Destination given bearing on a great circle

Average Error: 0.1 → 0.1

Time: 33.6s

Precision: binary64

Cost: 72064

Math

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

↓

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

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
   (* (sin theta) (* (sin delta) (cos phi1)))
   (+
    (cos delta)
    (-
     (* (/ (cos delta) 2.0) (+ (cos (+ phi1 phi1)) -1.0))
     (* (cos phi1) (* (sin delta) (* (sin phi1) (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((sin(theta) * (sin(delta) * cos(phi1))), (cos(delta) + (((cos(delta) / 2.0) * (cos((phi1 + phi1)) + -1.0)) - (cos(phi1) * (sin(delta) * (sin(phi1) * 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((sin(theta) * (sin(delta) * cos(phi1))), (cos(delta) + (((cos(delta) / 2.0d0) * (cos((phi1 + phi1)) + (-1.0d0))) - (cos(phi1) * (sin(delta) * (sin(phi1) * 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.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), (Math.cos(delta) + (((Math.cos(delta) / 2.0) * (Math.cos((phi1 + phi1)) + -1.0)) - (Math.cos(phi1) * (Math.sin(delta) * (Math.sin(phi1) * 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.sin(theta) * (math.sin(delta) * math.cos(phi1))), (math.cos(delta) + (((math.cos(delta) / 2.0) * (math.cos((phi1 + phi1)) + -1.0)) - (math.cos(phi1) * (math.sin(delta) * (math.sin(phi1) * 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(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(cos(delta) + Float64(Float64(Float64(cos(delta) / 2.0) * Float64(cos(Float64(phi1 + phi1)) + -1.0)) - Float64(cos(phi1) * Float64(sin(delta) * Float64(sin(phi1) * 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((sin(theta) * (sin(delta) * cos(phi1))), (cos(delta) + (((cos(delta) / 2.0) * (cos((phi1 + phi1)) + -1.0)) - (cos(phi1) * (sin(delta) * (sin(phi1) * 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[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] + N[(N[(N[(N[Cos[delta], $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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

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

   [Start] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   associate-*l* [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   cancel-sign-sub-inv [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta + \left(-\sin \phi_1\right) \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}} \]
   +-commutative [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\left(-\sin \phi_1\right) \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) + \cos delta}} \]
   *-commutative [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \left(-\sin \phi_1\right)} + \cos delta} \]
   fma-def [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\mathsf{fma}\left(\sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right), -\sin \phi_1, \cos delta\right)}} \]

3. Taylor expanded in delta around inf 0.1

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

4. Simplified 0.1

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

   [Start] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta + -1 \cdot \left(\left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \sin \phi_1\right)} \]
   mul-1-neg [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta + \color{blue}{\left(-\left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \sin \phi_1\right)}} \]
   *-commutative [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta + \left(-\color{blue}{\sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)}\right)} \]
   +-commutative [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta + \left(-\sin \phi_1 \cdot \color{blue}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \cos delta \cdot \sin \phi_1\right)}\right)} \]
   fma-def [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta + \left(-\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)}\right)} \]
   unsub-neg [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\color{blue}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin delta, \cos \phi_1 \cdot \cos theta, \cos delta \cdot \sin \phi_1\right)}} \]
   fma-def [<=] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \color{blue}{\left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \cos delta \cdot \sin \phi_1\right)}} \]
   +-commutative [<=] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \color{blue}{\left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)}} \]
   *-commutative [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \left(\color{blue}{\sin \phi_1 \cdot \cos delta} + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)} \]

5. Applied egg-rr 0.1

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

6. Applied egg-rr 0.1

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

7. Simplified 0.1

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

   [Start] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right) \cdot \sin \phi_1 + \frac{\cos delta \cdot \left(\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)\right)}{2}\right)} \]
   associate-/l* [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right) \cdot \sin \phi_1 + \color{blue}{\frac{\cos delta}{\frac{2}{\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)}}}\right)} \]
   associate-/r/ [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right) \cdot \sin \phi_1 + \color{blue}{\frac{\cos delta}{2} \cdot \left(\cos \left(\phi_1 - \phi_1\right) - \cos \left(\phi_1 + \phi_1\right)\right)}\right)} \]
   +-inverses [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right) \cdot \sin \phi_1 + \frac{\cos delta}{2} \cdot \left(\cos \color{blue}{0} - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]
   cos-0 [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right) \cdot \sin \phi_1 + \frac{\cos delta}{2} \cdot \left(\color{blue}{1} - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]

