Bearing on a great circle

Average Error: 13.2 → 0.2

Time: 1.1min

Precision: binary64

Cost: 97472

Math

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

↓

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

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 phi2)
   (fma (cos lambda1) (- (sin lambda2)) (* (sin lambda1) (cos lambda2))))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (* (cos phi2) (sin phi1))
    (+ (* (sin lambda2) (sin lambda1)) (* (cos lambda1) (cos 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) * fma(cos(lambda1), -sin(lambda2), (sin(lambda1) * cos(lambda2)))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda1) * cos(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) * fma(cos(lambda1), Float64(-sin(lambda2)), Float64(sin(lambda1) * cos(lambda2)))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda1) * cos(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[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision]) + N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[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[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $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 13.1

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

3. Applied egg-rr 0.2

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

4. Taylor expanded in lambda2 around inf 0.2

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

5. Simplified 0.2

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, \sin \lambda_1 \cdot \cos \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)} \]
   Proof
   (*.f64 (cos.f64 phi2) (fma.f64 (cos.f64 lambda1) (neg.f64 (sin.f64 lambda2)) (*.f64 (sin.f64 lambda1) (cos.f64 lambda2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 phi2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 lambda1) (neg.f64 (sin.f64 lambda2))) (*.f64 (sin.f64 lambda1) (cos.f64 lambda2))))): 8 points increase in error, 12 points decrease in error
   (*.f64 (cos.f64 phi2) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (sin.f64 lambda2)) (cos.f64 lambda1))) (*.f64 (sin.f64 lambda1) (cos.f64 lambda2)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 phi2) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (sin.f64 lambda1) (cos.f64 lambda2)) (*.f64 (neg.f64 (sin.f64 lambda2)) (cos.f64 lambda1))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 phi2) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 (sin.f64 lambda1) (cos.f64 lambda2)) (*.f64 (sin.f64 lambda2) (cos.f64 lambda1))))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	91136

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

Alternative 2
Error	0.2
Cost	91136

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

Alternative 3
Error	3.8
Cost	85384

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t_0 - \cos \lambda_1 \cdot \sin \lambda_2\\ t_3 := \cos \phi_2 \cdot \sin \phi_1\\ t_4 := t_1 - t_3 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -81677253.88112386:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, t_0\right)}{t_4}\\ \mathbf{elif}\;\phi_2 \leq 0.019355332082803024:\\ \;\;\;\;\tan^{-1}_* \frac{\left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right) \cdot t_2}{t_1 - t_3 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_4}\\ \end{array} \]

Alternative 4
Error	3.9
Cost	84872

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := t_0 - \cos \lambda_1 \cdot \sin \lambda_2\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := \cos \phi_2 \cdot \sin \phi_1\\ t_4 := t_2 - t_3 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -81677253.88112386:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, t_0\right)}{t_4}\\ \mathbf{elif}\;\phi_2 \leq 0.019355332082803024:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - t_3 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_4}\\ \end{array} \]

Alternative 5
Error	3.8
Cost	84872

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \phi_2 \cdot \left(t_0 - \cos \lambda_1 \cdot \sin \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := t_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1981929946810901000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_1, -\sin \lambda_2, t_0\right)}{t_3}\\ \mathbf{elif}\;\phi_2 \leq 0.019355332082803024:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \sin \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_3}\\ \end{array} \]

Alternative 6
Error	6.7
Cost	78016

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

Alternative 7
Error	7.0
Cost	71816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{if}\;\lambda_1 \leq -32801.91730503366:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 1.3066090103934743 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{t_0 - t_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	6.7
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 9
Error	7.6
Cost	65416

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot t_2\\ \mathbf{if}\;\phi_1 \leq -1.3042461613231323 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_3\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.1948233241408792 \cdot 10^{-24}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - t_1\right)}{t_0 - t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_1\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_2}\\ \end{array} \]

Alternative 10
Error	11.5
Cost	65288

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot t_1\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -81677253.88112386:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \cos \phi_2 \cdot t_2}\\ \mathbf{elif}\;\phi_2 \leq 0.019355332082803024:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(t_3\right)\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \end{array} \]

Alternative 11
Error	11.5
Cost	65288

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot t_2\\ \mathbf{if}\;\phi_2 \leq -81677253.88112386:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_3\right)\right)}\\ \mathbf{elif}\;\phi_2 \leq 0.019355332082803024:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right)}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_2}\\ \end{array} \]

Alternative 12
Error	12.3
Cost	65152

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

Alternative 13
Error	11.5
Cost	52744

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot t_1\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -81677253.88112386:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \cos \phi_2 \cdot t_2}\\ \mathbf{elif}\;\phi_2 \leq 0.019355332082803024:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 + \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(1 + \left(-1 - t_1\right)\right)}\\ \end{array} \]

Alternative 14
Error	18.6
Cost	52492

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{t_2 - t_0 \cdot t_1}\\ \mathbf{if}\;\lambda_1 \leq -4.80769666369875 \cdot 10^{-33}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 \leq 2.3415116928841094 \cdot 10^{-143}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_2 - \cos \lambda_2 \cdot t_0}\\ \mathbf{elif}\;\lambda_1 \leq 0.0003857462648331872:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_2 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 15
Error	11.5
Cost	52488

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot t_0}\\ \mathbf{if}\;\phi_2 \leq -81677253.88112386:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 0.019355332082803024:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	11.5
Cost	52488

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot t_1\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -81677253.88112386:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \cos \phi_2 \cdot t_2}\\ \mathbf{elif}\;\phi_2 \leq 0.019355332082803024:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2 - t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \end{array} \]

Alternative 17
Error	13.4
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\lambda_1 \leq -27.289597731134332:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 0.0003857462648331872:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	13.4
Cost	52360

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \lambda_1\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -27.289597731134332:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_0}\\ \mathbf{elif}\;\lambda_1 \leq 0.0003857462648331872:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_0\right)}\\ \end{array} \]

Alternative 19
Error	13.4
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \lambda_1\\ t_3 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -27.289597731134332:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_3 - t_0 \cdot t_1}\\ \mathbf{elif}\;\lambda_1 \leq 0.0003857462648331872:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_3 - \cos \lambda_2 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_3 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \end{array} \]

Alternative 20
Error	18.9
Cost	52296

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -4.813117380164443 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 272515031245752.56:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	23.8
Cost	45828

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t_1\\ \mathbf{if}\;\phi_2 \leq -1.60161406145746 \cdot 10^{-27}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \mathbf{elif}\;\phi_2 \leq 0.019355332082803024:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]

Alternative 22
Error	21.8
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 23
Error	23.2
Cost	39560

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \tan^{-1}_* \frac{t_0}{t_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -0.00015474683582399938:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 0.0010203432060950021:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{t_1 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 24
Error	23.2
Cost	39560

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.00015474683582399938:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\left(-1 + \left(t_0 + 1\right)\right) - t_1}\\ \mathbf{elif}\;\phi_1 \leq 0.0010203432060950021:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - t_1}\\ \end{array} \]

Alternative 25
Error	32.7
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 26
Error	33.4
Cost	32776

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -4.558812452871192 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 0.0010203432060950021:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 27
Error	33.2
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 28
Error	33.4
Cost	26504

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -4.558812452871192 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 0.0010203432060950021:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 29
Error	33.9
Cost	26440

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -2.7221599890740422 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 4.7370059059951334 \cdot 10^{-133}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 30
Error	43.5
Cost	19456

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

Reproduce

herbie shell --seed 2022313 
(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))))))