Bearing on a great circle

Average Error: 13.5 → 0.2

Time: 1.1min

Precision: binary64

Cost: 110208

Math

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

↓

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

FPCore

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

↓

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (cos phi2)
   (- (* (cos lambda2) (sin lambda1)) (* (cos lambda1) (sin lambda2))))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (* (cos phi2) (sin phi1))
    (fma
     (cos lambda2)
     (cos lambda1)
     (log1p (expm1 (* (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) {
	return atan2((cos(phi2) * ((cos(lambda2) * sin(lambda1)) - (cos(lambda1) * sin(lambda2)))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * fma(cos(lambda2), cos(lambda1), log1p(expm1((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)
	return atan(Float64(cos(phi2) * Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(cos(lambda1) * sin(lambda2)))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * fma(cos(lambda2), cos(lambda1), log1p(expm1(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_] := N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $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[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[Log[1 + N[(Exp[N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 13.5

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

2. Applied egg-rr 6.8

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
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 2
Error	3.4
Cost	91144

\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ t_3 := t_1 - t_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_4 := \cos \lambda_2 \cdot \sin \lambda_1\\ t_5 := t_4 - t_0\\ \mathbf{if}\;\phi_2 \leq -0.13868472721609698:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_4 - t_0 \cdot \cos \phi_2}{t_3}\\ \mathbf{elif}\;\phi_2 \leq 3.1262509619573026 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_5}{t_1 - t_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_5}{t_3}\\ \end{array} \]

Alternative 3
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 4
Error	3.4
Cost	84872

\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \lambda_2 \cdot \sin \lambda_1\\ t_4 := \cos \phi_2 \cdot \left(t_3 - t_0\right)\\ \mathbf{if}\;\phi_2 \leq -0.13868472721609698:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_3 - t_0 \cdot \cos \phi_2}{t_2}\\ \mathbf{elif}\;\phi_2 \leq 3.1262509619573026 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{t_1 - \sin \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{t_2}\\ \end{array} \]

Alternative 5
Error	6.8
Cost	78016

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

Alternative 6
Error	7.5
Cost	71944

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{if}\;\lambda_2 \leq -2332986643408826:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 4.449993759281544 \cdot 10^{+37}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - t_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	7.6
Cost	71816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{if}\;\lambda_2 \leq -2332986643408826:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 2.5788176126146425 \cdot 10^{-55}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)\right)}{t_0 - t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	6.8
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 9
Error	7.7
Cost	65416

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1\\ t_3 := \cos \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_1 \leq -2.1659897069598852 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_3\right)}{t_2}\\ \mathbf{elif}\;\phi_1 \leq 1.0402383813124898 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - t_3\right)}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)\right)}{t_2}\\ \end{array} \]

Alternative 10
Error	7.9
Cost	65288

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -0.004041547321906854:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 1.0402383813124898 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	8.0
Cost	65288

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.004041547321906854:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.0402383813124898 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(t_3\right)\right)}{t_0 - t_1 \cdot t_2}\\ \end{array} \]

Alternative 12
Error	8.9
Cost	52488

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -0.004041547321906854:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.0402383813124898 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	17.6
Cost	52428

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\lambda_1 \leq -2.821639137483923 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_1 \leq -9.405326626301977 \cdot 10^{-171}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 2.941330371783467 \cdot 10^{-81}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_2 \cdot \left(-\cos \phi_2\right)}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_1 \leq 1.4108974979130542 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 14
Error	13.4
Cost	52360

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -111587406176125.33:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_1 \leq 1.4108974979130542 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 15
Error	13.6
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -111587406176125.33:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{elif}\;\lambda_1 \leq 1.4108974979130542 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 16
Error	13.7
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -111587406176125.33:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_1 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 1.4108974979130542 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_1 - \cos \lambda_2 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 17
Error	16.8
Cost	52040

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -5.94406762976297:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.0402383813124898 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	16.8
Cost	45960

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -5.94406762976297:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.0402383813124898 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 19
Error	18.0
Cost	45828

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -5.94406762976297:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_0}\\ \mathbf{elif}\;\phi_1 \leq 1.0402383813124898 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \sin \phi_1 \cdot t_0}\\ \end{array} \]

Alternative 20
Error	18.0
Cost	45704

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -5.94406762976297:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.0402383813124898 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 21
Error	22.2
Cost	40264

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.3939622009326639:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 0.2518800323297903:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\left(\sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(-1 + 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right) + \cos \phi_1 \cdot \left(\phi_2 + -0.16666666666666666 \cdot {\phi_2}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Error	22.2
Cost	39944

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.3939622009326639:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 0.2518800323297903:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \phi_1 \cdot \sin \phi_2 + \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(-1 + 0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 23
Error	22.1
Cost	33032

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.3939622009326639:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 0.2518800323297903:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 24
Error	23.5
Cost	26440

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.3939622009326639:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 5.226236128273037 \cdot 10^{-45}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 25
Error	35.3
Cost	26184

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\lambda_1 \leq -2.0341806512522074 \cdot 10^{-50}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 4.732337290864429 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_2 \cdot \left(-\cos \phi_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 26
Error	39.5
Cost	26120

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.117285450813237 \cdot 10^{-29}:\\ \;\;\;\;-1 + \left(1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.796704756432475 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 27
Error	32.3
Cost	25984

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

Alternative 28
Error	44.9
Cost	19656

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -15711461846453028:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 3.40445126817864 \cdot 10^{-73}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 29
Error	43.2
Cost	19456

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

Alternative 30
Error	48.3
Cost	19328

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

Reproduce

herbie shell --seed 2022312 
(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))))))