Bearing on a great circle

Average Error: 13.5 → 0.2

Time: 2.4min

Precision: binary64

Cost: 91328

Math

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

↓

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

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 (sin (- lambda2))))
   (atan2
    (* (cos phi2) (+ (* (cos lambda1) t_0) (* (sin lambda1) (cos lambda2))))
    (-
     (* (cos phi1) (sin phi2))
     (*
      (* (sin phi1) (cos phi2))
      (- (* (cos lambda1) (cos (- lambda2))) (* (sin lambda1) t_0)))))))

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 = sin(-lambda2);
	return atan2((cos(phi2) * ((cos(lambda1) * t_0) + (sin(lambda1) * cos(lambda2)))), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * ((cos(lambda1) * cos(-lambda2)) - (sin(lambda1) * t_0)))));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

↓

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = sin(-lambda2)
    code = atan2((cos(phi2) * ((cos(lambda1) * t_0) + (sin(lambda1) * cos(lambda2)))), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * ((cos(lambda1) * cos(-lambda2)) - (sin(lambda1) * t_0)))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.sin((lambda1 - lambda2)) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
}

↓

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(-lambda2);
	return Math.atan2((Math.cos(phi2) * ((Math.cos(lambda1) * t_0) + (Math.sin(lambda1) * Math.cos(lambda2)))), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.sin(phi1) * Math.cos(phi2)) * ((Math.cos(lambda1) * Math.cos(-lambda2)) - (Math.sin(lambda1) * t_0)))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.sin((lambda1 - lambda2)) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - ((math.sin(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))

↓

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(-lambda2)
	return math.atan2((math.cos(phi2) * ((math.cos(lambda1) * t_0) + (math.sin(lambda1) * math.cos(lambda2)))), ((math.cos(phi1) * math.sin(phi2)) - ((math.sin(phi1) * math.cos(phi2)) * ((math.cos(lambda1) * math.cos(-lambda2)) - (math.sin(lambda1) * t_0)))))

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 = sin(Float64(-lambda2))
	return atan(Float64(cos(phi2) * Float64(Float64(cos(lambda1) * t_0) + Float64(sin(lambda1) * cos(lambda2)))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(sin(phi1) * cos(phi2)) * Float64(Float64(cos(lambda1) * cos(Float64(-lambda2))) - Float64(sin(lambda1) * t_0)))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
end

↓

function tmp = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(-lambda2);
	tmp = atan2((cos(phi2) * ((cos(lambda1) * t_0) + (sin(lambda1) * cos(lambda2)))), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * ((cos(lambda1) * cos(-lambda2)) - (sin(lambda1) * t_0)))));
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[Sin[(-lambda2)], $MachinePrecision]}, N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $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[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda1], $MachinePrecision] * t$95$0), $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 7.0

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2 + 0}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \]

5. Simplified 0.2

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

   [Start] 0.2	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2 + 0}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \]
   rational_best-simplify-3 [=>] 0.2	\[ \tan^{-1}_* \frac{\color{blue}{0 + \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \]
   rational_best-simplify-6 [=>] 0.2	\[ \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \]
   rational_best-simplify-1 [=>] 0.2	\[ \tan^{-1}_* \frac{\color{blue}{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \]
   rational_best-simplify-1 [=>] 0.2	\[ \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \left(-\lambda_2\right) \cdot \cos \lambda_1}\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \]
   rational_best-simplify-3 [=>] 0.2	\[ \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\left(\sin \left(-\lambda_2\right) \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \cos \lambda_2\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \]
   rational_best-simplify-1 [<=] 0.2	\[ \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \sin \left(-\lambda_2\right)} + \sin \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)} \]

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	91328

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

Alternative 2
Error	3.8
Cost	85064

\[\begin{array}{l} t_0 := \sin \left(-\lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \sin \phi_1 \cdot \cos \phi_2\\ t_3 := \tan^{-1}_* \frac{\left(\cos \lambda_2 \cdot \sin \lambda_1 - \sin \lambda_2 \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{t_1 - t_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -6.7 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-42}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \cos \lambda_2}{t_1 - t_2 \cdot \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	7.1
Cost	71944

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \left(-\lambda_2\right) + \sin \lambda_1 \cdot \cos \lambda_2\right)}{t_0 - \cos \left(-\lambda_2\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -0.00027:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 + \left(-\lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{t_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	7.1
Cost	71880

\[\begin{array}{l} t_0 := \cos \left(-\lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \sin \phi_1 \cdot \cos \phi_2\\ t_3 := \sin \left(-\lambda_2\right)\\ t_4 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot t_3 + \sin \lambda_1 \cdot \cos \lambda_2\right)}{t_1 - t_2 \cdot \cos \lambda_1}\\ \mathbf{if}\;\lambda_1 \leq -2.5 \cdot 10^{-5}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\lambda_1 \leq 9 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(t_3 + t_0 \cdot \lambda_1\right) \cdot \cos \phi_2}{t_1 - t_2 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 5
Error	7.0
Cost	71680

