Bearing on a great circle

Average Accuracy: 80.3% → 99.7%

Time: 52.2s

Precision: binary64

Cost: 116864

Math

\[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

↓

\[\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right) - \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

↓

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1))))
   (cos phi2))
  (-
   (fma
    (cos phi1)
    (sin phi2)
    (* (cos phi2) (* (- (sin phi1)) (* (cos lambda2) (cos lambda1)))))
   (* (sin lambda1) (* (sin lambda2) (* (cos phi2) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}

↓

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1))) * cos(phi2)), (fma(cos(phi1), sin(phi2), (cos(phi2) * (-sin(phi1) * (cos(lambda2) * cos(lambda1))))) - (sin(lambda1) * (sin(lambda2) * (cos(phi2) * sin(phi1))))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(sin(Float64(lambda1 - lambda2)) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))))
end

↓

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1)))) * cos(phi2)), Float64(fma(cos(phi1), sin(phi2), Float64(cos(phi2) * Float64(Float64(-sin(phi1)) * Float64(cos(lambda2) * cos(lambda1))))) - Float64(sin(lambda1) * Float64(sin(lambda2) * Float64(cos(phi2) * sin(phi1))))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

↓

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[((-N[Sin[phi1], $MachinePrecision]) * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 80.3%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

2. Applied egg-rr 89.9%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   Proof

   [Start] 80.3	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   sin-diff [=>] 89.9	\[ \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   fma-neg [=>] 89.9	\[ \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

3. Applied egg-rr 99.7%

   \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\cos \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right) + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}} \]
   Proof

   [Start] 89.9	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   cancel-sign-sub-inv [=>] 89.9	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]
   cos-diff [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]
   distribute-lft-in [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(-\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}} \]
   associate-+r+ [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\left(\cos \phi_1 \cdot \sin \phi_2 + \left(-\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(-\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)}} \]
   *-commutative [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(\cos \phi_1 \cdot \sin \phi_2 + \left(-\color{blue}{\cos \phi_2 \cdot \sin \phi_1}\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(-\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   distribute-rgt-neg-in [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right) + \left(-\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   *-commutative [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(\cos \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1\right)}\right) + \left(-\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   *-commutative [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(\cos \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right) + \left(-\color{blue}{\cos \phi_2 \cdot \sin \phi_1}\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   distribute-rgt-neg-in [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(\cos \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right) + \color{blue}{\left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)} \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]

4. Simplified 99.7%

   \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right) + \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)\right)}} \]
   Proof

   [Start] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(\cos \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right) + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   *-commutative [<=] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\left(\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)}\right) + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   fma-def [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)\right)} + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   *-commutative [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)}\right) + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   associate-*l* [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)}\right) + \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   *-commutative [<=] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right) + \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)}} \]
   associate-*l* [=>] 99.7	\[ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right) + \color{blue}{\sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)\right)}} \]

5. Final simplification 99.7%

   \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right) - \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	110528

\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\left(-\sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right) - \sin \lambda_1 \cdot \left(\sin \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)} \]

Alternative 2
Accuracy	99.7%
Cost	97472

\[\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)} \]

Alternative 3
Accuracy	99.7%
Cost	91136

\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)} \]

Alternative 4
Accuracy	89.9%
Cost	78016

\[\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

Alternative 5
Accuracy	89.8%
Cost	71817

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -1.3 \cdot 10^{-5} \lor \neg \left(\lambda_1 \leq 8 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{t_0 - t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 6
Accuracy	89.8%
Cost	71817

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -0.0027 \lor \neg \left(\lambda_1 \leq 0.00017\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_2 \cdot t_2}\\ \end{array} \]

Alternative 7
Accuracy	89.9%
Cost	71680

\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

Alternative 8
Accuracy	88.7%
Cost	65545

\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.45 \lor \neg \left(\phi_1 \leq 30000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_0\right)}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - t_0\right)}{t_1 - t_2 \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \end{array} \]

