Bearing on a great circle

Average Error: 12.8 → 0.2

Time: 41.5s

Precision: binary64

Cost: 155776

Math

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

↓

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

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))))
   (atan2
    (fma
     (* (cos lambda2) (sin lambda1))
     (cos phi2)
     (*
      (cos phi2)
      (fma
       (cos lambda1)
       t_0
       (fma t_0 (cos lambda1) (* (cos lambda1) (sin lambda2))))))
    (-
     (* (cos phi1) (sin phi2))
     (+
      (* (* (cos phi2) (* (cos lambda2) (cos lambda1))) (sin phi1))
      (* (sin phi1) (* (cos phi2) (* (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((cos(lambda2) * sin(lambda1)), cos(phi2), (cos(phi2) * fma(cos(lambda1), t_0, fma(t_0, cos(lambda1), (cos(lambda1) * sin(lambda2)))))), ((cos(phi1) * sin(phi2)) - (((cos(phi2) * (cos(lambda2) * cos(lambda1))) * sin(phi1)) + (sin(phi1) * (cos(phi2) * (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 = Float64(-sin(lambda2))
	return atan(fma(Float64(cos(lambda2) * sin(lambda1)), cos(phi2), Float64(cos(phi2) * fma(cos(lambda1), t_0, fma(t_0, cos(lambda1), Float64(cos(lambda1) * sin(lambda2)))))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(Float64(cos(phi2) * Float64(cos(lambda2) * cos(lambda1))) * sin(phi1)) + Float64(sin(phi1) * Float64(cos(phi2) * 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[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(N[Cos[phi2], $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]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 12.8

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

2. Applied egg-rr 6.4

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

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

5. Simplified 0.2

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

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	104192

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

Alternative 2
Error	0.2
Cost	104192

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

Alternative 3
Error	0.2
Cost	97408

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

Alternative 4
Error	0.2
Cost	91136

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

Alternative 5
Error	6.4
Cost	78208

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

Alternative 6
Error	6.7
Cost	71945

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_2 \leq -2650000 \lor \neg \left(\lambda_2 \leq 2300000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - t_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \end{array} \]

Alternative 7
Error	6.4
Cost	71817

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -0.062 \lor \neg \left(\lambda_1 \leq 8.5 \cdot 10^{-8}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \lambda_1 - \sin \lambda_2\right)}{t_0 - t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 8
Error	6.6
Cost	71817

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -650000 \lor \neg \left(\lambda_2 \leq 0.078\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]

Alternative 9
Error	6.4
Cost	71680

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

Alternative 10
Error	7.5
Cost	59016

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 2 \cdot 10^{-24}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\frac{\cos \phi_2}{\frac{1}{t_2}}}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \end{array} \]

Alternative 11
Error	7.5
Cost	58760

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6.6 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.5 \cdot 10^{-20}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\frac{\cos \phi_2}{\frac{1}{t_2}}}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \end{array} \]

Alternative 12
Error	18.4
Cost	52361

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -195000 \lor \neg \left(\lambda_1 \leq 750\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 13
Error	16.2
Cost	52361

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -46 \lor \neg \left(\phi_2 \leq 0.035\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1 + -0.5 \cdot \left(t_1 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 14
Error	12.9
Cost	52361

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -195000 \lor \neg \left(\lambda_1 \leq 750\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \end{array} \]

Alternative 15
Error	12.8
Cost	52224

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

Alternative 16
Error	24.7
Cost	45833

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -195000 \lor \neg \left(\lambda_1 \leq 7.5 \cdot 10^{+15}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1}\\ \end{array} \]

Alternative 17
Error	21.4
Cost	45833

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -195000 \lor \neg \left(\lambda_1 \leq 3.1 \cdot 10^{+16}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 18
Error	21.4
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 19
Error	24.7
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 20
Error	27.9
Cost	39040

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

Alternative 21
Error	28.4
Cost	32512

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

Reproduce

herbie shell --seed 2023056 
(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))))))