ab-angle->ABCF B

Average Error: 31.6 → 22.1

Time: 59.4s

Precision: binary64

Cost: 78660

Math

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

↓

\[\begin{array}{l} t_0 := {\left(\sqrt[3]{b}\right)}^{2}\\ t_1 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\sin t_1 \cdot \left(\left(-2 \cdot \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} + \mathsf{fma}\left(-{t_0}^{2}, t_0, b \cdot b\right)\right)\right) \cdot \cos t_1\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]

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 (pow (cbrt b) 2.0)) (t_1 (* (/ angle 180.0) PI)))
   (if (<= (/ angle 180.0) -1e+25)
     (*
      (sin t_1)
      (*
       (* -2.0 (+ (pow (hypot a b) 2.0) (fma (- (pow t_0 2.0)) t_0 (* b b))))
       (cos t_1)))
     (if (<= (/ angle 180.0) 1e-48)
       (*
        (*
         angle
         (* PI (fma -0.011111111111111112 a (* b 0.011111111111111112))))
        (+ a b))
       (*
        (sin (* 0.005555555555555556 (* angle PI)))
        (*
         (* -2.0 (fma a a (* b (- b))))
         (cos (* PI (* angle 0.005555555555555556)))))))))

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 = pow(cbrt(b), 2.0);
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double tmp;
	if ((angle / 180.0) <= -1e+25) {
		tmp = sin(t_1) * ((-2.0 * (pow(hypot(a, b), 2.0) + fma(-pow(t_0, 2.0), t_0, (b * b)))) * cos(t_1));
	} else if ((angle / 180.0) <= 1e-48) {
		tmp = (angle * (((double) M_PI) * fma(-0.011111111111111112, a, (b * 0.011111111111111112)))) * (a + b);
	} else {
		tmp = sin((0.005555555555555556 * (angle * ((double) M_PI)))) * ((-2.0 * fma(a, a, (b * -b))) * cos((((double) M_PI) * (angle * 0.005555555555555556))));
	}
	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 = cbrt(b) ^ 2.0
	t_1 = Float64(Float64(angle / 180.0) * pi)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -1e+25)
		tmp = Float64(sin(t_1) * Float64(Float64(-2.0 * Float64((hypot(a, b) ^ 2.0) + fma(Float64(-(t_0 ^ 2.0)), t_0, Float64(b * b)))) * cos(t_1)));
	elseif (Float64(angle / 180.0) <= 1e-48)
		tmp = Float64(Float64(angle * Float64(pi * fma(-0.011111111111111112, a, Float64(b * 0.011111111111111112)))) * Float64(a + b));
	else
		tmp = Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(Float64(-2.0 * fma(a, a, Float64(b * Float64(-b)))) * cos(Float64(pi * Float64(angle * 0.005555555555555556)))));
	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[Power[N[Power[b, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -1e+25], N[(N[Sin[t$95$1], $MachinePrecision] * N[(N[(-2.0 * N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision] + N[((-N[Power[t$95$0, 2.0], $MachinePrecision]) * t$95$0 + N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-48], N[(N[(angle * N[(Pi * N[(-0.011111111111111112 * a + N[(b * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(-2.0 * N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -1.00000000000000009e25

   1. Initial program 51.8

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

   2. Simplified 51.8

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 1 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))))): 9 points increase in error, 18 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 51.5

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

   if -1.00000000000000009e25 < (/.f64 angle 180) < 9.9999999999999997e-49

   1. Initial program 19.1

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

   2. Simplified 19.1

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 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))))): 1 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))))): 9 points increase in error, 18 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 19.9

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

   4. Simplified 19.9

      \[\leadsto \color{blue}{\pi \cdot \left(\left(a \cdot a - b \cdot b\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]
      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))): 1 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)))): 24 points increase in error, 21 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))))): 26 points increase in error, 26 points decrease in error

   5. Taylor expanded in a around 0 19.9

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

   6. Simplified 2.1

      \[\leadsto \color{blue}{\left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right) \cdot \left(b + a\right)} \]
      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)): 24 points increase in error, 19 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, 37 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)))): 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))))): 26 points increase in error, 30 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, 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)))): 13 points increase in error, 20 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)))))): 12 points increase in error, 7 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

