Bearing on a great circle

Average Error: 13.3 → 0.2

Time: 2.2min

Precision: binary64

Cost: 110784

Math

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

↓

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

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

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) * cos(lambda2)) * sin(lambda1)) + ((sin(lambda2) * -cos(lambda1)) * cos(phi2))), ((cos(phi1) * sin(phi2)) - ((((cos(lambda1) * cos(lambda2)) * cos(phi2)) * sin(phi1)) + ((sin(phi1) * sin(lambda1)) * (sin(lambda2) * cos(phi2))))));
}

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

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

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(Float64(Float64(cos(phi2) * cos(lambda2)) * sin(lambda1)) + Float64(Float64(sin(lambda2) * Float64(-cos(lambda1))) * cos(phi2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(Float64(Float64(cos(lambda1) * cos(lambda2)) * cos(phi2)) * sin(phi1)) + Float64(Float64(sin(phi1) * sin(lambda1)) * Float64(sin(lambda2) * cos(phi2))))))
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)
	tmp = atan2((((cos(phi2) * cos(lambda2)) * sin(lambda1)) + ((sin(lambda2) * -cos(lambda1)) * cos(phi2))), ((cos(phi1) * sin(phi2)) - ((((cos(lambda1) * cos(lambda2)) * cos(phi2)) * sin(phi1)) + ((sin(phi1) * sin(lambda1)) * (sin(lambda2) * cos(phi2))))));
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[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 13.3

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

2. Applied egg-rr 6.9

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

3. Applied egg-rr 0.2

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

4. Taylor expanded in lambda1 around inf 0.2

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

5. Simplified 0.2

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

6. Taylor expanded in lambda1 around inf 0.2

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

7. Simplified 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	104000

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

Alternative 2
Error	0.2
Cost	104000

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

Alternative 3
Error	0.2
Cost	97728

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

Alternative 4
Error	6.9
Cost	78272

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

Alternative 5
Error	7.0
Cost	71816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -0.00058:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 5.5 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	6.9
Cost	71680

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

Alternative 7
Error	7.9
Cost	65416

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 8.2 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \left(\cos \phi_2 \cdot t_1\right) \cdot \sin \phi_1}\\ \end{array} \]

Alternative 8
Error	7.9
Cost	59016

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -2.2 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot t_1}\\ \mathbf{elif}\;\phi_1 \leq 4.1 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - t_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \phi_1}\\ \end{array} \]

Alternative 9
Error	18.5
Cost	52624

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\lambda_1 \leq -7.8 \cdot 10^{+14}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq -1.35 \cdot 10^{-109}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\lambda_1 \leq 1.42 \cdot 10^{-235}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_1 - \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_1 \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	19.4
Cost	52560

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_1 - \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -3.9 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 1.02 \cdot 10^{-247}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 7.5 \cdot 10^{-82}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_2 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \sin \phi_1 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	14.3
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{t_0 - \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -3.9 \cdot 10^{+56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 5 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	14.2
Cost	52360

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_1 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 8.5 \cdot 10^{-83}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \lambda_1\right)}\\ \end{array} \]

Alternative 13
Error	13.3
Cost	52224

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

Alternative 14
Error	19.4
Cost	46216

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 0.1:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	22.4
Cost	45832

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 2.4 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\cos \phi_2 \cdot \cos \lambda_2\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	20.3
Cost	45832

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t_1 \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 1100000000:\\ \;\;\;\;\tan^{-1}_* \frac{t_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	23.9
Cost	39944

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t_1 \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 165000000000:\\ \;\;\;\;\tan^{-1}_* \frac{t_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	29.5
Cost	39816

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t_1 \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 165000000000:\\ \;\;\;\;\tan^{-1}_* \frac{t_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Error	38.3
Cost	39432

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \phi_1}\\ \mathbf{if}\;\phi_2 \leq -1.85:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1100000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	32.6
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 21
Error	43.6
Cost	33284

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 1.15 \cdot 10^{-247}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{t_1 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]

Alternative 22
Error	43.2
Cost	32768

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

Reproduce

herbie shell --seed 2023033 
(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))))))