Bearing on a great circle

Average Error: 20.14% → 0.29%

Time: 53.5s

Precision: binary64

Cost: 182400

Math

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

↓

\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1\\ t_1 := \sin \lambda_1 \cdot \sin \lambda_2\\ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\left({t_0}^{3} + {t_1}^{3}\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}{t_0 \cdot t_0 + \left(t_1 \cdot t_1 - t_0 \cdot t_1\right)}} \end{array} \]

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 lambda2) (cos lambda1)))
        (t_1 (* (sin lambda1) (sin lambda2))))
   (atan2
    (*
     (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
     (cos phi2))
    (-
     (* (cos phi1) (sin phi2))
     (/
      (* (+ (pow t_0 3.0) (pow t_1 3.0)) (* (cos phi2) (sin phi1)))
      (+ (* t_0 t_0) (- (* t_1 t_1) (* t_0 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 = cos(lambda2) * cos(lambda1);
	double t_1 = sin(lambda1) * sin(lambda2);
	return atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (((pow(t_0, 3.0) + pow(t_1, 3.0)) * (cos(phi2) * sin(phi1))) / ((t_0 * t_0) + ((t_1 * t_1) - (t_0 * 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 = cos(lambda2) * cos(lambda1)
    t_1 = sin(lambda1) * sin(lambda2)
    code = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((((t_0 ** 3.0d0) + (t_1 ** 3.0d0)) * (cos(phi2) * sin(phi1))) / ((t_0 * t_0) + ((t_1 * t_1) - (t_0 * 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.cos(lambda2) * Math.cos(lambda1);
	double t_1 = Math.sin(lambda1) * Math.sin(lambda2);
	return Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - (((Math.pow(t_0, 3.0) + Math.pow(t_1, 3.0)) * (Math.cos(phi2) * Math.sin(phi1))) / ((t_0 * t_0) + ((t_1 * t_1) - (t_0 * 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.cos(lambda2) * math.cos(lambda1)
	t_1 = math.sin(lambda1) * math.sin(lambda2)
	return math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - (((math.pow(t_0, 3.0) + math.pow(t_1, 3.0)) * (math.cos(phi2) * math.sin(phi1))) / ((t_0 * t_0) + ((t_1 * t_1) - (t_0 * 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(cos(lambda2) * cos(lambda1))
	t_1 = Float64(sin(lambda1) * sin(lambda2))
	return atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(Float64((t_0 ^ 3.0) + (t_1 ^ 3.0)) * Float64(cos(phi2) * sin(phi1))) / Float64(Float64(t_0 * t_0) + Float64(Float64(t_1 * t_1) - Float64(t_0 * 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 = cos(lambda2) * cos(lambda1);
	t_1 = sin(lambda1) * sin(lambda2);
	tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((((t_0 ^ 3.0) + (t_1 ^ 3.0)) * (cos(phi2) * sin(phi1))) / ((t_0 * t_0) + ((t_1 * t_1) - (t_0 * 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[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $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[(N[Power[t$95$0, 3.0], $MachinePrecision] + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 20.14

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

2. Applied egg-rr 10.22

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

3. Simplified 10.22

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

   [Start] 10.22	\[ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 + \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   cancel-sign-sub-inv [<=] 10.22	\[ \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

4. Applied egg-rr 0.29

   \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\frac{\left({\left(\cos \lambda_2 \cdot \cos \lambda_1\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}{\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}} \]

5. Final simplification 0.29

   \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\left({\left(\cos \lambda_2 \cdot \cos \lambda_1\right)}^{3} + {\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}{\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) - \left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}} \]

Alternatives

Alternative 1
Error	0.28%
Cost	103936

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

Alternative 2
Error	0.28%
Cost	97408

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

Alternative 3
Error	5.82%
Cost	91145

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 1.15 \cdot 10^{-71}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_2 \cdot \cos \phi_2}{t_1 - \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - t_0 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]

Alternative 4
Error	0.28%
Cost	91136

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

Alternative 5
Error	5.83%
Cost	84873

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -3.4 \cdot 10^{-6} \lor \neg \left(\phi_2 \leq 2 \cdot 10^{-72}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_2 \cdot \sin \phi_1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]

Alternative 6
Error	10.22%
Cost	84480

\[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_2 \cdot \sin \phi_1\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

Alternative 7
Error	10.22%
Cost	78208

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

Alternative 8
Error	10.83%
Cost	71817

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -2.8 \cdot 10^{-7} \lor \neg \left(\lambda_1 \leq 122000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - t_1 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]