   7. Applied egg-rr 2.1

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

   8. Taylor expanded in a around 0 2.1

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

   9. Simplified 2.1

      \[\leadsto \left(0 + \color{blue}{angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)}\right) \cdot \left(b + a\right) \]
      Proof
      (*.f64 angle (*.f64 (PI.f64) (fma.f64 -1/90 a (*.f64 b 1/90)))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (*.f64 (PI.f64) (fma.f64 -1/90 a (Rewrite<= *-commutative_binary64 (*.f64 1/90 b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (*.f64 (PI.f64) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 -1/90 a) (*.f64 1/90 b))))): 2 points increase in error, 0 points decrease in error
      (*.f64 angle (*.f64 (PI.f64) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/90 b) (*.f64 -1/90 a))))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 1/90 b) (PI.f64)) (*.f64 (*.f64 -1/90 a) (PI.f64))))): 0 points increase in error, 3 points decrease in error
      (*.f64 angle (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/90 (*.f64 b (PI.f64)))) (*.f64 (*.f64 -1/90 a) (PI.f64)))): 19 points increase in error, 17 points decrease in error
      (*.f64 angle (+.f64 (*.f64 1/90 (*.f64 b (PI.f64))) (Rewrite<= associate-*r*_binary64 (*.f64 -1/90 (*.f64 a (PI.f64)))))): 25 points increase in error, 25 points decrease in error
      (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 1/90 (*.f64 b (PI.f64))) angle) (*.f64 (*.f64 -1/90 (*.f64 a (PI.f64))) angle))): 1 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/90 (*.f64 (*.f64 b (PI.f64)) angle))) (*.f64 (*.f64 -1/90 (*.f64 a (PI.f64))) angle)): 16 points increase in error, 18 points decrease in error
      (+.f64 (*.f64 1/90 (Rewrite<= *-commutative_binary64 (*.f64 angle (*.f64 b (PI.f64))))) (*.f64 (*.f64 -1/90 (*.f64 a (PI.f64))) angle)): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 1/90 (*.f64 angle (*.f64 b (PI.f64)))) (Rewrite<= associate-*r*_binary64 (*.f64 -1/90 (*.f64 (*.f64 a (PI.f64)) angle)))): 30 points increase in error, 30 points decrease in error
      (+.f64 (*.f64 1/90 (*.f64 angle (*.f64 b (PI.f64)))) (*.f64 -1/90 (Rewrite<= *-commutative_binary64 (*.f64 angle (*.f64 a (PI.f64)))))): 0 points increase in error, 0 points decrease in error

   if 9.9999999999999997e-49 < (/.f64 angle 180)

   1. Initial program 43.8

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

   2. Simplified 43.8

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 1 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))))): 9 points increase in error, 18 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 44.0

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

   4. Simplified 43.8

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \color{blue}{\cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)}\right) \]
      Proof
      (cos.f64 (*.f64 (PI.f64) (*.f64 1/180 angle))): 0 points increase in error, 0 points decrease in error
      (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/180 angle) (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (cos.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/180 (*.f64 angle (PI.f64))))): 23 points increase in error, 18 points decrease in error

   5. Taylor expanded in angle around inf 44.0

      \[\leadsto \color{blue}{\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)} \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(-2 \cdot \left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} + \mathsf{fma}\left(-{\left({\left(\sqrt[3]{b}\right)}^{2}\right)}^{2}, {\left(\sqrt[3]{b}\right)}^{2}, b \cdot b\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.1
Cost	33672

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\sin t_0 \cdot \left(\cos t_0 \cdot \left(-2 \cdot \left(b \cdot b + a \cdot a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	22.0
Cost	27336

\[\begin{array}{l} t_0 := \cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	22.0
Cost	27336

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-82}:\\ \;\;\;\;\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \left(\sin t_0 \cdot \left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	22.0
Cost	27336

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\sin t_1 \cdot \left(\cos t_1 \cdot \left(-2 \cdot \left(b \cdot b + a \cdot a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \left(\sin t_0 \cdot \left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	22.9
Cost	26820

\[\begin{array}{l} t_0 := \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+25}:\\ \;\;\;\;t_0 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(-2 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	22.9
Cost	26820

