ab-angle->ABCF B

Average Error: 31.5 → 22.2

Time: 55.0s

Precision: binary64

Cost: 52804

Math

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

↓

\[\begin{array}{l} t_0 := -2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\\ t_1 := \frac{angle}{180} \cdot \pi\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\sin \left(\frac{\sqrt{\pi}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle}}\right) \cdot \left(t_0 \cdot \cos t_1\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-69}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t_1 \cdot \left(t_0 \cdot \sqrt{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}}\right)\\ \end{array} \]

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 (* -2.0 (fma a a (* b (- b))))) (t_1 (* (/ angle 180.0) PI)))
   (if (<= (/ angle 180.0) -5e-8)
     (*
      (sin (* (/ (sqrt PI) 180.0) (/ (sqrt PI) (/ 1.0 angle))))
      (* t_0 (cos t_1)))
     (if (<= (/ angle 180.0) 1e-69)
       (* (+ a b) (* (* PI (- a b)) (* angle -0.011111111111111112)))
       (*
        (sin t_1)
        (*
         t_0
         (sqrt (pow (cos (* angle (* PI 0.005555555555555556))) 2.0))))))))

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 = -2.0 * fma(a, a, (b * -b));
	double t_1 = (angle / 180.0) * ((double) M_PI);
	double tmp;
	if ((angle / 180.0) <= -5e-8) {
		tmp = sin(((sqrt(((double) M_PI)) / 180.0) * (sqrt(((double) M_PI)) / (1.0 / angle)))) * (t_0 * cos(t_1));
	} else if ((angle / 180.0) <= 1e-69) {
		tmp = (a + b) * ((((double) M_PI) * (a - b)) * (angle * -0.011111111111111112));
	} else {
		tmp = sin(t_1) * (t_0 * sqrt(pow(cos((angle * (((double) M_PI) * 0.005555555555555556))), 2.0)));
	}
	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(-2.0 * fma(a, a, Float64(b * Float64(-b))))
	t_1 = Float64(Float64(angle / 180.0) * pi)
	tmp = 0.0
	if (Float64(angle / 180.0) <= -5e-8)
		tmp = Float64(sin(Float64(Float64(sqrt(pi) / 180.0) * Float64(sqrt(pi) / Float64(1.0 / angle)))) * Float64(t_0 * cos(t_1)));
	elseif (Float64(angle / 180.0) <= 1e-69)
		tmp = Float64(Float64(a + b) * Float64(Float64(pi * Float64(a - b)) * Float64(angle * -0.011111111111111112)));
	else
		tmp = Float64(sin(t_1) * Float64(t_0 * sqrt((cos(Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0))));
	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[(-2.0 * N[(a * a + N[(b * (-b)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(angle / 180.0), $MachinePrecision], -5e-8], N[(N[Sin[N[(N[(N[Sqrt[Pi], $MachinePrecision] / 180.0), $MachinePrecision] * N[(N[Sqrt[Pi], $MachinePrecision] / N[(1.0 / angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(angle / 180.0), $MachinePrecision], 1e-69], N[(N[(a + b), $MachinePrecision] * N[(N[(Pi * N[(a - b), $MachinePrecision]), $MachinePrecision] * N[(angle * -0.011111111111111112), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[t$95$1], $MachinePrecision] * N[(t$95$0 * N[Sqrt[N[Power[N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 angle 180) < -4.9999999999999998e-8

   1. Initial program 49.0

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

   2. Simplified 49.0

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 19 points increase in error, 13 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 49.1

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

   4. Applied egg-rr 49.3

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

   if -4.9999999999999998e-8 < (/.f64 angle 180) < 9.9999999999999996e-70

   1. Initial program 19.2

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

   2. Simplified 19.2

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 19 points increase in error, 13 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 19.2

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

   4. Simplified 19.2

      \[\leadsto \color{blue}{\pi \cdot \left(\left(a \cdot a - b \cdot b\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]
      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)))): 22 points increase in error, 27 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))))): 21 points increase in error, 28 points decrease in error

   5. Taylor expanded in a around 0 19.2

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

   6. Simplified 0.4

      \[\leadsto \color{blue}{\left(b + a\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \]
      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))): 76 points increase in error, 31 points decrease in error
      (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (+.f64 a b) (-.f64 a b)) (PI.f64))) (*.f64 -1/90 angle)): 20 points increase in error, 26 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, 0 points decrease in error
      (Rewrite=> distribute-lft-in_binary64 (+.f64 (*.f64 (*.f64 -1/90 angle) (*.f64 (pow.f64 a 2) (PI.f64))) (*.f64 (*.f64 -1/90 angle) (*.f64 (*.f64 -1 (pow.f64 b 2)) (PI.f64))))): 1 points increase in error, 1 points decrease in error
      (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64))))) (*.f64 (*.f64 -1/90 angle) (*.f64 (*.f64 -1 (pow.f64 b 2)) (PI.f64)))): 11 points increase in error, 15 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)))))): 10 points increase in error, 13 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 1/90 (*.f64 angle (*.f64 (pow.f64 b 2) (PI.f64)))) (*.f64 -1/90 (*.f64 angle (*.f64 (pow.f64 a 2) (PI.f64)))))): 0 points increase in error, 0 points decrease in error

   if 9.9999999999999996e-70 < (/.f64 angle 180)

