Destination given bearing on a great circle

Average Accuracy: 99.8% → 99.9%

Time: 40.9s

Precision: binary64

Cost: 71616

Math

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

↓

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

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 delta) (sin theta)))
   (-
    (* (pow (cos phi1) 2.0) (cos delta))
    (* (sin phi1) (* (* (cos phi1) (sin delta)) (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) {
	return lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), ((pow(cos(phi1), 2.0) * cos(delta)) - (sin(phi1) * ((cos(phi1) * sin(delta)) * cos(theta)))));
}

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(delta) * sin(theta))), (((cos(phi1) ** 2.0d0) * cos(delta)) - (sin(phi1) * ((cos(phi1) * sin(delta)) * cos(theta)))))
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(delta) * Math.sin(theta))), ((Math.pow(Math.cos(phi1), 2.0) * Math.cos(delta)) - (Math.sin(phi1) * ((Math.cos(phi1) * Math.sin(delta)) * Math.cos(theta)))));
}

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(delta) * math.sin(theta))), ((math.pow(math.cos(phi1), 2.0) * math.cos(delta)) - (math.sin(phi1) * ((math.cos(phi1) * math.sin(delta)) * math.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)
	return Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), Float64(Float64((cos(phi1) ^ 2.0) * cos(delta)) - Float64(sin(phi1) * Float64(Float64(cos(phi1) * sin(delta)) * cos(theta))))))
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(delta) * sin(theta))), (((cos(phi1) ^ 2.0) * cos(delta)) - (sin(phi1) * ((cos(phi1) * sin(delta)) * 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_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Simplified 99.8%

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

   [Start] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   associate-*l* [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   *-commutative [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1}} \]
   cancel-sign-sub-inv [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta + \left(-\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \phi_1}} \]
   cancel-sign-sub [<=] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta - \left(-\left(-\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1}} \]
   *-commutative [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\sin \phi_1 \cdot \left(-\left(-\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}} \]
   remove-double-neg [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \color{blue}{\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}} \]

3. Applied egg-rr 99.8%

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

   [Start] 99.8

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

   flip-- [=>] 99.8

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

4. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\frac{{\cos delta}^{2} - {\left(\sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)}^{2}}{\cos delta + \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)}} \]
   unpow2 [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\frac{\color{blue}{\cos delta \cdot \cos delta} - {\left(\sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)}^{2}}{\cos delta + \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)}} \]
   unpow2 [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\frac{\cos delta \cdot \cos delta - \color{blue}{\left(\sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right) \cdot \left(\sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)\right)}}{\cos delta + \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)}} \]
   flip-- [<=] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)}} \]
   fma-udef [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos delta + \sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right)}} \]
   distribute-rgt-in [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \color{blue}{\left(\left(\sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1 + \left(\sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right) \cdot \sin \phi_1\right)}} \]
   associate--r+ [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(\cos delta - \left(\sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1\right) - \left(\sin delta \cdot \left(\cos theta \cdot \cos \phi_1\right)\right) \cdot \sin \phi_1}} \]

5. Taylor expanded in delta around inf 99.8%

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

6. Simplified 99.9%

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

   [Start] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(\cos delta - \cos delta \cdot {\sin \phi_1}^{2}\right) - \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]
   *-commutative [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(\cos delta - \color{blue}{{\sin \phi_1}^{2} \cdot \cos delta}\right) - \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]
   cancel-sign-sub-inv [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(\cos delta + \left(-{\sin \phi_1}^{2}\right) \cdot \cos delta\right)} - \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]
   distribute-rgt1-in [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(\left(-{\sin \phi_1}^{2}\right) + 1\right) \cdot \cos delta} - \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]
   +-commutative [<=] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(1 + \left(-{\sin \phi_1}^{2}\right)\right)} \cdot \cos delta - \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]
   sub-neg [<=] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(1 - {\sin \phi_1}^{2}\right)} \cdot \cos delta - \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]
   unpow2 [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\left(1 - \color{blue}{\sin \phi_1 \cdot \sin \phi_1}\right) \cdot \cos delta - \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]
   1-sub-sin [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_1\right)} \cdot \cos delta - \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1} \]

7. Taylor expanded in theta around 0 99.9%

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

8. Simplified 99.9%

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

   [Start] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta \cdot {\cos \phi_1}^{2} - \sin \phi_1 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)} \]
   *-commutative [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}}{\cos delta \cdot {\cos \phi_1}^{2} - \sin \phi_1 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)} \]
   associate-*r* [<=] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}}{\cos delta \cdot {\cos \phi_1}^{2} - \sin \phi_1 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)} \]
   *-commutative [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \color{blue}{\left(\sin delta \cdot \sin theta\right)}}{\cos delta \cdot {\cos \phi_1}^{2} - \sin \phi_1 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)} \]
   *-commutative [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\color{blue}{{\cos \phi_1}^{2} \cdot \cos delta} - \sin \phi_1 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)} \]
   associate-*r* [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2} \cdot \cos delta - \sin \phi_1 \cdot \color{blue}{\left(\left(\sin delta \cdot \cos \phi_1\right) \cdot \cos theta\right)}} \]

9. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	65152

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

Alternative 2
Accuracy	92.1%
Cost	45504

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

Alternative 3
Accuracy	91.2%
Cost	39305

\[\begin{array}{l} \mathbf{if}\;delta \leq -62000000 \lor \neg \left(delta \leq 8.4 \cdot 10^{+18}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \]

Alternative 4
Accuracy	88.8%
Cost	32512

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

Alternative 5
Accuracy	86.4%
Cost	25984

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

Alternative 6
Accuracy	80.5%
Cost	19849

\[\begin{array}{l} \mathbf{if}\;delta \leq -0.0018 \lor \neg \left(delta \leq 7 \cdot 10^{+29}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \end{array} \]

Alternative 7
Accuracy	76.3%
Cost	19848

\[\begin{array}{l} \mathbf{if}\;theta \leq -1.32 \cdot 10^{+36}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;theta \leq 1.55 \cdot 10^{+122}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 8
Accuracy	69.9%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;theta \leq -2.1 \cdot 10^{-101}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;theta \leq 1.1 \cdot 10^{+145}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 9
Accuracy	69.9%
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023136 
(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))))))))))