Bearing on a great circle

Average Error: 13.3 → 0.2

Time: 1.0min

Precision: binary64

Cost: 110208

Math

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

↓

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

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

Derivation

1. Initial program 13.3

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

2. Applied egg-rr 7.0

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

3. Applied egg-rr 0.2

   \[\leadsto \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\frac{{\left(\cos \lambda_1 \cdot \cos \lambda_2\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}{\mathsf{fma}\left(\cos \lambda_1 \cdot \cos \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}} \]

4. Applied egg-rr 0.2

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

5. Applied egg-rr 0.2

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

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	91136

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

Alternative 2
Error	3.7
Cost	84872

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -9.431632257910712:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - t_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_3\right)\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.9419727562144953 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - t_2 \cdot \log \left(e^{t_3}\right)}\\ \end{array} \]

Alternative 3
Error	7.0
Cost	84480

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

Alternative 4
Error	7.0
Cost	84480

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

Alternative 5
Error	6.9
Cost	71816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{if}\;\lambda_1 \leq -8.395999175767996 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 0.0002248603846744988:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \lambda_1 - \sin \lambda_2\right)}{t_0 - t_1 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	7.6
Cost	71816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{if}\;\lambda_2 \leq -487347165656087040:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 0.00018002346995709106:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - t_1 \cdot \left(\cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	7.0
Cost	71680

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

Alternative 8
Error	8.1
Cost	65544

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -98168.33257247868:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.8145633734018635 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - t_2 \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \log \left(e^{t_2}\right)}\\ \end{array} \]

Alternative 9
Error	8.1
Cost	65416

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -98168.33257247868:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot t_3\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.8145633734018635 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \log \left(e^{t_2}\right)}\\ \end{array} \]

Alternative 10
Error	11.2
Cost	58888

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.004236244786467129:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_2}\\ \mathbf{elif}\;\phi_2 \leq 1.9419727562144953 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 + 3 \cdot \left(t_2 \cdot -0.3333333333333333\right)}\\ \end{array} \]

Alternative 11
Error	11.2
Cost	52744

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -7.906530349460656 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_2}\\ \mathbf{elif}\;\phi_2 \leq 1.9419727562144953 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 + 3 \cdot \left(t_2 \cdot -0.3333333333333333\right)}\\ \end{array} \]

Alternative 12
Error	11.2
Cost	52488

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot t_0}\\ \mathbf{if}\;\phi_2 \leq -7.906530349460656 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.9419727562144953 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	11.2
Cost	52488

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot t_1\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -7.906530349460656 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \mathbf{elif}\;\phi_2 \leq 1.9419727562144953 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \cos \phi_2 \cdot t_2}\\ \end{array} \]

Alternative 14
Error	13.7
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -8.395999175767996 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 1062071330517902.4:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	13.7
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -8.395999175767996 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 1062071330517902.4:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	14.2
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -7.906530349460656 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.9419727562144953 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 17
Error	19.6
Cost	52232

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -8.395999175767996 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 1062071330517902.4:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	19.6
Cost	46472

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_2 \leq -6.98366077742319 \cdot 10^{-14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 1.9419727562144953 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{\frac{2}{\sin \left(\phi_1 - \phi_2\right) + \sin \left(\phi_2 + \phi_1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Error	19.6
Cost	45832

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_2 \leq -6.98366077742319 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.9419727562144953 \cdot 10^{-36}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	32.4
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
Error	33.4
Cost	32776

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -101741.45868516705:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 7.579663651176602 \cdot 10^{-62}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 22
Error	33.0
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 23
Error	36.1
Cost	32512

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

Alternative 24
Error	33.1
Cost	26504

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -98168.33257247868:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 0.0009381299764466194:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 25
Error	33.5
Cost	26504

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -98168.33257247868:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\phi_2 - \sin \phi_1 \cdot t_0}\\ \mathbf{elif}\;\phi_1 \leq 0.0009381299764466194:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 \cdot \left(-\sin \phi_1\right)}\\ \end{array} \]

Alternative 26
Error	33.7
Cost	26440

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -399407.83901773934:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 1.1965812214406813 \cdot 10^{-55}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 27
Error	43.5
Cost	19456

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

Reproduce

herbie shell --seed 2022302 
(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))))))