ab-angle->ABCF B

Average Error: 31.2 → 22.1

Time: 56.1s

Precision: binary64

Cost: 52484

Math

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

↓

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_1 := \cos t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;{\left(\sqrt[3]{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(-2 \cdot t_1\right)\right) \cdot \sin t_0}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \end{array} \]

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 (* angle (* 0.005555555555555556 PI))) (t_1 (cos t_0)))
   (if (<= (/ angle 180.0) -100000.0)
     (pow (cbrt (* (* (pow (hypot a b) 2.0) (* -2.0 t_1)) (sin t_0))) 3.0)
     (if (<= (/ angle 180.0) 4e-82)
       (* (- a b) (* (* angle PI) (* -0.011111111111111112 (+ a b))))
       (*
        t_1
        (*
         2.0
         (*
          (sin (* 0.005555555555555556 (* angle PI)))
          (- (* b b) (* a 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 * (0.005555555555555556 * ((double) M_PI));
	double t_1 = cos(t_0);
	double tmp;
	if ((angle / 180.0) <= -100000.0) {
		tmp = pow(cbrt(((pow(hypot(a, b), 2.0) * (-2.0 * t_1)) * sin(t_0))), 3.0);
	} else if ((angle / 180.0) <= 4e-82) {
		tmp = (a - b) * ((angle * ((double) M_PI)) * (-0.011111111111111112 * (a + b)));
	} else {
		tmp = t_1 * (2.0 * (sin((0.005555555555555556 * (angle * ((double) M_PI)))) * ((b * b) - (a * a))));
	}
	return tmp;
}

Java

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

↓

public static double code(double a, double b, double angle) {
	double t_0 = angle * (0.005555555555555556 * Math.PI);
	double t_1 = Math.cos(t_0);
	double tmp;
	if ((angle / 180.0) <= -100000.0) {
		tmp = Math.pow(Math.cbrt(((Math.pow(Math.hypot(a, b), 2.0) * (-2.0 * t_1)) * Math.sin(t_0))), 3.0);
	} else if ((angle / 180.0) <= 4e-82) {
		tmp = (a - b) * ((angle * Math.PI) * (-0.011111111111111112 * (a + b)));
	} else {
		tmp = t_1 * (2.0 * (Math.sin((0.005555555555555556 * (angle * Math.PI))) * ((b * b) - (a * 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(0.005555555555555556 * pi))
	t_1 = cos(t_0)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -100000.0)
		tmp = cbrt(Float64(Float64((hypot(a, b) ^ 2.0) * Float64(-2.0 * t_1)) * sin(t_0))) ^ 3.0;
	elseif (Float64(angle / 180.0) <= 4e-82)
		tmp = Float64(Float64(a - b) * Float64(Float64(angle * pi) * Float64(-0.011111111111111112 * Float64(a + b))));
	else
		tmp = Float64(t_1 * Float64(2.0 * Float64(sin(Float64(0.005555555555555556 * Float64(angle * pi))) * Float64(Float64(b * b) - Float64(a * 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[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -100000.0], N[Power[N[Power[N[(N[(N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision] * N[(-2.0 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 4e-82], N[(N[(a - b), $MachinePrecision] * N[(N[(angle * Pi), $MachinePrecision] * N[(-0.011111111111111112 * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(2.0 * N[(N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -1e5

   1. Initial program 51.1

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

   2. Simplified 51.1

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

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

   if -1e5 < (/.f64 angle 180) < 4e-82

   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))))): 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))))): 9 points increase in error, 7 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 19.1

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

   4. Simplified 19.1

      \[\leadsto \color{blue}{\pi \cdot \left(\left(a \cdot a - b \cdot b\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]
      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)))): 31 points increase in error, 30 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, 24 points decrease in error

   5. Taylor expanded in a around 0 19.1

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

   6. Simplified 0.9

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]
      Proof
      (*.f64 (+.f64 b a) (*.f64 (*.f64 (PI.f64) (-.f64 a b)) (*.f64 angle -1/90))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 a b)) (*.f64 (*.f64 (PI.f64) (-.f64 a b)) (*.f64 angle -1/90))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 a b) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 a b) (PI.f64))) (*.f64 angle -1/90))): 0 points increase in error, 0 points decrease in error
      (*.f64 (+.f64 a b) (*.f64 (*.f64 (-.f64 a b) (PI.f64)) (Rewrite<= *-commutative_binary64 (*.f64 -1/90 angle)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (+.f64 a b) (*.f64 (-.f64 a b) (PI.f64))) (*.f64 -1/90 angle))): 69 points increase in error, 30 points decrease in error
      (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (+.f64 a b) (-.f64 a b)) (PI.f64))) (*.f64 -1/90 angle)): 32 points increase in error, 21 points decrease in error
      (*.f64 (*.f64 (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 a a) (*.f64 b b))) (PI.f64)) (*.f64 -1/90 angle)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 a 2)) (*.f64 b b)) (PI.f64)) (*.f64 -1/90 angle)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (-.f64 (pow.f64 a 2) (Rewrite<= unpow2_binary64 (pow.f64 b 2))) (PI.f64)) (*.f64 -1/90 angle)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1/90 angle) (*.f64 (-.f64 (pow.f64 a 2) (pow.f64 b 2)) (PI.f64)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 -1/90 angle) (Rewrite=> *-commutative_binary64 (*.f64 (PI.f64) (-.f64 (pow.f64 a 2) (pow.f64 b 2))))): 0 points increase in error, 0 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))))): 1 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))))): 2 points increase in error, 2 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)))): 15 points increase in error, 9 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)))))): 13 points increase in error, 15 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. Taylor expanded in angle around 0 19.1

