Bearing on a great circle

Average Error: 13.4 → 0.2

Time: 1.0min

Precision: binary64

Cost: 110464

Math

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

↓

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

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 (* (cos phi2) (sin phi1))))
   (atan2
    (*
     (- (* (cos lambda2) (sin lambda1)) (* (cos lambda1) (sin lambda2)))
     (cos phi2))
    (-
     (* (cos phi1) (sin phi2))
     (fma
      (* (cos lambda2) (cos lambda1))
      t_0
      (* t_0 (* (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) {
	double t_0 = cos(phi2) * sin(phi1);
	return atan2((((cos(lambda2) * sin(lambda1)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - fma((cos(lambda2) * cos(lambda1)), t_0, (t_0 * (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)
	t_0 = Float64(cos(phi2) * sin(phi1))
	return atan(Float64(Float64(Float64(cos(lambda2) * sin(lambda1)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - fma(Float64(cos(lambda2) * cos(lambda1)), t_0, Float64(t_0 * 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_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, 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[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(t$95$0 * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 13.4

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

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

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	91136

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

Alternative 2
Error	4.3
Cost	84872

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -225810138990364060:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 3.950959731480658 \cdot 10^{-59}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	7.1
Cost	71816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{if}\;\lambda_1 \leq -0.00040543331513377543:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 0.0020355851799748223:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - t_1 \cdot \left(-0.5 \cdot \left(\lambda_1 \cdot \left(\cos \lambda_2 \cdot \lambda_1\right)\right) + \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
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 5
Error	7.9
Cost	65416

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ t_3 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_1\right)}{t_0 - t_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -2.342841789657611 \cdot 10^{-10}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_1 \leq 1.9009847948408425 \cdot 10^{-30}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - t_1\right) \cdot \cos \phi_2}{t_0 - t_2}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	8.3
Cost	65288

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.342841789657611 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_2}\\ \mathbf{elif}\;\phi_1 \leq 1.9009847948408425 \cdot 10^{-30}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - 0.3333333333333333 \cdot \left(t_2 \cdot 3\right)}\\ \end{array} \]

Alternative 7
Error	19.1
Cost	52824

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_3 := \cos \phi_2 \cdot \sin \phi_1\\ t_4 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}{t_1 - \cos \lambda_2 \cdot t_3}\\ \mathbf{if}\;\lambda_1 \leq -1.6863291930797635 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq -9.370129132807804 \cdot 10^{-69}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq -5.193275912817547 \cdot 10^{-108}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - t_3}\\ \mathbf{elif}\;\lambda_1 \leq 4.879255117849937 \cdot 10^{-230}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\lambda_1 \leq 2.7454439400685026 \cdot 10^{-114}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 0.0020355851799748223:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	19.1
Cost	52824

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_3 := \cos \phi_2 \cdot \sin \phi_1\\ t_4 := \cos \phi_1 \cdot \sin \phi_2\\ t_5 := \cos \phi_2 \cdot \left(-\sin \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -1.6863291930797635 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq -9.370129132807804 \cdot 10^{-69}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_4 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq -5.193275912817547 \cdot 10^{-108}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_4 - t_3}\\ \mathbf{elif}\;\lambda_1 \leq 4.879255117849937 \cdot 10^{-230}:\\ \;\;\;\;\tan^{-1}_* \frac{t_5}{t_4 - \cos \lambda_2 \cdot t_3}\\ \mathbf{elif}\;\lambda_1 \leq 2.7454439400685026 \cdot 10^{-114}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - t_0}\\ \mathbf{elif}\;\lambda_1 \leq 0.0020355851799748223:\\ \;\;\;\;\tan^{-1}_* \frac{t_5}{t_4 - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	9.0
Cost	52744

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.944502075579319 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - t_1}\\ \mathbf{elif}\;\phi_1 \leq 1.9009847948408425 \cdot 10^{-30}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - 0.3333333333333333 \cdot \left(t_1 \cdot 3\right)}\\ \end{array} \]

Alternative 10
Error	13.7
Cost	52556

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_2 \cdot \left(-\sin \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1.135310521088871 \cdot 10^{+111}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq -1088.6807375602846:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \lambda_2 \cdot t_0}\\ \mathbf{elif}\;\lambda_2 \leq 0.0010595551098740857:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_2 - \cos \lambda_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 11
Error	13.7
Cost	52492

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}{t_1 - \cos \lambda_2 \cdot t_0}\\ \mathbf{if}\;\lambda_2 \leq -1.135310521088871 \cdot 10^{+111}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq -1088.6807375602846:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 0.0010595551098740857:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_1 - \cos \lambda_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	9.0
Cost	52488

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_1 \leq -3.944502075579319 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.9009847948408425 \cdot 10^{-30}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	13.9
Cost	52360

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\lambda_1 \leq -0.283154649424769:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 208.59428394515004:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	19.0
Cost	46736

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_3 := \tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1.5 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -0.0005:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 4000:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 5 \cdot 10^{+115}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	21.2
Cost	45968

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -9.247275976581543 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq -2.9665093614620762 \cdot 10^{-232}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 \cdot \left(-\sin \phi_1\right)}\\ \mathbf{elif}\;\phi_2 \leq -4.495587701110191 \cdot 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.950959731480658 \cdot 10^{-59}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	23.7
Cost	39824

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -190416507.92711568:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq -2.9665093614620762 \cdot 10^{-232}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq -4.495587701110191 \cdot 10^{-296}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 1.3115844729984523 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 17
Error	25.2
Cost	39312

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_2 \leq -1.4512442058417437 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq -2.9665093614620762 \cdot 10^{-232}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq -4.495587701110191 \cdot 10^{-296}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 3.950959731480658 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 18
Error	24.3
Cost	39048

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.4512442058417437 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 3.950959731480658 \cdot 10^{-59}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 19
Error	24.3
Cost	32968

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -1.4512442058417437 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 3.950959731480658 \cdot 10^{-59}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	36.1
Cost	26184

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\sin \lambda_2\right)}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -8.408315255131446 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.1945678002728541 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 21
Error	39.9
Cost	26120

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.408315255131446 \cdot 10^{-58}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 1.1945678002728541 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 22
Error	32.8
Cost	25984

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

Alternative 23
Error	45.2
Cost	19656

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \mathbf{if}\;\lambda_1 \leq -1.7180132157674363 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 5.747445352339396 \cdot 10^{+67}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 24
Error	43.6
Cost	19456

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

Alternative 25
Error	48.6
Cost	19328

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

Reproduce

herbie shell --seed 2022241 
(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))))))