   1. Initial program 40.9

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

   2. Simplified 40.9

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
      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))))): 19 points increase in error, 13 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 41.9

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

4. Final simplification 22.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\sin \left(\frac{\sqrt{\pi}}{180} \cdot \frac{\sqrt{\pi}}{\frac{1}{angle}}\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-69}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sqrt{{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	21.9
Cost	46536

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \cos t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t_1 \cdot \left(\left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right) \cdot \sin t_0\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-55}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sqrt{{t_1}^{2}}\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\\ \end{array} \]

Alternative 2
Error	22.1
Cost	46536

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := \cos t_0\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t_1 \cdot \left(\left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right) \cdot \sin t_0\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-69}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \left(\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sqrt{{t_1}^{2}}\right)\\ \end{array} \]

Alternative 3
Error	22.4
Cost	27076

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\cos t_0 \cdot \left(\left(-2 \cdot \left(a \cdot a - b \cdot b\right)\right) \cdot \sin t_0\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-69}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \end{array} \]

Alternative 4
Error	23.2
Cost	26820

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -2000000000000:\\ \;\;\;\;\sin t_0 \cdot \left(-2 \cdot \left(\cos t_0 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-69}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \end{array} \]

Alternative 5
Error	22.9
Cost	20552

\[\begin{array}{l} t_0 := \left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\\ \mathbf{if}\;\frac{angle}{180} \leq -5 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{-69}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	23.1
Cost	20296

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a \cdot \left(-2 \cdot a\right)\right)\\ \mathbf{elif}\;angle \leq 10^{-50}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-2 \cdot \mathsf{fma}\left(a, a, b \cdot \left(-b\right)\right)\right) \cdot \sin \left(\frac{\pi}{\frac{180}{angle}}\right)\\ \end{array} \]

Alternative 7
Error	23.6
Cost	13704

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \left(a \cdot \left(-2 \cdot a\right)\right)\\ \mathbf{if}\;angle \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 3.8 \cdot 10^{-19}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	32.8
Cost	7568

\[\begin{array}{l} t_0 := \left(a - b\right) \cdot \left(-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{if}\;b \leq -4.800269968970421 \cdot 10^{+77}:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(angle \cdot \left(b \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq -3.165098777430972 \cdot 10^{-178}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 2.1419099186543603 \cdot 10^{-184}:\\ \;\;\;\;\pi \cdot \left(\left(angle \cdot -0.011111111111111112\right) \cdot \left(a \cdot a\right)\right)\\ \mathbf{elif}\;b \leq 1.3105761158191551 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	24.2
Cost	7432

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;angle \leq 6.546295114286323 \cdot 10^{+50}:\\ \;\;\;\;\left(a - b\right) \cdot \left(\left(a + b\right) \cdot \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(1 + b \cdot \left(\pi \cdot b\right)\right) + -1\right)\right)\\ \end{array} \]

Alternative 10
Error	24.2
Cost	7432

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.45 \cdot 10^{-5}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{elif}\;angle \leq 6.546295114286323 \cdot 10^{+50}:\\ \;\;\;\;\left(a + b\right) \cdot \left(\left(\pi \cdot \left(a - b\right)\right) \cdot \left(angle \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\left(1 + b \cdot \left(\pi \cdot b\right)\right) + -1\right)\right)\\ \end{array} \]

Alternative 11
Error	34.3
Cost	7176

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;b \leq -4.800269968970421 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.3105761158191551 \cdot 10^{-90}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	34.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -4.800269968970421 \cdot 10^{+77}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.3105761158191551 \cdot 10^{-90}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	34.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -4.800269968970421 \cdot 10^{+77}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.3105761158191551 \cdot 10^{-90}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 14
Error	34.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -4.800269968970421 \cdot 10^{+77}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.3105761158191551 \cdot 10^{-90}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]

Alternative 15
Error	34.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -4.800269968970421 \cdot 10^{+77}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(b \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.3105761158191551 \cdot 10^{-90}:\\ \;\;\;\;\pi \cdot \left(angle \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]

Alternative 16
Error	34.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -4.800269968970421 \cdot 10^{+77}:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(angle \cdot \left(b \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.3105761158191551 \cdot 10^{-90}:\\ \;\;\;\;\pi \cdot \left(angle \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(\left(\pi \cdot 0.011111111111111112\right) \cdot \left(angle \cdot b\right)\right)\\ \end{array} \]

Alternative 17
Error	34.2
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -4.800269968970421 \cdot 10^{+77}:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(angle \cdot \left(b \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.3105761158191551 \cdot 10^{-90}:\\ \;\;\;\;\pi \cdot \left(angle \cdot \left(a \cdot \left(a \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 18
Error	32.6
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -4.800269968970421 \cdot 10^{+77}:\\ \;\;\;\;\pi \cdot \left(b \cdot \left(angle \cdot \left(b \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.3105761158191551 \cdot 10^{-90}:\\ \;\;\;\;\pi \cdot \left(-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(b \cdot \left(angle \cdot \left(\pi \cdot b\right)\right)\right)\\ \end{array} \]

Alternative 19
Error	39.9
Cost	6912

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

Reproduce

herbie shell --seed 2022297 
(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)))))