ab-angle->ABCF B

Average Error: 31.2 → 22.1

Time: 47.3s

Precision: binary64

Cost: 65348

Math

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

↓

\[\begin{array}{l} t_0 := \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\\ t_1 := \cos \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{{t_0}^{2}} \cdot \left(\left(-2 \cdot {\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}}\right)}^{3}\right) \cdot t_1\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\right)\\ \end{array} \]

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 (sin (* angle (* 0.005555555555555556 PI))))
        (t_1 (cos (* (/ angle 180.0) PI))))
   (if (<= (/ angle 180.0) -2e-9)
     (*
      (sqrt (pow t_0 2.0))
      (* (* -2.0 (pow (cbrt (pow (hypot a b) 2.0)) 3.0)) t_1))
     (if (<= (/ angle 180.0) 2e-41)
       (*
        PI
        (*
         (* angle (+ (* b 0.011111111111111112) (* a -0.011111111111111112)))
         (+ a b)))
       (* t_0 (* t_1 (* -2.0 (fma a a (* b (- b))))))))))

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 = sin((angle * (0.005555555555555556 * ((double) M_PI))));
	double t_1 = cos(((angle / 180.0) * ((double) M_PI)));
	double tmp;
	if ((angle / 180.0) <= -2e-9) {
		tmp = sqrt(pow(t_0, 2.0)) * ((-2.0 * pow(cbrt(pow(hypot(a, b), 2.0)), 3.0)) * t_1);
	} else if ((angle / 180.0) <= 2e-41) {
		tmp = ((double) M_PI) * ((angle * ((b * 0.011111111111111112) + (a * -0.011111111111111112))) * (a + b));
	} else {
		tmp = t_0 * (t_1 * (-2.0 * fma(a, a, (b * -b))));
	}
	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 = sin(Float64(angle * Float64(0.005555555555555556 * pi)))
	t_1 = cos(Float64(Float64(angle / 180.0) * pi))
	tmp = 0.0
	if (Float64(angle / 180.0) <= -2e-9)
		tmp = Float64(sqrt((t_0 ^ 2.0)) * Float64(Float64(-2.0 * (cbrt((hypot(a, b) ^ 2.0)) ^ 3.0)) * t_1));
	elseif (Float64(angle / 180.0) <= 2e-41)
		tmp = Float64(pi * Float64(Float64(angle * Float64(Float64(b * 0.011111111111111112) + Float64(a * -0.011111111111111112))) * Float64(a + b)));
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(-2.0 * fma(a, a, Float64(b * Float64(-b))))));
	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[Sin[N[(angle * N[(0.005555555555555556 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -2e-9], N[(N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision] * N[(N[(-2.0 * N[Power[N[Power[N[Power[N[Sqrt[a ^ 2 + b ^ 2], $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 2e-41], N[(Pi * N[(N[(angle * N[(N[(b * 0.011111111111111112), $MachinePrecision] + N[(a * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(-2.0 * N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -2.00000000000000012e-9

   1. Initial program 48.2

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

   2. Simplified 48.2

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 14 points increase in error, 19 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 49.8

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

   4. Applied egg-rr 49.7

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

   if -2.00000000000000012e-9 < (/.f64 angle 180) < 2.00000000000000001e-41

   1. Initial program 18.2

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

   2. Simplified 18.2

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 14 points increase in error, 19 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 18.2

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

   4. Simplified 18.2

      \[\leadsto \color{blue}{\pi \cdot \left(\left(a \cdot a - b \cdot b\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]
      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)))): 24 points increase in error, 23 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))))): 27 points increase in error, 34 points decrease in error

   5. Applied egg-rr 0.4

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

   6. Taylor expanded in angle around 0 0.3

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

   7. Simplified 0.4

      \[\leadsto \pi \cdot \left(0 + \left(a + b\right) \cdot \color{blue}{\left(angle \cdot \left(0.011111111111111112 \cdot b + -0.011111111111111112 \cdot a\right)\right)}\right) \]
      Proof
      (*.f64 angle (+.f64 (*.f64 1/90 b) (*.f64 -1/90 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 b 1/90)) (*.f64 -1/90 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (+.f64 (*.f64 b (Rewrite<= metadata-eval (neg.f64 -1/90))) (*.f64 -1/90 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 b -1/90))) (*.f64 -1/90 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (+.f64 (Rewrite<= distribute-lft-neg-out_binary64 (*.f64 (neg.f64 b) -1/90)) (*.f64 -1/90 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (+.f64 (*.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 b)) -1/90) (*.f64 -1/90 a))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (+.f64 (*.f64 (*.f64 -1 b) -1/90) (Rewrite=> *-commutative_binary64 (*.f64 a -1/90)))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (Rewrite<= distribute-rgt-in_binary64 (*.f64 -1/90 (+.f64 (*.f64 -1 b) a)))): 2 points increase in error, 2 points decrease in error
      (*.f64 angle (*.f64 -1/90 (Rewrite<= +-commutative_binary64 (+.f64 a (*.f64 -1 b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (*.f64 -1/90 (+.f64 a (Rewrite=> mul-1-neg_binary64 (neg.f64 b))))): 0 points increase in error, 0 points decrease in error
      (*.f64 angle (*.f64 -1/90 (Rewrite<= sub-neg_binary64 (-.f64 a b)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 -1/90 (-.f64 a b)) angle)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 -1/90 (*.f64 (-.f64 a b) angle))): 42 points increase in error, 34 points decrease in error
      (*.f64 -1/90 (Rewrite<= *-commutative_binary64 (*.f64 angle (-.f64 a b)))): 0 points increase in error, 0 points decrease in error

