Bearing on a great circle

Average Error: 12.9 → 0.2

Time: 42.7s

Precision: binary64

Cost: 110976

Math

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

↓

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

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 (* (sin phi1) (cos phi2))) (t_1 (sin (- lambda2))))
   (atan2
    (-
     (* (* (sin lambda1) (cos lambda2)) (cos phi2))
     (* t_1 (* (cos phi2) (- (cos lambda1)))))
    (-
     (* (cos phi1) (sin phi2))
     (-
      (* t_0 (* (cos lambda1) (cos (- lambda2))))
      (* t_0 (* (sin lambda1) t_1)))))))

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(phi1) * cos(phi2);
	double t_1 = sin(-lambda2);
	return atan2((((sin(lambda1) * cos(lambda2)) * cos(phi2)) - (t_1 * (cos(phi2) * -cos(lambda1)))), ((cos(phi1) * sin(phi2)) - ((t_0 * (cos(lambda1) * cos(-lambda2))) - (t_0 * (sin(lambda1) * t_1)))));
}

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
    real(8) :: t_1
    t_0 = sin(phi1) * cos(phi2)
    t_1 = sin(-lambda2)
    code = atan2((((sin(lambda1) * cos(lambda2)) * cos(phi2)) - (t_1 * (cos(phi2) * -cos(lambda1)))), ((cos(phi1) * sin(phi2)) - ((t_0 * (cos(lambda1) * cos(-lambda2))) - (t_0 * (sin(lambda1) * t_1)))))
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(phi1) * Math.cos(phi2);
	double t_1 = Math.sin(-lambda2);
	return Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) * Math.cos(phi2)) - (t_1 * (Math.cos(phi2) * -Math.cos(lambda1)))), ((Math.cos(phi1) * Math.sin(phi2)) - ((t_0 * (Math.cos(lambda1) * Math.cos(-lambda2))) - (t_0 * (Math.sin(lambda1) * t_1)))));
}

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(phi1) * math.cos(phi2)
	t_1 = math.sin(-lambda2)
	return math.atan2((((math.sin(lambda1) * math.cos(lambda2)) * math.cos(phi2)) - (t_1 * (math.cos(phi2) * -math.cos(lambda1)))), ((math.cos(phi1) * math.sin(phi2)) - ((t_0 * (math.cos(lambda1) * math.cos(-lambda2))) - (t_0 * (math.sin(lambda1) * t_1)))))

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(sin(phi1) * cos(phi2))
	t_1 = sin(Float64(-lambda2))
	return atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) * cos(phi2)) - Float64(t_1 * Float64(cos(phi2) * Float64(-cos(lambda1))))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(t_0 * Float64(cos(lambda1) * cos(Float64(-lambda2)))) - Float64(t_0 * Float64(sin(lambda1) * t_1)))))
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(phi1) * cos(phi2);
	t_1 = sin(-lambda2);
	tmp = atan2((((sin(lambda1) * cos(lambda2)) * cos(phi2)) - (t_1 * (cos(phi2) * -cos(lambda1)))), ((cos(phi1) * sin(phi2)) - ((t_0 * (cos(lambda1) * cos(-lambda2))) - (t_0 * (sin(lambda1) * t_1)))));
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[Sin[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[(-lambda2)], $MachinePrecision]}, N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[Cos[phi2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(N[Sin[lambda1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 12.9

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

2. Applied egg-rr 6.7

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	110912

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

Alternative 2
Error	0.2
Cost	104448

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

Alternative 3
Error	0.2
Cost	91392

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

Alternative 4
Error	4.6
Cost	78600

\[\begin{array}{l} t_0 := \sin \left(-\lambda_2\right)\\ t_1 := \cos \left(-\lambda_2\right)\\ t_2 := \left(t_1 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot t_0\right) \cdot \cos \phi_2\\ t_3 := \tan^{-1}_* \frac{t_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -2.9 \cdot 10^{-62}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_2 \leq 2.85 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\sin \lambda_1 \cdot \left(\sin \phi_1 \cdot t_0\right) - \sin \phi_1 \cdot \left(t_1 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	4.6
Cost	78600

\[\begin{array}{l} t_0 := \sin \left(-\lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 - t_0 \cdot \left(\cos \phi_2 \cdot \left(-\cos \lambda_1\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ t_2 := \cos \left(-\lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -4.1 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 4.1 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(t_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot t_0\right) \cdot \cos \phi_2}{\sin \lambda_1 \cdot \left(\sin \phi_1 \cdot t_0\right) - \sin \phi_1 \cdot \left(t_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	4.6
Cost	72136

\[\begin{array}{l} t_0 := \sin \left(-\lambda_2\right)\\ t_1 := \cos \left(-\lambda_2\right)\\ t_2 := \left(t_1 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot t_0\right) \cdot \cos \phi_2\\ t_3 := \tan^{-1}_* \frac{t_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\phi_2 \leq -5.5 \cdot 10^{-62}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_2 \leq 1.08 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\left(\cos \lambda_1 \cdot t_1 - \sin \lambda_1 \cdot t_0\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	6.7
Cost	72008

