Destination given bearing on a great circle

Average Accuracy: 99.8% → 99.9%

Time: 35.4s

Precision: binary64

Cost: 71680

Math

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

↓

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

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 delta) (sin theta)))
   (-
    (* (* (cos phi1) (cos 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(delta) * sin(theta))), (((cos(phi1) * cos(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(delta) * sin(theta))), (((cos(phi1) * cos(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(delta) * Math.sin(theta))), (((Math.cos(phi1) * Math.cos(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(delta) * math.sin(theta))), (((math.cos(phi1) * math.cos(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(delta) * sin(theta))), Float64(Float64(Float64(cos(phi1) * cos(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(delta) * sin(theta))), (((cos(phi1) * cos(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[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi1], $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 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%

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

   [Start] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   *-commutative [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   associate-*r* [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   *-commutative [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   *-commutative [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta - \color{blue}{\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1}} \]
   cancel-sign-sub-inv [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\color{blue}{\cos delta + \left(-\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right) \cdot \sin \phi_1}} \]
   cancel-sign-sub [<=] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\color{blue}{\cos delta - \left(-\left(-\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right) \cdot \sin \phi_1}} \]
   *-commutative [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta - \color{blue}{\sin \phi_1 \cdot \left(-\left(-\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)\right)}} \]

3. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)} \]
   cancel-sign-sub-inv [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\color{blue}{\cos delta + \left(-\sin \phi_1\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right)}} \]
   +-commutative [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\color{blue}{\left(-\sin \phi_1\right) \cdot \sin \sin^{-1} \left(\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)\right) + \cos delta}} \]
   sin-asin [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\left(-\sin \phi_1\right) \cdot \color{blue}{\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)} + \cos delta} \]
   fma-udef [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\left(-\sin \phi_1\right) \cdot \color{blue}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \cos delta \cdot \sin \phi_1\right)} + \cos delta} \]
   distribute-rgt-in [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\color{blue}{\left(\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos delta \cdot \sin \phi_1\right) \cdot \left(-\sin \phi_1\right)\right)} + \cos delta} \]
   associate-+l+ [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\color{blue}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\left(\cos delta \cdot \sin \phi_1\right) \cdot \left(-\sin \phi_1\right) + \cos delta\right)}} \]

4. Taylor expanded in delta around inf 99.8%

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

5. Simplified 99.9%

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

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

6. Applied egg-rr 85.5%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)} - 1}}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]
   Proof

   [Start] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]
   expm1-log1p-u [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]
   expm1-udef [=>] 85.5	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)} - 1}}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]

7. Simplified 99.9%

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

   [Start] 85.5	\[ \lambda_1 + \tan^{-1}_* \frac{e^{\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)} - 1}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]
   expm1-def [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\right)\right)}}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]
   expm1-log1p [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]
   associate-*r* [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\sin delta \cdot \cos \phi_1\right) \cdot \sin theta}}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]
   *-commutative [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\cos \phi_1 \cdot \sin delta\right)} \cdot \sin theta}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]
   associate-*l* [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right) + \left(\cos \phi_1 \cdot \cos \phi_1\right) \cdot \cos delta} \]

8. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	99.9%
Cost	71616

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

Alternative 2
Accuracy	94.7%
Cost	65152

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

Alternative 3
Accuracy	94.7%
Cost	65152

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

Alternative 4
Accuracy	94.8%
Cost	65152

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

Alternative 5
Accuracy	91.9%
Cost	45576

\[\begin{array}{l} t_1 := \cos \phi_1 \cdot \sin theta\\ t_2 := \sin delta \cdot t_1\\ \mathbf{if}\;delta \leq -0.000315:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_2}{\cos delta}\\ \mathbf{elif}\;delta \leq 3.9 \cdot 10^{-25}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_2}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)}{\cos delta}\\ \end{array} \]

Alternative 6
Accuracy	92.4%
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 7
Accuracy	92.0%
Cost	39305

\[\begin{array}{l} t_1 := \sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)\\ \mathbf{if}\;delta \leq -1.5 \cdot 10^{-5} \lor \neg \left(delta \leq 3.9 \cdot 10^{-25}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \]

Alternative 8
Accuracy	86.5%
Cost	32648

\[\begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ \mathbf{if}\;\phi_1 \leq 1.25 \cdot 10^{-14}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos delta}\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{+243}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left|\sin theta\right|}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left|t_1\right|}{\cos delta}\\ \end{array} \]

Alternative 9
Accuracy	88.7%
Cost	32512

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

Alternative 10
Accuracy	87.2%
Cost	26377

\[\begin{array}{l} \mathbf{if}\;theta \leq -4 \cdot 10^{-126} \lor \neg \left(theta \leq 1.05 \cdot 10^{+26}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot theta\right)}{\cos delta}\\ \end{array} \]

Alternative 11
Accuracy	86.5%
Cost	25984

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

Alternative 12
Accuracy	76.8%
Cost	19849

\[\begin{array}{l} \mathbf{if}\;delta \leq -4.3 \cdot 10^{-66} \lor \neg \left(delta \leq 2.05 \cdot 10^{-78}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 13
Accuracy	80.3%
Cost	19849

\[\begin{array}{l} \mathbf{if}\;theta \leq -1.35 \cdot 10^{+34} \lor \neg \left(theta \leq 1.3 \cdot 10^{-47}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]

Alternative 14
Accuracy	69.0%
Cost	19720

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.7 \cdot 10^{-260}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\lambda_1 \leq 1.6 \cdot 10^{-123}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 15
Accuracy	70.1%
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023135 
(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))))))))))