Destination given bearing on a great circle

Average Error: 0.2 → 0.1

Time: 32.1s

Precision: binary64

Cost: 71680

Math

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

↓

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

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)))
   (-
    (* (* (cos phi1) (cos phi1)) (cos delta))
    (* (cos theta) (* (cos phi1) (* (sin delta) (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))), (((cos(phi1) * cos(phi1)) * cos(delta)) - (cos(theta) * (cos(phi1) * (sin(delta) * 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))), (((cos(phi1) * cos(phi1)) * cos(delta)) - (cos(theta) * (cos(phi1) * (sin(delta) * 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))), (((Math.cos(phi1) * Math.cos(phi1)) * Math.cos(delta)) - (Math.cos(theta) * (Math.cos(phi1) * (Math.sin(delta) * 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))), (((math.cos(phi1) * math.cos(phi1)) * math.cos(delta)) - (math.cos(theta) * (math.cos(phi1) * (math.sin(delta) * 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(cos(phi1) * cos(phi1)) * cos(delta)) - Float64(cos(theta) * Float64(cos(phi1) * Float64(sin(delta) * 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))), (((cos(phi1) * cos(phi1)) * cos(delta)) - (cos(theta) * (cos(phi1) * (sin(delta) * 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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[theta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $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. Applied egg-rr 0.2

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

3. Taylor expanded in delta around inf 0.2

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

4. Simplified 0.2

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \color{blue}{{\sin \phi_1}^{2} \cdot \cos delta}\right)} \]
   Proof
   (*.f64 (pow.f64 (sin.f64 phi1) 2) (cos.f64 delta)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))): 0 points increase in error, 0 points decrease in error

5. Taylor expanded in delta around inf 0.2

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

6. Simplified 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \cos \phi_1 \cdot \left(\sin delta \cdot \left(\sin \phi_1 \cdot \cos theta\right)\right)}} \]
   Proof
   (-.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi1)) (cos.f64 delta)) (*.f64 (cos.f64 phi1) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (cos.f64 theta))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (Rewrite<= 1-sub-sin_binary64 (-.f64 1 (*.f64 (sin.f64 phi1) (sin.f64 phi1)))) (cos.f64 delta)) (*.f64 (cos.f64 phi1) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (cos.f64 theta))))): 40 points increase in error, 38 points decrease in error
   (-.f64 (*.f64 (-.f64 1 (Rewrite<= unpow2_binary64 (pow.f64 (sin.f64 phi1) 2))) (cos.f64 delta)) (*.f64 (cos.f64 phi1) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (cos.f64 theta))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (Rewrite=> sub-neg_binary64 (+.f64 1 (neg.f64 (pow.f64 (sin.f64 phi1) 2)))) (cos.f64 delta)) (*.f64 (cos.f64 phi1) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (cos.f64 theta))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (pow.f64 (sin.f64 phi1) 2)) 1)) (cos.f64 delta)) (*.f64 (cos.f64 phi1) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (cos.f64 theta))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= distribute-rgt1-in_binary64 (+.f64 (cos.f64 delta) (*.f64 (neg.f64 (pow.f64 (sin.f64 phi1) 2)) (cos.f64 delta)))) (*.f64 (cos.f64 phi1) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (cos.f64 theta))))): 16 points increase in error, 13 points decrease in error
   (-.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (cos.f64 delta) (*.f64 (pow.f64 (sin.f64 phi1) 2) (cos.f64 delta)))) (*.f64 (cos.f64 phi1) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (cos.f64 theta))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)))) (*.f64 (cos.f64 phi1) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (cos.f64 theta))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))) (*.f64 (cos.f64 phi1) (*.f64 (sin.f64 delta) (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 theta) (sin.f64 phi1)))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (*.f64 (cos.f64 theta) (sin.f64 phi1))))): 7 points increase in error, 5 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 delta) (cos.f64 phi1))) (*.f64 (cos.f64 theta) (sin.f64 phi1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))) (Rewrite=> associate-*l*_binary64 (*.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (*.f64 (cos.f64 theta) (sin.f64 phi1)))))): 1 points increase in error, 8 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))) (*.f64 (sin.f64 delta) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 theta)) (sin.f64 phi1))))): 6 points increase in error, 1 points decrease in error
   (-.f64 (-.f64 (cos.f64 delta) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))) (*.f64 (sin.f64 delta) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi1) (cos.f64 theta)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--r+_binary64 (-.f64 (cos.f64 delta) (+.f64 (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)) (*.f64 (sin.f64 delta) (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi1) (cos.f64 theta))))))): 10 points increase in error, 17 points decrease in error
   (-.f64 (cos.f64 delta) (+.f64 (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)) (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (sin.f64 phi1) (*.f64 (cos.f64 phi1) (cos.f64 theta))) (sin.f64 delta))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (cos.f64 delta) (+.f64 (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)) (Rewrite=> associate-*l*_binary64 (*.f64 (sin.f64 phi1) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 theta)) (sin.f64 delta)))))): 3 points increase in error, 1 points decrease in error
   (-.f64 (cos.f64 delta) (+.f64 (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2)) (*.f64 (sin.f64 phi1) (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta))))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (cos.f64 delta) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (sin.f64 phi1) (*.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta)))) (*.f64 (cos.f64 delta) (pow.f64 (sin.f64 phi1) 2))))): 0 points increase in error, 0 points decrease in error

