Bearing on a great circle

Average Error: 13.2 → 0.2

Time: 57.2s

Precision: binary64

Cost: 161728

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\sin \lambda_2}\\ t_1 := \sqrt[3]{t_0}\\ \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left({\left(t_1 \cdot {t_1}^{2}\right)}^{2}, \sin \lambda_1 \cdot t_0, \cos \lambda_2 \cdot \cos \lambda_1\right)} \end{array} \]

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 (cbrt (sin lambda2))) (t_1 (cbrt t_0)))
   (atan2
    (*
     (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
     (cos phi2))
    (-
     (* (cos phi1) (sin phi2))
     (*
      (* (cos phi2) (sin phi1))
      (fma
       (pow (* t_1 (pow t_1 2.0)) 2.0)
       (* (sin lambda1) t_0)
       (* (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) {
	double t_0 = cbrt(sin(lambda2));
	double t_1 = cbrt(t_0);
	return atan2((fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * fma(pow((t_1 * pow(t_1, 2.0)), 2.0), (sin(lambda1) * t_0), (cos(lambda2) * 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)
	t_0 = cbrt(sin(lambda2))
	t_1 = cbrt(t_0)
	return atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * fma((Float64(t_1 * (t_1 ^ 2.0)) ^ 2.0), Float64(sin(lambda1) * t_0), 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_] := Block[{t$95$0 = N[Power[N[Sin[lambda2], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $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[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(t$95$1 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 13.2

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

2. Applied egg-rr 6.8

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

   \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin \lambda_2}\right)}^{2}, \sqrt[3]{\sin \lambda_2} \cdot \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)}} \]

5. Applied egg-rr 0.2

   \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left({\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\sin \lambda_2}}\right)}^{2} \cdot \sqrt[3]{\sqrt[3]{\sin \lambda_2}}\right)}}^{2}, \sqrt[3]{\sin \lambda_2} \cdot \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)} \]

6. Final simplification 0.2

   \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\sqrt[3]{\sin \lambda_2}} \cdot {\left(\sqrt[3]{\sqrt[3]{\sin \lambda_2}}\right)}^{2}\right)}^{2}, \sin \lambda_1 \cdot \sqrt[3]{\sin \lambda_2}, \cos \lambda_2 \cdot \cos \lambda_1\right)} \]

Alternatives

Alternative 1
Error	12.4
Cost	157320

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;t_1 \leq 1.56:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 1.575:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right)}{t_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	0.2
Cost	116672

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

Alternative 3
Error	0.2
Cost	103744

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

Alternative 4
Error	3.5
Cost	97480

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.0021041536001674918:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1}\\ \mathbf{elif}\;\phi_2 \leq 7630.642219697093:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_1}\\ \end{array} \]

Alternative 5
Error	0.2
Cost	97472

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

Alternative 6
Error	6.8
Cost	78016

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

Alternative 7
Error	8.0
Cost	71944

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -1997563359812.636:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 737545068070071:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	8.1
Cost	71752

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := t_1 \cdot t_2\\ t_4 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1997563359812.636:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{t_0 - t_3}\\ \mathbf{elif}\;\phi_1 \leq 737545068070071:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{t_0 - t_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right)}\\ \end{array} \]

Alternative 9
Error	6.8
Cost	71680

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

Alternative 10
Error	8.1
Cost	65416

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.3570023675791569 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_1 \cdot t_2}\\ \mathbf{elif}\;\phi_1 \leq 737545068070071:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_2\right)\right)}\\ \end{array} \]

Alternative 11
Error	8.0
Cost	65352

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.3570023675791569 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_1}\\ \mathbf{elif}\;\phi_1 \leq 227.1875159028481:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{t_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_2\right)\right)}{t_1}\\ \end{array} \]

Alternative 12
Error	8.0
Cost	65288

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.3570023675791569 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_1}\\ \mathbf{elif}\;\phi_1 \leq 227.1875159028481:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_2\right)\right)}{t_1}\\ \end{array} \]

Alternative 13
Error	8.0
Cost	59016

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -1.3570023675791569 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 227.1875159028481:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	13.2
Cost	52488

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_3 := t_0 - t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq 7632021163.757232:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_3}\\ \mathbf{elif}\;\lambda_1 \leq 7.351002619359391 \cdot 10^{+199}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 + \lambda_2\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_1 \cdot t_1}\\ \end{array} \]

Alternative 15
Error	13.3
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_1 - \cos \lambda_1 \cdot t_0}\\ \mathbf{if}\;\lambda_1 \leq -0.08069846666760098:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 1.1138426723995172 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_1 - \cos \lambda_2 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	13.3
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_1 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\lambda_1 \leq -0.08069846666760098:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 1.1138426723995172 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_1 - \cos \lambda_2 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	19.6
Cost	52232

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -0.08069846666760098:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 3.145679290450188 \cdot 10^{-43}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	13.2
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 19
Error	25.8
Cost	45964

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t_1\\ \mathbf{if}\;\phi_2 \leq -31302279980.901627:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_2 \cdot \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 4.9262277234775025 \cdot 10^{+34}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1 \cdot \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.7036143957348309 \cdot 10^{+165}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 20
Error	22.2
Cost	45832

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -0.08069846666760098:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 7.528250286415661 \cdot 10^{+30}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	22.2
Cost	45696

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

Alternative 22
Error	24.6
Cost	39944

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t_1\\ \mathbf{if}\;\phi_2 \leq -31302279980.901627:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_2 \cdot \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 4.9262277234775025 \cdot 10^{+34}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1 \cdot \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]

Alternative 23
Error	33.5
Cost	39168

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

Alternative 24
Error	33.4
Cost	39168

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

Alternative 25
Error	44.2
Cost	33284

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 3.1459239156615094 \cdot 10^{-227}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0 \cdot \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right)}{t_1 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]

Alternative 26
Error	43.9
Cost	32768

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

Reproduce

herbie shell --seed 2022310 
(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))))))