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

   8. Simplified 0.9

      \[\leadsto \color{blue}{\left(a - b\right) \cdot \left(\left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right) \cdot \left(b + a\right)\right)} \]
      Proof
      (*.f64 (-.f64 a b) (*.f64 (*.f64 (PI.f64) (*.f64 angle -1/90)) (+.f64 b a))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 a b) (*.f64 (*.f64 (PI.f64) (Rewrite<= *-commutative_binary64 (*.f64 -1/90 angle))) (+.f64 b a))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 a b) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1/90 angle) (PI.f64))) (+.f64 b a))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 a b) (*.f64 (*.f64 (*.f64 -1/90 angle) (PI.f64)) (Rewrite<= +-commutative_binary64 (+.f64 a b)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (-.f64 a b) (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 -1/90 angle) (*.f64 (PI.f64) (+.f64 a b))))): 35 points increase in error, 27 points decrease in error
      (*.f64 (-.f64 a b) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (PI.f64) (+.f64 a b)) (*.f64 -1/90 angle)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (-.f64 a b) (*.f64 (PI.f64) (+.f64 a b))) (*.f64 -1/90 angle))): 70 points increase in error, 29 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1/90 angle) (*.f64 (-.f64 a b) (*.f64 (PI.f64) (+.f64 a b))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1/90 (*.f64 angle (*.f64 (-.f64 a b) (*.f64 (PI.f64) (+.f64 a b)))))): 33 points increase in error, 22 points decrease in error

   9. Taylor expanded in angle around 0 0.9

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

   10. Simplified 0.9

       \[\leadsto \left(a - b\right) \cdot \color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)} \]
       Proof
       (*.f64 (*.f64 angle (PI.f64)) (*.f64 -1/90 (+.f64 a b))): 0 points increase in error, 0 points decrease in error
       (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 angle (PI.f64)) -1/90) (+.f64 a b))): 46 points increase in error, 34 points decrease in error
       (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/90 (*.f64 angle (PI.f64)))) (+.f64 a b)): 0 points increase in error, 0 points decrease in error
       (Rewrite<= associate-*r*_binary64 (*.f64 -1/90 (*.f64 (*.f64 angle (PI.f64)) (+.f64 a b)))): 38 points increase in error, 37 points decrease in error
       (*.f64 -1/90 (Rewrite<= associate-*r*_binary64 (*.f64 angle (*.f64 (PI.f64) (+.f64 a b))))): 38 points increase in error, 36 points decrease in error

   if 4e-82 < (/.f64 angle 180)

   1. Initial program 40.3

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

   2. Taylor expanded in b around 0 40.6

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

   3. Simplified 40.6

      \[\leadsto \color{blue}{\left(2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
      Proof
      (*.f64 2 (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (-.f64 (*.f64 b b) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 a a)))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (-.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 a 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))): 3 points increase in error, 1 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (*.f64 2 (Rewrite=> sub-neg_binary64 (+.f64 (pow.f64 b 2) (neg.f64 (pow.f64 a 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (*.f64 2 (+.f64 (pow.f64 b 2) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (pow.f64 a 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (*.f64 2 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (pow.f64 a 2)) (pow.f64 b 2))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))) (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 2 (*.f64 -1 (pow.f64 a 2))) (*.f64 2 (pow.f64 b 2))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 (*.f64 2 (*.f64 -1 (pow.f64 a 2))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 (*.f64 2 (pow.f64 b 2)) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))): 1 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 2 -1) (pow.f64 a 2))) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 (*.f64 2 (pow.f64 b 2)) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (*.f64 (Rewrite=> metadata-eval -2) (pow.f64 a 2)) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (*.f64 (*.f64 2 (pow.f64 b 2)) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (*.f64 -2 (pow.f64 a 2)) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 (pow.f64 b 2) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))))): 1 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -2 (*.f64 (pow.f64 a 2) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))) (*.f64 2 (*.f64 (pow.f64 b 2) (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64))))))): 0 points increase in error, 1 points decrease in error

   4. Applied egg-rr 40.6

      \[\leadsto \left(2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \color{blue}{\left(0 + \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;{\left(\sqrt[3]{\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(-2 \cdot \cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right) \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}\right)}^{3}\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.1
Cost	39620

\[\begin{array}{l} t_0 := angle \cdot \left(0.005555555555555556 \cdot \pi\right)\\ t_1 := \cos t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;\left({\left(\mathsf{hypot}\left(a, b\right)\right)}^{2} \cdot \left(-2 \cdot t_1\right)\right) \cdot \sin t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	22.1
Cost	27336