   if 2.00000000000000001e-41 < (/.f64 angle 180)

   1. Initial program 44.4

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

   2. Simplified 44.4

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 14 points increase in error, 19 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 44.4

      \[\leadsto \color{blue}{\left(0 + \sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)\right)} \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{{\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right)}^{2}} \cdot \left(\left(-2 \cdot {\left(\sqrt[3]{{\left(\mathsf{hypot}\left(a, b\right)\right)}^{2}}\right)}^{3}\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(angle \cdot \left(0.005555555555555556 \cdot \pi\right)\right) \cdot \left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	21.8
Cost	59528

\[\begin{array}{l} t_0 := {b}^{2} - {a}^{2}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;\left(\cos \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\left(a + b\right) \cdot \left(\left(a - b\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	22.2
Cost	27336

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-11}:\\ \;\;\;\;t_1 \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-37}:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \left(t_1 \cdot \left(2 \cdot \left(\left(a + b\right) \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	22.7
Cost	27080

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\sin t_0 \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\cos t_0 \cdot \left(-2 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	22.7
Cost	27080

\[\begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\sin t_0 \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \left(\left(a \cdot a\right) \cdot \left(-2 \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	23.0
Cost	20296

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -1 \cdot 10^{-11}:\\ \;\;\;\;\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 5:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	23.0
Cost	20164

\[\begin{array}{l} t_0 := \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{if}\;angle \leq -1 \cdot 10^{-32}:\\ \;\;\;\;t_0 \cdot \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right)\\ \mathbf{elif}\;angle \leq 620:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left({a}^{2} \cdot t_0\right)\\ \end{array} \]

Alternative 7
Error	23.2
Cost	20040

\[\begin{array}{l} t_0 := -2 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;angle \leq -58000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 620:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	23.6
Cost	13828

\[\begin{array}{l} \mathbf{if}\;angle \leq -1:\\ \;\;\;\;\pi \cdot \left(\left(a \cdot a - b \cdot b\right) \cdot \log \left(angle \cdot -0.011111111111111112 + 1\right)\right)\\ \mathbf{elif}\;angle \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(-1 + \left(1 + a \cdot \left(\pi \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	29.2
Cost	7696

\[\begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(angle \cdot \left(\left(a - b\right) \cdot \left(\pi \cdot \left(a + b\right)\right)\right)\right)\\ \mathbf{if}\;b \leq -7.612355695145208 \cdot 10^{+150}:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(b \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq -2.8500299846995752 \cdot 10^{-101}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.8964782537162696 \cdot 10^{-257}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.261262069577623 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	25.2
Cost	7428

\[\begin{array}{l} \mathbf{if}\;angle \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot \left(b \cdot 0.011111111111111112 + a \cdot -0.011111111111111112\right)\right) \cdot \left(a + b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(-1 + \left(1 + a \cdot \left(\pi \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	25.2
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq 2.3 \cdot 10^{+18}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(\left(a + b\right) \cdot \left(angle \cdot \left(a - b\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left(-1 + \left(1 + a \cdot \left(\pi \cdot a\right)\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	33.0
Cost	7176

\[\begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;a \leq -4.2088501315267896 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7.310284536136748 \cdot 10^{-7}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	33.0
Cost	7176

\[\begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;a \leq -4.2088501315267896 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 7.310284536136748 \cdot 10^{-7}:\\ \;\;\;\;\pi \cdot \left(angle \cdot \left(0.011111111111111112 \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	31.9
Cost	7176

\[\begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;a \leq -7.352469059706928 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 0.0026326916707995773:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	31.9
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -7.352469059706928 \cdot 10^{+23}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(angle \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;a \leq 0.0026326916707995773:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 16
Error	31.9
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -7.352469059706928 \cdot 10^{+23}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(angle \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;a \leq 0.0026326916707995773:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(b \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	31.9
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -7.352469059706928 \cdot 10^{+23}:\\ \;\;\;\;-2 \cdot \left(a \cdot \left(\left(angle \cdot \pi\right) \cdot \left(0.005555555555555556 \cdot a\right)\right)\right)\\ \mathbf{elif}\;a \leq 0.0026326916707995773:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(b \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(a \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 18
Error	43.2
Cost	6912

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

Reproduce

herbie shell --seed 2022313 
(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)))))