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\cos t_0 \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(a \cdot \left(-2 \cdot a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	22.9
Cost	26820

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{+25}:\\ \;\;\;\;t_1 \cdot \left(\cos t_0 \cdot \left(-2 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	22.6
Cost	20552

\[\begin{array}{l} t_0 := \sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -10:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-48}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	23.7
Cost	13832

\[\begin{array}{l} t_0 := 2 \cdot \left(\left(b \cdot b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{if}\;angle \leq -6.5 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \mathsf{fma}\left(-0.011111111111111112, a, b \cdot 0.011111111111111112\right)\right)\right) \cdot \left(a + b\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	23.7
Cost	13704

\[\begin{array}{l} t_0 := 2 \cdot \left(\left(b \cdot b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{if}\;angle \leq -6.5 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	32.3
Cost	7568

\[\begin{array}{l} t_0 := \left(a + b\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{if}\;b \leq -5.69024834492528 \cdot 10^{+38}:\\ \;\;\;\;b \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq -4.2397296816166974 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.8026229701629253 \cdot 10^{-95}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.716325521396268 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	32.3
Cost	7568

\[\begin{array}{l} t_0 := \left(a + b\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{if}\;b \leq -5.69024834492528 \cdot 10^{+38}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot b\right) \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq -4.2397296816166974 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.8026229701629253 \cdot 10^{-95}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.716325521396268 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	32.3
Cost	7568

\[\begin{array}{l} t_0 := \left(a + b\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{if}\;b \leq -5.69024834492528 \cdot 10^{+38}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq -4.2397296816166974 \cdot 10^{-109}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.8026229701629253 \cdot 10^{-95}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.716325521396268 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 14
Error	32.3
Cost	7568

\[\begin{array}{l} \mathbf{if}\;b \leq -5.69024834492528 \cdot 10^{+38}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq -4.2397296816166974 \cdot 10^{-109}:\\ \;\;\;\;\left(a + b\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.8026229701629253 \cdot 10^{-95}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.716325521396268 \cdot 10^{-6}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 15
Error	32.3
Cost	7568

\[\begin{array}{l} \mathbf{if}\;b \leq -5.69024834492528 \cdot 10^{+38}:\\ \;\;\;\;\left(a + b\right) \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq -4.2397296816166974 \cdot 10^{-109}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot -0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq 1.8026229701629253 \cdot 10^{-95}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.716325521396268 \cdot 10^{-6}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 16
Error	32.3
Cost	7568

\[\begin{array}{l} \mathbf{if}\;b \leq -5.69024834492528 \cdot 10^{+38}:\\ \;\;\;\;\left(a + b\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq -4.2397296816166974 \cdot 10^{-109}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(a \cdot -0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq 1.8026229701629253 \cdot 10^{-95}:\\ \;\;\;\;angle \cdot \left(a \cdot \left(a \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq 2.716325521396268 \cdot 10^{-6}:\\ \;\;\;\;\left(a + b\right) \cdot \left(a \cdot \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	25.2
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq 2 \cdot 10^{+80}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(b \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 18
Error	24.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq 1.7 \cdot 10^{+58}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(1 + -0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot a\right)\right)\right) + -1\right)\\ \end{array} \]

Alternative 19
Error	24.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq 1.7 \cdot 10^{+58}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(a - b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\left(1 + -0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot a\right)\right)\right) + -1\right)\\ \end{array} \]

Alternative 20
Error	31.6
Cost	7176

\[\begin{array}{l} t_0 := b \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)\\ \mathbf{if}\;b \leq -5.69024834492528 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.052729593743892 \cdot 10^{-8}:\\ \;\;\;\;a \cdot \left(a \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 21
Error	31.7
Cost	7176

\[\begin{array}{l} t_0 := b \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)\\ \mathbf{if}\;b \leq -5.69024834492528 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.052729593743892 \cdot 10^{-8}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 22
Error	31.7
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -5.69024834492528 \cdot 10^{+38}:\\ \;\;\;\;b \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.052729593743892 \cdot 10^{-8}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 23
Error	39.5
Cost	6912

\[b \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot b\right)\right)\right) \]

Reproduce

herbie shell --seed 2022318 
(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)))))