\[\begin{array}{l} t_0 := \cos \left(-\lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{\left(t_0 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_0\right)}\\ \mathbf{if}\;\lambda_2 \leq -0.000108:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 + \lambda_2 \cdot \left(-\cos \lambda_1\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	6.8
Cost	71944

\[\begin{array}{l} t_0 := \sin \left(-\lambda_2\right)\\ t_1 := \cos \left(-\lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\left(t_1 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot t_0\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \lambda_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -7 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 4.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_1 + t_0 \cdot \left(-\lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	6.7
Cost	71808

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

Alternative 10
Error	13.0
Cost	52688

\[\begin{array}{l} t_0 := \sin \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}\\ t_3 := \cos \left(-\lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 7.3 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(t_3 \cdot \cos \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 8.2 \cdot 10^{+101}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 9 \cdot 10^{+297}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(-\lambda_2\right) \cdot \cos \lambda_1\right) - \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \left(-t_3\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \end{array} \]

Alternative 11
Error	13.0
Cost	52492

\[\begin{array}{l} t_0 := \sin \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}\\ t_3 := \cos \left(-\lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 7.3 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(t_3 \cdot \cos \phi_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 1.45 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 2.75 \cdot 10^{+298}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(t_3 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \end{array} \]

Alternative 12
Error	8.9
Cost	52488

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{if}\;\phi_1 \leq -4.1 \cdot 10^{-35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 5 \cdot 10^{-48}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(-\lambda_2\right) \cdot \cos \lambda_1\right) - \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \left(-\cos \left(-\lambda_2\right)\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	8.9
Cost	52488

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -9.5 \cdot 10^{-31}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t_1}\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-48}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(-\lambda_2\right) \cdot \cos \lambda_1\right) - \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \left(-\cos \left(-\lambda_2\right)\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t_1\right)}\\ \end{array} \]

Alternative 14
Error	13.1
Cost	52424

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\lambda_2 \leq -0.215:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \cos \lambda_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	17.3
Cost	52364

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\left(\cos \left(-\lambda_2\right) \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_1 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -1.8 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq -2 \cdot 10^{-144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.1 \cdot 10^{-155}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\sin \phi_2 \cdot \cos \phi_1 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \lambda_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.3 \cdot 10^{+291}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	13.5
Cost	52360

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\left(\cos \left(-\lambda_2\right) \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -3.4 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 0.5:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 \cdot \cos \phi_1 - \cos \lambda_1 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	17.1
Cost	46024

\[\begin{array}{l} t_0 := \sin \left(-\lambda_2\right)\\ t_1 := \cos \left(-\lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{\left(t_1 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot t_0\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -1900:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 3750000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot t_1 + t_0 \cdot \cos \lambda_1}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	16.9
Cost	45960

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \left(-\lambda_2\right) \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 19
Error	18.9
Cost	45832

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.0044:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 0.00072:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	18.1
Cost	45832

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -9.6 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1550:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \left(-\lambda_2\right) \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 21
Error	18.4
Cost	45832

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -9.6 \cdot 10^{-31}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t_0}\\ \mathbf{elif}\;\phi_1 \leq 1550:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \left(-\lambda_2\right) \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1 \cdot \cos \phi_2}{t_0 \cdot \left(-\sin \phi_1\right)}\\ \end{array} \]

Alternative 22
Error	21.8
Cost	39560

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -0.032:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 0.035:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - t_0 \cdot \left(\phi_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 23
Error	21.8
Cost	39560

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -0.012:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 0.0003:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\phi_2 \cdot \cos \phi_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 24
Error	21.9
Cost	39432

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -0.00064:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 0.00125:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \phi_2 \cdot \left(\phi_1 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 25
Error	21.9
Cost	33032

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -0.0007:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 0.0036:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - t_0 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 26
Error	23.1
Cost	32968

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{if}\;\phi_1 \leq -9.6 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 1600:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 27
Error	23.1
Cost	26440

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -1900:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 3750000:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 28
Error	26.3
Cost	26312

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \tan^{-1}_* \frac{t_0 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\phi_2 \leq -5.6 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_1 \cdot \left(-\cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 29
Error	36.1
Cost	26184

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -3.9 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 1.65 \cdot 10^{-129}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 30
Error	39.2
Cost	26120

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 0.0021:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 31
Error	32.2
Cost	25984

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

Alternative 32
Error	45.1
Cost	19656

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -1.06 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 6.3 \cdot 10^{-56}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 33
Error	43.5
Cost	19456

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

Alternative 34
Error	48.7
Cost	19328

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

Reproduce

herbie shell --seed 2023104 
(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))))))