Destination given bearing on a great circle

Average Error: 0.1 → 0.1

Time: 33.8s

Precision: binary64

Cost: 65536

Math

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

↓

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\left(0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)\right) \cdot \cos delta - \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (cos delta)
    (*
     (sin phi1)
     (sin
      (asin
       (+
        (* (sin phi1) (cos delta))
        (* (* (cos phi1) (sin delta)) (cos theta))))))))))

↓

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin theta) (sin delta)))
   (-
    (* (+ 0.5 (* 0.5 (cos (+ phi1 phi1)))) (cos delta))
    (* (* (sin delta) (* (cos phi1) (cos theta))) (sin phi1))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
}

↓

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (((0.5 + (0.5 * cos((phi1 + phi1)))) * cos(delta)) - ((sin(delta) * (cos(phi1) * cos(theta))) * sin(phi1))));
}

Fortran

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

↓

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (((0.5d0 + (0.5d0 * cos((phi1 + phi1)))) * cos(delta)) - ((sin(delta) * (cos(phi1) * cos(theta))) * sin(phi1))))
end function

Java

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

↓

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), (((0.5 + (0.5 * Math.cos((phi1 + phi1)))) * Math.cos(delta)) - ((Math.sin(delta) * (Math.cos(phi1) * Math.cos(theta))) * Math.sin(phi1))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.sin(phi1) * math.cos(delta)) + ((math.cos(phi1) * math.sin(delta)) * math.cos(theta))))))))

↓

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), (((0.5 + (0.5 * math.cos((phi1 + phi1)))) * math.cos(delta)) - ((math.sin(delta) * (math.cos(phi1) * math.cos(theta))) * math.sin(phi1))))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(sin(phi1) * cos(delta)) + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)))))))))
end

↓

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(phi1 + phi1)))) * cos(delta)) - Float64(Float64(sin(delta) * Float64(cos(phi1) * cos(theta))) * sin(phi1)))))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (sin(phi1) * sin(asin(((sin(phi1) * cos(delta)) + ((cos(phi1) * sin(delta)) * cos(theta))))))));
end

↓

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (((0.5 + (0.5 * cos((phi1 + phi1)))) * cos(delta)) - ((sin(delta) * (cos(phi1) * cos(theta))) * sin(phi1))));
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

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

2. Applied egg-rr 0.1

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

3. Taylor expanded in delta around inf 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos delta + -1 \cdot \left(\cos delta \cdot {\sin \phi_1}^{2}\right)\right)} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]

4. Simplified 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos delta - {\sin \phi_1}^{2} \cdot \cos delta\right)} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   Proof

   [Start] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos delta + -1 \cdot \left(\cos delta \cdot {\sin \phi_1}^{2}\right)\right) + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   mul-1-neg [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos delta + \color{blue}{\left(-\cos delta \cdot {\sin \phi_1}^{2}\right)}\right) + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   sub-neg [<=] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos delta - \cos delta \cdot {\sin \phi_1}^{2}\right)} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   *-commutative [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos delta - \color{blue}{{\sin \phi_1}^{2} \cdot \cos delta}\right) + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]

5. Taylor expanded in delta around inf 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos delta - \cos delta \cdot {\sin \phi_1}^{2}\right)} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]

6. Simplified 0.1

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

   [Start] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos delta - \cos delta \cdot {\sin \phi_1}^{2}\right) + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   *-commutative [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos delta - \color{blue}{{\sin \phi_1}^{2} \cdot \cos delta}\right) + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   cancel-sign-sub-inv [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos delta + \left(-{\sin \phi_1}^{2}\right) \cdot \cos delta\right)} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   distribute-rgt1-in [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\left(-{\sin \phi_1}^{2}\right) + 1\right) \cdot \cos delta} + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   +-commutative [<=] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(1 + \left(-{\sin \phi_1}^{2}\right)\right)} \cdot \cos delta + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   sub-neg [<=] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(1 - {\sin \phi_1}^{2}\right)} \cdot \cos delta + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   unpow2 [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(1 - \color{blue}{\sin \phi_1 \cdot \sin \phi_1}\right) \cdot \cos delta + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   1-sub-sin [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_1\right)} \cdot \cos delta + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]

7. Applied egg-rr 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(0.5 \cdot \cos \left(\phi_1 + \phi_1\right) + 0.5\right)} \cdot \cos delta + \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]

8. Final simplification 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\left(0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)\right) \cdot \cos delta - \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]

Alternatives

Alternative 1
Error	3.4
Cost	65152

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

Alternative 2
Error	3.4
Cost	65152

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

Alternative 3
Error	3.4
Cost	65152

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

Alternative 4
Error	3.5
Cost	59017

\[\begin{array}{l} \mathbf{if}\;theta \leq -0.000195 \lor \neg \left(theta \leq 0.0001\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta + \left(0.5 \cdot \cos \left(\phi_1 + \phi_1\right) + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \left(\sin delta \cdot \cos \phi_1 + \cos delta \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 5
Error	5.0
Cost	45696

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

Alternative 6
Error	5.0
Cost	39424

\[\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta + \left(0.5 \cdot \cos \left(\phi_1 + \phi_1\right) + -0.5\right)} \]

Alternative 7
Error	5.2
Cost	39304

\[\begin{array}{l} t_1 := \sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)\\ \mathbf{if}\;delta \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{elif}\;delta \leq 10^{-15}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]

Alternative 8
Error	5.2
Cost	39304

\[\begin{array}{l} t_1 := \sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\\ \mathbf{if}\;delta \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 10^{-15}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \end{array} \]

Alternative 9
Error	7.1
Cost	32512

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

Alternative 10
Error	7.7
Cost	26505

\[\begin{array}{l} \mathbf{if}\;theta \leq -3.5 \cdot 10^{-6} \lor \neg \left(theta \leq 2.7 \cdot 10^{-9}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{\sin theta}{\frac{2}{\sin delta \cdot 2}}}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]

Alternative 11
Error	7.7
Cost	26377

\[\begin{array}{l} \mathbf{if}\;theta \leq -7.5 \cdot 10^{-6} \lor \neg \left(theta \leq 1.8 \cdot 10^{-9}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]

Alternative 12
Error	7.7
Cost	26376

\[\begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ \mathbf{if}\;theta \leq -7.6 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{1}{\frac{1}{t_1}}}{\cos delta}\\ \mathbf{elif}\;theta \leq 3.2 \cdot 10^{-9}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \end{array} \]

Alternative 13
Error	13.5
Cost	26116

\[\begin{array}{l} \mathbf{if}\;delta \leq -3:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left|\sin delta\right|}{\cos delta}\\ \mathbf{elif}\;delta \leq 10^{+38}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 14
Error	8.5
Cost	25984

\[\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta} \]

Alternative 15
Error	12.8
Cost	19849

\[\begin{array}{l} \mathbf{if}\;theta \leq -3.5 \cdot 10^{-38} \lor \neg \left(theta \leq 0.00046\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 16
Error	17.0
Cost	19584

\[\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta} \]

Alternative 17
Error	20.9
Cost	13184

\[\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta} \]

Reproduce

herbie shell --seed 2023034 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))