Alternative 9
Error	10.26%
Cost	71817

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_2 \leq -6.8 \cdot 10^{-6} \lor \neg \left(\lambda_2 \leq 0.06\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - t_1 \cdot \left(\cos \lambda_1 + \sin \lambda_1 \cdot \lambda_2\right)}\\ \end{array} \]

Alternative 10
Error	10.22%
Cost	71680

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

Alternative 11
Error	11.7%
Cost	65544

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right) \cdot t_3}\\ \mathbf{elif}\;\phi_1 \leq 0.118:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - t_3 \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - t_1 \cdot t_3}\\ \end{array} \]

Alternative 12
Error	12.44%
Cost	65284

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1\\ t_3 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\frac{\cos \phi_2}{\frac{1}{t_3}}}{t_0 - \mathsf{expm1}\left(\mathsf{log1p}\left(t_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 5.6 \cdot 10^{-58}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_3}{t_0 - t_2}\\ \end{array} \]

Alternative 13
Error	12.44%
Cost	65156

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right) \cdot t_3}\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - t_1 \cdot t_3}\\ \end{array} \]

Alternative 14
Error	12.43%
Cost	58888

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_0\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\frac{\cos \phi_2}{\frac{1}{t_2}}}{t_1}\\ \mathbf{elif}\;\phi_1 \leq 5.6 \cdot 10^{-58}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \sin \phi_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_1}\\ \end{array} \]

Alternative 15
Error	20.54%
Cost	52492

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{if}\;\lambda_2 \leq -1.35 \cdot 10^{+127}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 0.063:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	20.55%
Cost	52492

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -3.4 \cdot 10^{+128}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_2 \cdot t_1}\\ \mathbf{elif}\;\lambda_2 \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 0.06:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 17
Error	20.54%
Cost	52492

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1.08 \cdot 10^{+127}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_2 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 0.06:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \lambda_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 18
Error	13.33%
Cost	52489

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.8 \cdot 10^{-41} \lor \neg \left(\phi_1 \leq 4.2 \cdot 10^{-60}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 - \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]

Alternative 19
Error	29.01%
Cost	52364

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t_1\\ t_3 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq -5.3 \cdot 10^{-223}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_3 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 8.5 \cdot 10^{-224}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_3 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 3.1 \cdot 10^{+128}:\\ \;\;\;\;\tan^{-1}_* \frac{\frac{\cos \phi_2}{\frac{1}{t_1}}}{t_3 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 1.85 \cdot 10^{+263}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_3 - \cos \lambda_2 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 20
Error	20.71%
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -820:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 102000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 - \left(\cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]

Alternative 21
Error	26.88%
Cost	45961

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-40} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-58}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]

Alternative 22
Error	27.77%
Cost	45704

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.2 \cdot 10^{-40}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - \sin \phi_1 \cdot t_0}\\ \mathbf{elif}\;\phi_1 \leq 0.118:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 \cdot \left(-\sin \phi_1\right)}\\ \end{array} \]

Alternative 23
Error	36.05%
Cost	39432

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t_0\\ \mathbf{if}\;\phi_2 \leq -9.2 \cdot 10^{+21}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 1.45 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2}\\ \end{array} \]

Alternative 24
Error	35.1%
Cost	39305

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.72 \lor \neg \left(\phi_1 \leq 65000\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 25
Error	34.82%
Cost	39168

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

Alternative 26
Error	37.68%
Cost	33033

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -4 \cdot 10^{+122} \lor \neg \left(\phi_2 \leq 3.1 \cdot 10^{-13}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 27
Error	35.01%
Cost	33033

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 0.118\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - \phi_1 \cdot t_0}\\ \end{array} \]

Alternative 28
Error	36.38%
Cost	32969

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.72 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 1.45 \cdot 10^{-6}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \end{array} \]

Alternative 29
Error	55.33%
Cost	26185

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.6 \cdot 10^{-28} \lor \neg \left(\lambda_2 \leq 1.12 \cdot 10^{-28}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]

Alternative 30
Error	60.23%
Cost	26121

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -13800000000000 \lor \neg \left(\phi_2 \leq 3.2 \cdot 10^{+19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 31
Error	50.5%
Cost	25984

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

Alternative 32
Error	67.6%
Cost	19456

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

Reproduce

herbie shell --seed 2023121 
(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))))))