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

Alternative 6
Error	8.5
Cost	65416

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -300000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \left(-\lambda_2\right) \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	8.2
Cost	59144

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \left(-\lambda_2\right) \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{t_0 - t_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	18.0
Cost	52624

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \lambda_1\right)\\ t_1 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - t_0}\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ t_3 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\lambda_2\right)}\\ \mathbf{if}\;\lambda_2 \leq -5 \cdot 10^{-19}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_2 \leq -2.5 \cdot 10^{-189}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 8.8 \cdot 10^{-136}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_2 - t_0}\\ \mathbf{elif}\;\lambda_2 \leq 1.75 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	13.7
Cost	52424

\[\begin{array}{l} t_0 := \sin \left(-\lambda_2\right) \cdot \cos \phi_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -24.5:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t_1 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]

Alternative 10
Error	13.7
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\lambda_2\right)}\\ \mathbf{if}\;\lambda_2 \leq -25:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	21.0
Cost	52232

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \lambda_1\right)\\ t_3 := t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_2 \leq -1.05 \cdot 10^{-188}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_3}\\ \mathbf{elif}\;\lambda_2 \leq 4.2 \cdot 10^{-136}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - t_2}\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_3}\\ \end{array} \]

Alternative 12
Error	13.5
Cost	52224

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

Alternative 13
Error	23.2
Cost	46424

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \sin \phi_1 \cdot \cos \lambda_1\\ t_3 := \cos \phi_1 \cdot \sin \phi_2\\ t_4 := \tan^{-1}_* \frac{t_1}{t_3 - t_0}\\ t_5 := \tan^{-1}_* \frac{t_1}{\sin \phi_2 - t_0 \cdot \cos \left(-\lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_2 \leq -5.9 \cdot 10^{-293}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_3 - t_2}\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-196}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-133}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - \cos \phi_2 \cdot t_2}\\ \mathbf{elif}\;\phi_2 \leq 2.5 \cdot 10^{-93}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_3 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{+25}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 14
Error	23.2
Cost	46424

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(-\lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \cos \phi_2\\ t_3 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_4 := \tan^{-1}_* \frac{t_3}{t_0 - t_2}\\ t_5 := \sin \phi_1 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -1.12 \cdot 10^{-7}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_2 \leq -2.5 \cdot 10^{-291}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - t_5}\\ \mathbf{elif}\;\phi_2 \leq 5.2 \cdot 10^{-198}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{\sin \phi_2 - t_2 \cdot t_1}\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-133}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{\sin \phi_2 - \cos \phi_2 \cdot t_5}\\ \mathbf{elif}\;\phi_2 \leq 2.5 \cdot 10^{-93}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 4.6 \cdot 10^{+25}:\\ \;\;\;\;\tan^{-1}_* \frac{t_3}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 15
Error	20.2
Cost	46216

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{+52}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 50000000:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \cos \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	31.0
Cost	46096

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - t_2 \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -1.3 \cdot 10^{-188}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \left(-\lambda_2\right) \cdot \phi_1}\\ \mathbf{elif}\;\lambda_2 \leq 1.06 \cdot 10^{-135}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\lambda_2 \leq 4.2 \cdot 10^{-38}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{elif}\;\lambda_2 \leq 9.5 \cdot 10^{-15}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_0 - t_2 \cdot \phi_1}\\ \end{array} \]

Alternative 17
Error	22.1
Cost	45896

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -860:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.95 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	22.8
Cost	45832

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \cos \lambda_1}\\ \mathbf{if}\;\lambda_1 \leq -7 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 3.2 \cdot 10^{-28}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \cos \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Error	25.0
Cost	45568

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

Alternative 20
Error	43.6
Cost	39368

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \lambda_1}{t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -1.45 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 9.5 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \left(-\lambda_2\right) \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	32.8
Cost	39296

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

Alternative 22
Error	33.0
Cost	39168

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

Alternative 23
Error	43.5
Cost	33032

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right) \cdot \phi_1\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \lambda_1}{\cos \phi_1 \cdot \sin \phi_2 - t_0}\\ \mathbf{if}\;\lambda_2 \leq -1.08 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.35 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 24
Error	48.9
Cost	32640

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

Alternative 25
Error	43.6
Cost	32640

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

Reproduce

herbie shell --seed 2023099 
(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))))))