Bearing on a great circle

Average Error: 13.4 → 0.2

Time: 58.8s

Precision: binary64

Cost: 116864

Math

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

↓

\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \cos \lambda_1\\ \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{fma}\left(\cos \lambda_2, \sin \lambda_1, t_0\right) - 2 \cdot t_0\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)} \end{array} \]

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
 (let* ((t_0 (* (sin lambda2) (cos lambda1))))
   (atan2
    (* (cos phi2) (- (fma (cos lambda2) (sin lambda1) t_0) (* 2.0 t_0)))
    (-
     (* (cos phi1) (sin phi2))
     (*
      (sin phi1)
      (*
       (cos phi2)
       (fma (cos lambda1) (cos lambda2) (* (sin lambda2) (sin lambda1)))))))))

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) {
	double t_0 = sin(lambda2) * cos(lambda1);
	return atan2((cos(phi2) * (fma(cos(lambda2), sin(lambda1), t_0) - (2.0 * t_0))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * (cos(phi2) * fma(cos(lambda1), cos(lambda2), (sin(lambda2) * sin(lambda1)))))));
}

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)
	t_0 = Float64(sin(lambda2) * cos(lambda1))
	return atan(Float64(cos(phi2) * Float64(fma(cos(lambda2), sin(lambda1), t_0) - Float64(2.0 * t_0))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * Float64(cos(phi2) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda2) * sin(lambda1)))))))
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_] := Block[{t$95$0 = N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + t$95$0), $MachinePrecision] - N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 13.4

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

2. Simplified 13.4

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

   [Start] 13.4	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   *-commutative [=>] 13.4	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   associate-*l* [=>] 13.4	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

3. Applied egg-rr 6.6

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

4. Applied egg-rr 0.2

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

5. Simplified 0.2

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

   [Start] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]
   distribute-lft-out [=>] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]
   +-commutative [<=] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}} \]
   associate-*l* [=>] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)}} \]
   +-commutative [=>] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)} \]
   *-commutative [=>] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]
   fma-udef [<=] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)} \]

6. Applied egg-rr 0.2

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

7. Simplified 0.2

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \left(2 \cdot \left(\sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) + \mathsf{fma}\left(\cos \lambda_2, \sin \lambda_1, \sin \lambda_2 \cdot \cos \lambda_1\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]
   Proof

   [Start] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) + \cos \phi_2 \cdot \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]
   distribute-lft-in [<=] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \color{blue}{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \left(-\sin \lambda_2\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]
   *-commutative [=>] 0.2	\[ \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2\right)} + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \left(-\sin \lambda_2\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]
   distribute-lft-out [=>] 0.2	\[ \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \left(\cos \lambda_1 \cdot \left(-\sin \lambda_2\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]
   +-commutative [<=] 0.2	\[ \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\left(\cos \lambda_1 \cdot \left(-\sin \lambda_2\right) + \mathsf{fma}\left(-\sin \lambda_2, \cos \lambda_1, \cos \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \lambda_1 \cdot \cos \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]

8. Final simplification 0.2

   \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{fma}\left(\cos \lambda_2, \sin \lambda_1, \sin \lambda_2 \cdot \cos \lambda_1\right) - 2 \cdot \left(\sin \lambda_2 \cdot \cos \lambda_1\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)} \]

Alternatives

Alternative 1
Error	0.2
Cost	91136

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

Alternative 2
Error	0.2
Cost	91136

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

Alternative 3
Error	3.4
Cost	84872

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right)\right)}\\ \end{array} \]

Alternative 4
Error	7.0
Cost	71945

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -400000 \lor \neg \left(\lambda_2 \leq 9 \cdot 10^{-40}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 5
Error	6.7
Cost	71817

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -5.9 \cdot 10^{-5} \lor \neg \left(\lambda_1 \leq 1.85 \cdot 10^{-11}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\lambda_1 - \lambda_2\right)\right)\right)}{t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 6
Error	7.0
Cost	71817