Alternative 9
Accuracy	88.8%
Cost	65417

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.45 \lor \neg \left(\phi_1 \leq 2.3 \cdot 10^{-12}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \cos \phi_2 \cdot \left(\phi_1 \cdot t_0\right)}\\ \end{array} \]

Alternative 10
Accuracy	88.4%
Cost	65352

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.45:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \cos \phi_2 \cdot \left(\phi_1 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\frac{\cos \phi_2}{\frac{1}{t_2}}}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \end{array} \]

Alternative 11
Accuracy	86.5%
Cost	52616

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.45:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-70}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\frac{\cos \phi_2}{\frac{1}{t_2}}}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \end{array} \]

Alternative 12
Accuracy	86.2%
Cost	52489

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.45 \lor \neg \left(\phi_1 \leq 1.95 \cdot 10^{-100}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]

Alternative 13
Accuracy	86.2%
Cost	52488

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.45:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.95 \cdot 10^{-100}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \end{array} \]

Alternative 14
Accuracy	78.9%
Cost	52361

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1500000000000 \lor \neg \left(\lambda_1 \leq 0.0042\right):\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 15
Accuracy	72.7%
Cost	52041

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.45 \lor \neg \left(\phi_1 \leq 1.95 \cdot 10^{-100}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]

Alternative 16
Accuracy	66.2%
Cost	45833

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.45 \lor \neg \left(\lambda_1 \leq 200000000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 17
Accuracy	66.7%
Cost	45696

\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

Alternative 18
Accuracy	61.3%
Cost	45568

\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1} \]

Alternative 19
Accuracy	58.6%
Cost	40009

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{-11} \lor \neg \left(\phi_1 \leq 1.5 \cdot 10^{+29}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \left(1 + -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right) - \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_1 \cdot \left(-t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot t_0}\\ \end{array} \]

Alternative 20
Accuracy	55.6%
Cost	39561

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -140000000 \lor \neg \left(\phi_1 \leq 1.9 \cdot 10^{+65}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot t_0}\\ \end{array} \]

Alternative 21
Accuracy	45.3%
Cost	32841

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.3 \cdot 10^{-35} \lor \neg \left(\phi_1 \leq 3.3 \cdot 10^{-20}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \left(1 + -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right) - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 22
Accuracy	39.0%
Cost	26888

\[\begin{array}{l} t_0 := 1 + -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq 4.2:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot t_0 - \lambda_2\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 9.5 \cdot 10^{+270}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \left(\lambda_2 \cdot 0.5\right) - \lambda_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot t_0 - \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 23
Accuracy	38.8%
Cost	26628

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.3 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \left(1 + -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right) - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 24
Accuracy	33.9%
Cost	26372

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.2 \cdot 10^{+82}:\\ \;\;\;\;\tan^{-1}_* \frac{-0.5 \cdot \left(\sin \lambda_1 \cdot \left(\cos \phi_2 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 1.26 \cdot 10^{-81}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\lambda_2 \cdot \left(-\cos \phi_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 25
Accuracy	33.9%
Cost	26120

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.55 \cdot 10^{+95}:\\ \;\;\;\;\tan^{-1}_* \frac{\lambda_2 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 1.24 \cdot 10^{-82}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\lambda_2 \cdot \left(-\cos \phi_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 26
Accuracy	20.3%
Cost	19721

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.6 \cdot 10^{-65} \lor \neg \left(\phi_2 \leq 3 \cdot 10^{+48}\right):\\ \;\;\;\;\tan^{-1}_* \frac{-\lambda_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\lambda_2 \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]

Alternative 27
Accuracy	22.9%
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4.2 \cdot 10^{+119}:\\ \;\;\;\;\tan^{-1}_* \frac{\lambda_2 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\lambda_2 \cdot \left(-\cos \phi_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 28
Accuracy	20.3%
Cost	12992

\[\tan^{-1}_* \frac{-\lambda_2}{\sin \phi_2} \]

Reproduce

herbie shell --seed 2023141 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))