Destination given bearing on a great circle

Average Error: 0.2 → 0.2

Time: 27.2s

Precision: binary64

Cost: 77952

Math

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

↓

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

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
 (let* ((t_1 (* (sin delta) (cos phi1))))
   (+
    lambda1
    (atan2
     (* (sin theta) t_1)
     (-
      (cos delta)
      (* (sin phi1) (fma (sin phi1) (cos delta) (* t_1 (cos theta)))))))))

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) {
	double t_1 = sin(delta) * cos(phi1);
	return lambda1 + atan2((sin(theta) * t_1), (cos(delta) - (sin(phi1) * fma(sin(phi1), cos(delta), (t_1 * cos(theta))))));
}

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)
	t_1 = Float64(sin(delta) * cos(phi1))
	return Float64(lambda1 + atan(Float64(sin(theta) * t_1), Float64(cos(delta) - Float64(sin(phi1) * fma(sin(phi1), cos(delta), Float64(t_1 * cos(theta)))))))
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_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(t$95$1 * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)}} \]
   Proof
   (+.f64 lambda1 (atan2.f64 (*.f64 (sin.f64 theta) (*.f64 (sin.f64 delta) (cos.f64 phi1))) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (fma.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta)) (*.f64 (cos.f64 delta) (sin.f64 phi1))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 lambda1 (atan2.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1))) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (fma.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta)) (*.f64 (cos.f64 delta) (sin.f64 phi1))))))))): 3 points increase in error, 2 points decrease in error
   (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (sin.f64 phi1)))) (sin.f64 (asin.f64 (fma.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta)) (*.f64 (cos.f64 delta) (sin.f64 phi1))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (neg.f64 (neg.f64 (sin.f64 phi1))) (sin.f64 (asin.f64 (fma.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta)) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 phi1) (cos.f64 delta)))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (neg.f64 (neg.f64 (sin.f64 phi1))) (sin.f64 (asin.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta))) (*.f64 (sin.f64 phi1) (cos.f64 delta)))))))))): 1 points increase in error, 0 points decrease in error
   (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (neg.f64 (neg.f64 (sin.f64 phi1))) (sin.f64 (asin.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (sin.f64 delta) (cos.f64 phi1)) (cos.f64 theta))) (*.f64 (sin.f64 phi1) (cos.f64 delta))))))))): 0 points increase in error, 1 points decrease in error
   (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (neg.f64 (neg.f64 (sin.f64 phi1))) (sin.f64 (asin.f64 (+.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 phi1) (sin.f64 delta))) (cos.f64 theta)) (*.f64 (sin.f64 phi1) (cos.f64 delta))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (neg.f64 (neg.f64 (sin.f64 phi1))) (sin.f64 (asin.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (Rewrite=> cancel-sign-sub_binary64 (+.f64 (cos.f64 delta) (*.f64 (neg.f64 (sin.f64 phi1)) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in delta around inf 0.2

