Destination given bearing on a great circle

Average Error: 0.1 → 0.2

Time: 1.5min

Precision: binary64

Cost: 71936

Math

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

↓

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

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
   (* (sin theta) (* (sin delta) (cos phi1)))
   (+
    (-
     -1.0
     (-
      (*
       (sin phi1)
       (+
        (* (cos delta) (sin phi1))
        (* (sin delta) (* (cos phi1) (cos theta)))))
      (cos delta)))
    1.0))))

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((sin(theta) * (sin(delta) * cos(phi1))), ((-1.0 - ((sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * (cos(phi1) * cos(theta))))) - cos(delta))) + 1.0));
}

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((sin(theta) * (sin(delta) * cos(phi1))), (((-1.0d0) - ((sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * (cos(phi1) * cos(theta))))) - cos(delta))) + 1.0d0))
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.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), ((-1.0 - ((Math.sin(phi1) * ((Math.cos(delta) * Math.sin(phi1)) + (Math.sin(delta) * (Math.cos(phi1) * Math.cos(theta))))) - Math.cos(delta))) + 1.0));
}

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.sin(theta) * (math.sin(delta) * math.cos(phi1))), ((-1.0 - ((math.sin(phi1) * ((math.cos(delta) * math.sin(phi1)) + (math.sin(delta) * (math.cos(phi1) * math.cos(theta))))) - math.cos(delta))) + 1.0))

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(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(Float64(-1.0 - Float64(Float64(sin(phi1) * Float64(Float64(cos(delta) * sin(phi1)) + Float64(sin(delta) * Float64(cos(phi1) * cos(theta))))) - cos(delta))) + 1.0)))
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((sin(theta) * (sin(delta) * cos(phi1))), ((-1.0 - ((sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * (cos(phi1) * cos(theta))))) - cos(delta))) + 1.0));
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[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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. Simplified 0.1

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

   [Start] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \color{blue}{\left(\sin delta \cdot \sin theta\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   rational_best-simplify-50 [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\color{blue}{\cos delta \cdot \sin \phi_1} + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   rational_best-simplify-1 [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \color{blue}{\left(\sin delta \cdot \cos \phi_1\right)} \cdot \cos theta\right)} \]

3. Taylor expanded in delta around inf 0.1

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

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.1
Cost	71680

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

Alternative 2
Error	3.3
Cost	65152

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

Alternative 3
Error	4.7
Cost	58624

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

Alternative 4
Error	5.1
Cost	45696

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

Alternative 5
Error	5.0
Cost	45504

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

Alternative 6
Error	5.3
Cost	39368

\[\begin{array}{l} t_1 := \sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\\ \mathbf{if}\;delta \leq -0.21:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{elif}\;delta \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{1 - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{-1 + \left(1 - \left(-\cos delta\right)\right)}\\ \end{array} \]

Alternative 7
Error	5.3
Cost	33096

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -0.21:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 2.65 \cdot 10^{-20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(delta \cdot \sin theta\right)}{-0.5 - \left({\sin \phi_1}^{2} + -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	5.3
Cost	33096

\[\begin{array}{l} t_1 := \sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\\ \mathbf{if}\;delta \leq -0.21:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{elif}\;delta \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(delta \cdot \sin theta\right)}{-0.5 - \left({\sin \phi_1}^{2} + -1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{-1 + \left(1 - \left(-\cos delta\right)\right)}\\ \end{array} \]

Alternative 9
Error	5.3
Cost	32968

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -0.21:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 2.5 \cdot 10^{-20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(delta \cdot \sin theta\right)}{1 - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	7.2
Cost	32512

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

Alternative 11
Error	8.0
Cost	26568

\[\begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ \mathbf{if}\;theta \leq -11.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{elif}\;theta \leq 2 \cdot 10^{-224}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\left(-1 - \left(-\cos delta\right)\right) + 1}\\ \end{array} \]

Alternative 12
Error	8.0
Cost	26568

\[\begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ \mathbf{if}\;theta \leq -11.5:\\ \;\;\;\;\left(\lambda_1 + \lambda_1\right) - \left(\left(-\tan^{-1}_* \frac{t_1}{\cos delta}\right) + \lambda_1\right)\\ \mathbf{elif}\;theta \leq 4 \cdot 10^{-223}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\left(-1 - \left(-\cos delta\right)\right) + 1}\\ \end{array} \]

Alternative 13
Error	8.0
Cost	26376

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;theta \leq -15.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 10^{-226}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	8.5
Cost	25984

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

Alternative 15
Error	12.6
Cost	20168

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{if}\;theta \leq -1.1 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 3.5 \cdot 10^{-101}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\left(-1 - \left(-\cos delta\right)\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	13.2
Cost	19848

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1}\\ \mathbf{if}\;theta \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 2.8 \cdot 10^{-101}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	13.0
Cost	19848

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{if}\;delta \leq -4.4 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 1.02 \cdot 10^{+153}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	12.6
Cost	19848

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1}\\ \mathbf{if}\;theta \leq -1.1 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 3.9 \cdot 10^{-101}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	18.8
Cost	13448

\[\begin{array}{l} \mathbf{if}\;theta \leq 7.5 \cdot 10^{-236}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;theta \leq 5.4 \cdot 10^{+88}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 20
Error	18.1
Cost	13448

\[\begin{array}{l} \mathbf{if}\;theta \leq -1.28 \cdot 10^{+79}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;theta \leq 5.4 \cdot 10^{+88}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 21
Error	15.7
Cost	13448

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1}\\ \mathbf{if}\;theta \leq -1.3 \cdot 10^{-43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 2.1 \cdot 10^{-126}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Error	18.9
Cost	64

\[\lambda_1 \]

Reproduce

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