8. Taylor expanded in phi1 around inf 0.1

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

9. Simplified 0.1

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

   [Start] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\sin \phi_1 \cdot \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) + \frac{\cos delta}{2} \cdot \left(1 - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]
   associate-*r* [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\color{blue}{\left(\sin \phi_1 \cdot \sin delta\right) \cdot \left(\cos \phi_1 \cdot \cos theta\right)} + \frac{\cos delta}{2} \cdot \left(1 - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]
   *-commutative [<=] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\color{blue}{\left(\sin delta \cdot \sin \phi_1\right)} \cdot \left(\cos \phi_1 \cdot \cos theta\right) + \frac{\cos delta}{2} \cdot \left(1 - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]
   *-commutative [<=] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\color{blue}{\left(\cos \phi_1 \cdot \cos theta\right) \cdot \left(\sin delta \cdot \sin \phi_1\right)} + \frac{\cos delta}{2} \cdot \left(1 - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]
   associate-*r* [<=] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\color{blue}{\cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)} + \frac{\cos delta}{2} \cdot \left(1 - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]
   *-commutative [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\cos \phi_1 \cdot \color{blue}{\left(\left(\sin delta \cdot \sin \phi_1\right) \cdot \cos theta\right)} + \frac{\cos delta}{2} \cdot \left(1 - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]
   associate-*l* [=>] 0.1	\[ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\cos \phi_1 \cdot \color{blue}{\left(\sin delta \cdot \left(\sin \phi_1 \cdot \cos theta\right)\right)} + \frac{\cos delta}{2} \cdot \left(1 - \cos \left(\phi_1 + \phi_1\right)\right)\right)} \]

10. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
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	3.2
Cost	65536

\[\begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t_1}{\cos delta + \left(\frac{\cos delta}{2} \cdot \left(\cos \left(\phi_1 + \phi_1\right) + -1\right) - t_1 \cdot \sin \phi_1\right)} \end{array} \]

Alternative 3
Error	3.3
Cost	65289

\[\begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \mathbf{if}\;theta \leq -7.5 \cdot 10^{-8} \lor \neg \left(theta \leq 0.0062\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t_1}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, t_1\right)}\\ \end{array} \]

Alternative 4
Error	3.3
Cost	65289

\[\begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \mathbf{if}\;theta \leq -8.2 \cdot 10^{-8} \lor \neg \left(theta \leq 0.0062\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t_1}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{\cos delta - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \phi_1, \cos delta, t_1\right)}\\ \end{array} \]

Alternative 5
Error	3.2
Cost	65152

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

Alternative 6
Error	4.7
Cost	45504

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

Alternative 7
Error	6.4
Cost	39436

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{if}\;\phi_1 \leq -13.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 31000000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta - \sin delta \cdot \phi_1}\\ \mathbf{elif}\;\phi_1 \leq 1.42 \cdot 10^{+59}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{\sin delta}{\frac{2}{\sin \left(theta - \phi_1\right) + \sin \left(theta + \phi_1\right)}}}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	6.4
Cost	39436

\[\begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{if}\;\phi_1 \leq -13.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 33000000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta - \sin delta \cdot \left(\phi_1 \cdot \cos theta\right)}\\ \mathbf{elif}\;\phi_1 \leq 4.5 \cdot 10^{+58}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{\sin delta}{\frac{2}{\sin \left(theta - \phi_1\right) + \sin \left(theta + \phi_1\right)}}}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	5.2
Cost	32905

\[\begin{array}{l} t_1 := \sin theta \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -6000000000000 \lor \neg \left(delta \leq 4.15 \cdot 10^{-50}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot t_1}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot t_1}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \]

Alternative 10
Error	6.8
Cost	32512

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

Alternative 11
Error	8.1
Cost	25984

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

Alternative 12
Error	12.9
Cost	19849

\[\begin{array}{l} \mathbf{if}\;delta \leq -5.2 \cdot 10^{-110} \lor \neg \left(delta \leq 1040000000\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \]

Alternative 13
Error	14.5
Cost	19848

\[\begin{array}{l} \mathbf{if}\;theta \leq -260000000000:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;theta \leq 3.3 \cdot 10^{+85}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 14
Error	18.3
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.3 \cdot 10^{-190}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\lambda_1 \leq 1.2 \cdot 10^{-242}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 15
Error	18.5
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023016 
(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))))))))))