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := \left(2 \cdot \left(\sin t_0 \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	22.1
Cost	27336

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ t_1 := 2 \cdot \left(\sin t_0 \cdot \left(b \cdot b - a \cdot a\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;t_1 \cdot \cos t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \end{array} \]

Alternative 4
Error	22.2
Cost	27336

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;\sin t_0 \cdot \left(\left(-2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos t_0\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\\ \end{array} \]

Alternative 5
Error	22.1
Cost	27336

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \cos t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;\sin t_0 \cdot \left(\left(-2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot t_1\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right) \cdot t_1\\ \end{array} \]

Alternative 6
Error	22.1
Cost	27336

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;\sin t_0 \cdot \left(\left(-2 \cdot \left(b \cdot b + a \cdot a\right)\right) \cdot \cos t_0\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 4 \cdot 10^{-82}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	22.6
Cost	27080

\[\begin{array}{l} t_0 := 0.005555555555555556 \cdot \left(angle \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-11}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \left(\left(a \cdot a\right) \cdot \left(-2 \cdot \sin t_0\right)\right)\\ \end{array} \]

Alternative 8
Error	22.7
Cost	27080

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot 0.005555555555555556\right)\\ t_1 := \cos t_0 \cdot \left(\sin t_0 \cdot \left(a \cdot \left(a \cdot -2\right)\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-11}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	22.5
Cost	20292

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;\sin \left(\frac{\pi}{\frac{180}{angle}}\right) \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-11}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(-2 \cdot \left(b \cdot b + a \cdot a\right)\right)\\ \end{array} \]

Alternative 10
Error	22.6
Cost	14216

\[\begin{array}{l} t_0 := \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-85}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	22.5
Cost	14216

\[\begin{array}{l} t_0 := \sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(-2 \cdot \left(b \cdot b + a \cdot a\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-11}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	23.1
Cost	13960

\[\begin{array}{l} t_0 := \sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(-2 \cdot \left(a \cdot a\right)\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -100000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-11}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	23.1
Cost	13704

\[\begin{array}{l} t_0 := a \cdot \left(-2 \cdot \left(a \cdot \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)\right)\\ \mathbf{if}\;angle \leq -185:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 1.65 \cdot 10^{-9}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	32.0
Cost	7568

\[\begin{array}{l} t_0 := \left(a - b\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq -4.492413293943486 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.887606783883774 \cdot 10^{-221}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 16335981000327385000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(angle \cdot \left(b \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 15
Error	32.0
Cost	7568

\[\begin{array}{l} t_0 := \left(a - b\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\\ t_1 := \left(a - b\right) \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot b\right)\right)\right)\\ \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.492413293943486 \cdot 10^{-266}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.887606783883774 \cdot 10^{-221}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 16335981000327385000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	32.0
Cost	7568

\[\begin{array}{l} t_0 := \left(a - b\right) \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot b\right)\right)\right)\\ \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -4.492413293943486 \cdot 10^{-266}:\\ \;\;\;\;\left(a - b\right) \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 4.887606783883774 \cdot 10^{-221}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 16335981000327385000:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot a\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	32.0
Cost	7568

\[\begin{array}{l} t_0 := \left(a + b\right) \cdot \left(angle \cdot \left(a \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\right)\\ t_1 := \left(a - b\right) \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(angle \cdot b\right)\right)\right)\\ \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.506862744319868 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.887606783883774 \cdot 10^{-221}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;b \leq 16335981000327385000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	23.9
Cost	7432

\[\begin{array}{l} \mathbf{if}\;angle \leq -3.332994799182996 \cdot 10^{+45}:\\ \;\;\;\;\pi \cdot \left(angle \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;angle \leq 20.5:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\left(1 + \pi \cdot b\right) + -1\right)\right)\right)\\ \end{array} \]

Alternative 19
Error	25.0
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq -3.332994799182996 \cdot 10^{+45}:\\ \;\;\;\;\pi \cdot \left(angle \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(angle \cdot \pi\right) \cdot \left(-0.011111111111111112 \cdot \left(a + b\right)\right)\right)\\ \end{array} \]

Alternative 20
Error	38.5
Cost	7176

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 83794.32472161677:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 21
Error	33.5
Cost	7176

\[\begin{array}{l} t_0 := b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 83794.32472161677:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 22
Error	33.5
Cost	7176

\[\begin{array}{l} t_0 := b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 83794.32472161677:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 23
Error	33.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq 83794.32472161677:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 24
Error	33.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq 83794.32472161677:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(angle \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(angle \cdot b\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 25
Error	33.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.0247567661015632 \cdot 10^{+52}:\\ \;\;\;\;b \cdot \left(b \cdot \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\right)\\ \mathbf{elif}\;b \leq 83794.32472161677:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(angle \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(angle \cdot \left(b \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 26
Error	43.4
Cost	6912

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

Reproduce

herbie shell --seed 2022312 
(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)))))