \cdot 10^{+98}:\\ \;\;\;\;b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 30
Error	32.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -2.262712903415839 \cdot 10^{+40}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.6480168738985596 \cdot 10^{+98}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 31
Error	32.8
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -2.262712903415839 \cdot 10^{+40}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;a \leq 1.6480168738985596 \cdot 10^{+98}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(angle \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 32
Error	40.1
Cost	6912

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

Reproduce

herbie shell --seed 2022306 
(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)))))