Bearing on a great circle

Average Error: 12.9 → 0.2

Time: 57.5s

Precision: binary64

Cost: 103936

Math

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

↓

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

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
  (*
   (-
    (* (cos lambda2) (sin lambda1))
    (expm1 (log1p (* (cos lambda1) (sin lambda2)))))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (* (cos phi2) (sin phi1))
    (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda2) (cos 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) {
	return atan2((((cos(lambda2) * sin(lambda1)) - expm1(log1p((cos(lambda1) * sin(lambda2))))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))));
}

Java

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

↓

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((((Math.cos(lambda2) * Math.sin(lambda1)) - Math.expm1(Math.log1p((Math.cos(lambda1) * Math.sin(lambda2))))) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos(phi2) * Math.sin(phi1)) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1))))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))

↓

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((((math.cos(lambda2) * math.sin(lambda1)) - math.expm1(math.log1p((math.cos(lambda1) * math.sin(lambda2))))) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.cos(phi2) * math.sin(phi1)) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(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)
	return atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - expm1(log1p(Float64(cos(lambda1) * sin(lambda2))))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(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_] := N[ArcTan[N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(Exp[N[Log[1 + N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 12.9

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

2. Applied egg-rr 6.3

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	91136

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

Alternative 2
Error	3.3
Cost	84872

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -0.08608647141831227:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 7.264018513639833 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	6.6
Cost	71944

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{if}\;\lambda_2 \leq -16933856550.50532:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 10.983021466556234:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - t_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	6.7
Cost	71816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{if}\;\lambda_1 \leq -7.424832858115512 \cdot 10^{+28}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 0.028887363745734366:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - t_1 \cdot \log \left(e^{\cos \left(\lambda_1 - \lambda_2\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	6.6
Cost	71816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{if}\;\lambda_2 \leq -16933856550.50532:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 10.983021466556234:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\lambda_1 - \lambda_2\right)\right)\right)}{t_0 - t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	6.3
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 7
Error	10.2
Cost	65416

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_3 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_2\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \mathbf{if}\;\phi_2 \leq -0.08608647141831227:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_2 \leq 7.264018513639833 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - t_2}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Error	7.7
Cost	65416

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \lambda_2 \cdot \sin \lambda_1\\ \mathbf{if}\;\phi_1 \leq -1.1444185548868805 \cdot 10^{+33}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_3 - \sin \lambda_2\right)}{t_0 - t_1 \cdot t_2}\\ \mathbf{elif}\;\phi_1 \leq 0.07719201905529957:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t_3 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \sin \phi_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - t_1 \cdot \log \left(e^{t_2}\right)}\\ \end{array} \]

Alternative 9
Error	10.6
Cost	65288

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.08608647141831227:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_2}\\ \mathbf{elif}\;\phi_2 \leq 7.264018513639833 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right)}\\ \end{array} \]

Alternative 10
Error	10.6
Cost	65288

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.08608647141831227:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_1 \cdot t_2}\\ \mathbf{elif}\;\phi_2 \leq 9.664529313599592 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{t_0 - \sin \phi_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_1 \cdot \log \left(e^{t_2}\right)}\\ \end{array} \]

Alternative 11
Error	10.6
Cost	58888

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.08608647141831227:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_1 \cdot t_2}\\ \mathbf{elif}\;\phi_2 \leq 7.264018513639833 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{t_0 - \sin \phi_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_1 \cdot \left(3 \cdot \left(t_2 \cdot 0.3333333333333333\right)\right)}\\ \end{array} \]

Alternative 12
Error	13.3
Cost	52424

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -16933856550.50532:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{elif}\;\lambda_2 \leq 369802.16391568596:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 13
Error	16.2
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -16933856550.50532:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 - \sin \lambda_2\right)}{t_1}\\ \mathbf{elif}\;\lambda_2 \leq 369802.16391568596:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_1}\\ \end{array} \]

Alternative 14
Error	12.9
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.0009263043778473284:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_2 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 0.028887363745734366:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \lambda_2 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \end{array} \]

Alternative 15
Error	20.2
Cost	52228

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1.5089028339890058 \cdot 10^{-31}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_1 - \sin \phi_1 \cdot t_0}\\ \end{array} \]

Alternative 16
Error	12.9
Cost	52224

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

Alternative 17
Error	21.7
Cost	45896

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -16933856550.50532:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_2 \leq 369802.16391568596:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 18
Error	22.1
Cost	45832

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_2 \leq -1596167106414.7578:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 161238318255.04105:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Error	22.1
Cost	45832

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_2 \leq -1596167106414.7578:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 161238318255.04105:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 20
Error	21.5
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 21
Error	29.1
Cost	39816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{t_0}\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.0005:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 - \lambda_2\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Error	22.8
Cost	39560

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := \tan^{-1}_* \frac{t_1}{t_2 - \sin \phi_1 \cdot t_0}\\ \mathbf{if}\;\phi_1 \leq -4.813574402070665 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_1 \leq 1.5452379259299872 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_2 - \phi_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 23
Error	31.0
Cost	39432

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -1.1760271697216962 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.051897588192805 \cdot 10^{-35}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 24
Error	31.6
Cost	39176

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -4.325966616567705 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.051897588192805 \cdot 10^{-35}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 25
Error	33.0
Cost	26504

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -384292.7310545303:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 1.5452379259299872 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 26
Error	33.9
Cost	26440

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -1.1760271697216962 \cdot 10^{-126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 4.9515292356059845 \cdot 10^{-177}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 27
Error	44.6
Cost	19656

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -1.1402927581291893 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 2.8836690192230006 \cdot 10^{+32}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 28
Error	43.3
Cost	19456

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

Alternative 29
Error	48.1
Cost	19328

\[\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2} \]

Reproduce

herbie shell --seed 2022319 
(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))))))