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -0.00095 \lor \neg \left(\lambda_2 \leq 9 \cdot 10^{-40}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1\right)}\\ \end{array} \]

Alternative 7
Error	6.6
Cost	71680

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

Alternative 8
Error	7.8
Cost	65416

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -3800:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_3\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 620:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 9
Error	7.8
Cost	65288

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -3800:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_3\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 620:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 10
Error	7.8
Cost	65156

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -3800:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)}{t_0 - t_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 620:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0 - \cos \phi_2 \cdot t_3}\\ \end{array} \]

Alternative 11
Error	7.8
Cost	65156

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3800:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 620:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot t_1}\\ \end{array} \]

Alternative 12
Error	7.8
Cost	58888

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -3800:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - t_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 620:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot t_3}\\ \end{array} \]

Alternative 13
Error	13.7
Cost	52688

\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \cos \lambda_1\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \lambda_2 \cdot \sin \lambda_1\\ t_3 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_2 - t_0\right)}{\sin \phi_2}\\ t_4 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1.3 \cdot 10^{+21}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_2 \leq 3.4 \cdot 10^{-31}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_4 - \cos \lambda_1 \cdot t_1}\\ \mathbf{elif}\;\lambda_2 \leq 1.42 \cdot 10^{+211}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2 - \cos \phi_2 \cdot t_0}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 1.3 \cdot 10^{+245}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_4 - \cos \left(\lambda_1 - \lambda_2\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 14
Error	19.5
Cost	52560

\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \cos \lambda_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \lambda_2 \cdot \sin \lambda_1\\ t_3 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_2 - t_0\right)}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -8 \cdot 10^{-100}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_2 \leq 2.8 \cdot 10^{-32}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{t_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 5.2 \cdot 10^{+213}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2 - \cos \phi_2 \cdot t_0}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 3.4 \cdot 10^{+249}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_1 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 15
Error	13.8
Cost	52560

\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \cos \lambda_1\\ t_1 := \cos \lambda_2 \cdot \sin \lambda_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_1 - t_0\right)}{\sin \phi_2}\\ t_3 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1.3 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 3.4 \cdot 10^{-31}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_3 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 1.3 \cdot 10^{+215}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_1 - \cos \phi_2 \cdot t_0}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 2.35 \cdot 10^{+245}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_3 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	8.5
Cost	52489

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-31} \lor \neg \left(\phi_1 \leq 1.5 \cdot 10^{-13}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 17
Error	8.5
Cost	52488

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{-32}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - t_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 7.8 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \end{array} \]

Alternative 18
Error	13.9
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -460:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_1 \leq 1.85 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 19
Error	17.1
Cost	45961

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.9 \cdot 10^{-31} \lor \neg \left(\phi_1 \leq 9.6 \cdot 10^{-14}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 20
Error	17.9
Cost	45705

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -4 \cdot 10^{-31} \lor \neg \left(\phi_1 \leq 2.9 \cdot 10^{-15}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 21
Error	22.8
Cost	39304

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.001:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 22
Error	22.8
Cost	39168

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

Alternative 23
Error	22.8
Cost	39048

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t_0\\ \mathbf{if}\;\phi_2 \leq -0.00085:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 24
Error	22.8
Cost	38916

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.21:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 8.6 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \end{array} \]

Alternative 25
Error	22.8
Cost	33033

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.00065 \lor \neg \left(\phi_2 \leq 9 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 26
Error	23.8
Cost	32969

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0013 \lor \neg \left(\phi_2 \leq 3.5 \cdot 10^{-6}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]

Alternative 27
Error	36.8
Cost	26185

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -880000000000 \lor \neg \left(\lambda_1 \leq 2.2 \cdot 10^{-120}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 28
Error	38.6
Cost	26121

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -7 \cdot 10^{+24} \lor \neg \left(\phi_2 \leq 0.122\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 29
Error	32.8
Cost	25984

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

Alternative 30
Error	43.6
Cost	19456

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

Reproduce

herbie shell --seed 2023041 
(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))))))