Bearing on a great circle

Average Accuracy: 79.1% → 99.7%

Time: 45.1s

Precision: binary64

Cost: 136192

Math

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

↓

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

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

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);
	return atan2((fma(sin(lambda1), cos(lambda2), fma(cos(lambda1), t_0, fma(t_0, cos(lambda1), (cos(lambda1) * sin(lambda2))))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
}

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

Derivation

1. Initial program 79.1%

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

2. Simplified 79.1%

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

   [Start] 79.1	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   associate-*l* [=>] 79.1	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

3. Applied egg-rr 79.3%

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

4. Simplified 79.3%

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

   [Start] 79.3	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \]
   distribute-lft-out [=>] 79.3	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]
   *-commutative [<=] 79.3	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)} \]
   *-commutative [=>] 79.3	\[ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)} \]

5. Applied egg-rr 99.7%

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

6. Simplified 99.7%

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

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

7. Final simplification 99.7%

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

Alternatives

Alternative 1
Accuracy	99.7%
Cost	116800

\[\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \mathsf{fma}\left(\cos \lambda_1, \sin \left(-\lambda_2\right), \lambda_2 \cdot 0\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \sin \phi_1\right)\right)} \]

Alternative 2
Accuracy	99.7%
Cost	91136

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

Alternative 3
Accuracy	88.9%
Cost	71817

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1.35 \cdot 10^{+25} \lor \neg \left(\lambda_2 \leq 6.5 \cdot 10^{-12}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	88.7%
Cost	71817

\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -14000000 \lor \neg \left(\lambda_1 \leq 6.2 \cdot 10^{-50}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - t_0\right)}{t_1 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_0\right)}{t_1 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]

Alternative 5
Accuracy	89.5%
Cost	71680

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

Alternative 6
Accuracy	83.1%
Cost	65417

\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -8.8 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 6 \cdot 10^{-62}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_0\right)}{t_1 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - t_0}{t_1 - \sin \phi_1 \cdot t_2}\\ \end{array} \]

Alternative 7
Accuracy	88.2%
Cost	65416

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := t_1 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_2\right)\\ t_4 := \cos \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_1 \leq -100:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_4\right)}{t_3}\\ \mathbf{elif}\;\phi_1 \leq 64:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_0 - t_4\right)}{t_1 - \sin \phi_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_0 - \sin \lambda_2\right)}{t_3}\\ \end{array} \]

Alternative 8
Accuracy	81.4%
Cost	65220

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.15 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sqrt[3]{{t_2}^{3}}}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_2 \leq 7 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \end{array} \]

Alternative 9
Accuracy	81.1%
Cost	65152

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

Alternative 10
Accuracy	82.6%
Cost	58889

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.8 \cdot 10^{-17} \lor \neg \left(\phi_2 \leq 2.3 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 11
Accuracy	82.6%
Cost	52489

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-14} \lor \neg \left(\phi_2 \leq 5.5 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - t_0}\\ \end{array} \]

Alternative 12
Accuracy	79.1%
Cost	52424

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 0.32:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]

Alternative 13
Accuracy	67.2%
Cost	52361

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -1.05 \cdot 10^{-73} \lor \neg \left(\lambda_2 \leq 4.9 \cdot 10^{-64}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \end{array} \]

Alternative 14
Accuracy	78.6%
Cost	52361

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.032 \lor \neg \left(\lambda_1 \leq 29000000000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \end{array} \]

Alternative 15
Accuracy	79.0%
Cost	52361

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -3.8 \cdot 10^{-5} \lor \neg \left(\lambda_2 \leq 7.8 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 16
Accuracy	65.9%
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 17
Accuracy	48.0%
Cost	39304

\[\begin{array}{l} t_0 := \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 3.6 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\frac{1}{\frac{1}{t_0}}}\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{+294}:\\ \;\;\;\;\tan^{-1}_* \frac{\left|t_1\right|}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2}\\ \end{array} \]

Alternative 18
Accuracy	48.0%
Cost	39304

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_2 - t_0\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 3.6 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\frac{1}{\frac{1}{t_1}}}\\ \mathbf{elif}\;\phi_2 \leq 6 \cdot 10^{+238}:\\ \;\;\;\;\tan^{-1}_* \frac{\left|t_2\right|}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\sin \phi_2 - \left|t_0\right|}\\ \end{array} \]

Alternative 19
Accuracy	48.1%
Cost	39304

\[\begin{array}{l} t_0 := \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 3.6 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\frac{1}{\frac{1}{t_0}}}\\ \mathbf{elif}\;\phi_2 \leq 6 \cdot 10^{+238}:\\ \;\;\;\;\tan^{-1}_* \frac{\left|t_1\right|}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 20
Accuracy	49.0%
Cost	39168

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

Alternative 21
Accuracy	48.1%
Cost	32896

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

Alternative 22
Accuracy	47.8%
Cost	32777

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+33} \lor \neg \left(\phi_1 \leq 6.2 \cdot 10^{-21}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 23
Accuracy	47.5%
Cost	32777

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 1.7 \cdot 10^{+56}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 24
Accuracy	48.1%
Cost	32640

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

Alternative 25
Accuracy	48.0%
Cost	26505

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 6.2 \cdot 10^{-21}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_1 \cdot \left(-t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - \phi_1 \cdot t_0}\\ \end{array} \]

Alternative 26
Accuracy	46.8%
Cost	26441

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{-16} \lor \neg \left(\phi_1 \leq 3.5 \cdot 10^{-42}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \end{array} \]

Alternative 27
Accuracy	31.4%
Cost	19456

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

Reproduce

herbie shell --seed 2023125 
(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))))))