7. Taylor expanded in phi1 around inf 0.1

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

8. Simplified 0.1

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \color{blue}{\cos theta \cdot \left(\left(\sin \phi_1 \cdot \sin delta\right) \cdot \cos \phi_1\right)}} \]
   Proof
   (*.f64 (cos.f64 theta) (*.f64 (*.f64 (sin.f64 phi1) (sin.f64 delta)) (cos.f64 phi1))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 theta) (Rewrite<= associate-*r*_binary64 (*.f64 (sin.f64 phi1) (*.f64 (sin.f64 delta) (cos.f64 phi1))))): 19 points increase in error, 24 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sin.f64 phi1) (*.f64 (sin.f64 delta) (cos.f64 phi1))) (cos.f64 theta))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r*_binary64 (*.f64 (sin.f64 phi1) (*.f64 (*.f64 (sin.f64 delta) (cos.f64 phi1)) (cos.f64 theta)))): 23 points increase in error, 16 points decrease in error
   (*.f64 (sin.f64 phi1) (Rewrite<= associate-*r*_binary64 (*.f64 (sin.f64 delta) (*.f64 (cos.f64 phi1) (cos.f64 theta))))): 8 points increase in error, 7 points decrease in error

9. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	71616

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

Alternative 2
Error	0.1
Cost	65536

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

Alternative 3
Error	3.5
Cost	65088

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

Alternative 4
Error	4.0
Cost	59016

\[\begin{array}{l} t_1 := {\sin \phi_1}^{2}\\ t_2 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;theta \leq -62753833563323.24:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_2}{\cos delta - \mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right)}\\ \mathbf{elif}\;theta \leq 6.208798581852973 \cdot 10^{-53}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta - \sin \phi_1 \cdot \left(\sin delta \cdot \cos \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_2}{\cos delta - t_1}\\ \end{array} \]

Alternative 5
Error	3.6
Cost	59008

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

Alternative 6
Error	5.0
Cost	58304

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \mathsf{expm1}\left(\mathsf{log1p}\left({\sin \phi_1}^{2}\right)\right)} \]

Alternative 7
Error	5.0
Cost	51904

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

Alternative 8
Error	5.0
Cost	45504

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

Alternative 9
Error	5.0
Cost	39424

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

Alternative 10
Error	5.8
Cost	39240

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -1.1889821949133493 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 1.4672717185711938 \cdot 10^{-24}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	7.4
Cost	32512

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

Alternative 12
Error	8.0
Cost	26376

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;theta \leq -4.065642888028582 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;theta \leq 0.1692667107167209:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	8.9
Cost	25984

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

Alternative 14
Error	13.2
Cost	19848

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -3.645967852510931 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;delta \leq 9.490748390709758 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	19.7
Cost	19720

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.163421819364092 \cdot 10^{-278}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\lambda_1 \leq 2.2071446287752207 \cdot 10^{-168}:\\ \;\;\;\;\tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 16
Error	16.6
Cost	19716

\[\begin{array}{l} \mathbf{if}\;delta \leq 9.016670098993389 \cdot 10^{+71}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 17
Error	19.7
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2022308 
(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))))))))))