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

4. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}} \]
   Proof
   (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (fma.f64 (sin.f64 phi1) (cos.f64 delta) (*.f64 (cos.f64 theta) (*.f64 (cos.f64 phi1) (sin.f64 delta)))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (fma.f64 (sin.f64 phi1) (cos.f64 delta) (*.f64 (cos.f64 theta) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 delta) (cos.f64 phi1))))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (fma.f64 (sin.f64 phi1) (cos.f64 delta) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 theta) (sin.f64 delta)) (cos.f64 phi1)))))): 2 points increase in error, 1 points decrease in error
   (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (fma.f64 (sin.f64 phi1) (cos.f64 delta) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 delta) (cos.f64 theta))) (cos.f64 phi1))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (sin.f64 delta) (cos.f64 theta)) (cos.f64 phi1)))))): 3 points increase in error, 5 points decrease in error
   (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 delta) (sin.f64 phi1))) (*.f64 (*.f64 (sin.f64 delta) (cos.f64 theta)) (cos.f64 phi1))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (Rewrite<= fma-udef_binary64 (fma.f64 (cos.f64 delta) (sin.f64 phi1) (*.f64 (*.f64 (sin.f64 delta) (cos.f64 theta)) (cos.f64 phi1)))))): 5 points increase in error, 3 points decrease in error
   (-.f64 (cos.f64 delta) (Rewrite=> *-commutative_binary64 (*.f64 (fma.f64 (cos.f64 delta) (sin.f64 phi1) (*.f64 (*.f64 (sin.f64 delta) (cos.f64 theta)) (cos.f64 phi1))) (sin.f64 phi1)))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> cancel-sign-sub-inv_binary64 (+.f64 (cos.f64 delta) (*.f64 (neg.f64 (fma.f64 (cos.f64 delta) (sin.f64 phi1) (*.f64 (*.f64 (sin.f64 delta) (cos.f64 theta)) (cos.f64 phi1)))) (sin.f64 phi1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cos.f64 delta) (*.f64 (neg.f64 (fma.f64 (cos.f64 delta) (sin.f64 phi1) (Rewrite=> associate-*l*_binary64 (*.f64 (sin.f64 delta) (*.f64 (cos.f64 theta) (cos.f64 phi1)))))) (sin.f64 phi1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cos.f64 delta) (*.f64 (neg.f64 (fma.f64 (cos.f64 delta) (sin.f64 phi1) (*.f64 (sin.f64 delta) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 phi1) (cos.f64 theta)))))) (sin.f64 phi1))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cos.f64 delta) (*.f64 (neg.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 delta) (sin.f64 phi1)) (*.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta)))))) (sin.f64 phi1))): 2 points increase in error, 5 points decrease in error
   (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (cos.f64 delta) (*.f64 (+.f64 (*.f64 (cos.f64 delta) (sin.f64 phi1)) (*.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta)))) (sin.f64 phi1)))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	71680

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

Alternative 2
Error	3.1
Cost	71616

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

Alternative 3
Error	4.5
Cost	65152

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

Alternative 4
Error	3.1
Cost	65152

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

Alternative 5
Error	4.4
Cost	39424

\[\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta + \left(\frac{\cos \left(\phi_1 + \phi_1\right)}{2} + -0.5\right)} \]

Alternative 6
Error	4.5
Cost	39424

\[\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(\cos delta + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)\right) + -0.5} \]

Alternative 7
Error	4.8
Cost	39304

\[\begin{array}{l} t_1 := \sin theta \cdot \cos \phi_1\\ t_2 := \sin delta \cdot t_1\\ \mathbf{if}\;delta \leq -48000000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(3 \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \left(t_1 \cdot 0.3333333333333333\right)\right)\right)\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{-41}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_2}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_2}{\cos delta}\\ \end{array} \]

Alternative 8
Error	4.7
Cost	33288

\[\begin{array}{l} t_1 := \sin theta \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -1.9 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(3 \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \left(t_1 \cdot 0.3333333333333333\right)\right)\right)\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{-41}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \cos \phi_1\right)}{\cos delta + \left(\frac{\cos \left(\phi_1 + \phi_1\right)}{2} + -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot t_1}{\cos delta}\\ \end{array} \]

Alternative 9
Error	4.9
Cost	33160

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -48000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{-41}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	4.9
Cost	33160

\[\begin{array}{l} t_1 := \sin theta \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -48000000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(3 \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \left(t_1 \cdot 0.3333333333333333\right)\right)\right)\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{-41}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot t_1}{\cos delta}\\ \end{array} \]

Alternative 11
Error	6.8
Cost	32512

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

Alternative 12
Error	8.2
Cost	25984

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

Alternative 13
Error	14.7
Cost	19848

\[\begin{array}{l} \mathbf{if}\;theta \leq -1.65 \cdot 10^{+119}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;theta \leq 12200:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 14
Error	11.8
Cost	19848

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{if}\;theta \leq -0.005:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 29000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	18.7
Cost	19720

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.4 \cdot 10^{-186}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\lambda_1 \leq 9.5 \cdot 10^{-234}:\\ \;\;\;\;\tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 16
Error	18.2
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.8 \cdot 10^{-253}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\lambda_1 \leq 1.36 \cdot 10^{-265}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 17
Error	18.6
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2022332 
(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))))))))))