Destination given bearing on a great circle

Average Accuracy: 99.7% → 99.9%

Time: 28.9s

Precision: binary64

Cost: 305344

Math

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

↓

\[\begin{array}{l} t_1 := \cos theta \cdot \left(-\sin \phi_1\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot t_1\right)\\ t_3 := \mathsf{fma}\left(t_1, \sin delta, \sin \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)\\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{t_2}^{3} + {t_3}^{3}}{\frac{{t_2}^{2} + t_3 \cdot \left(t_3 - t_2\right)}{\cos \phi_1}}} \end{array} \]

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 (* (cos theta) (- (sin phi1))))
        (t_2 (fma (cos phi1) (cos delta) (* (sin delta) t_1)))
        (t_3 (fma t_1 (sin delta) (* (sin phi1) (* (sin delta) (cos theta))))))
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (/
      (+ (pow t_2 3.0) (pow t_3 3.0))
      (/ (+ (pow t_2 2.0) (* t_3 (- t_3 t_2))) (cos 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) {
	double t_1 = cos(theta) * -sin(phi1);
	double t_2 = fma(cos(phi1), cos(delta), (sin(delta) * t_1));
	double t_3 = fma(t_1, sin(delta), (sin(phi1) * (sin(delta) * cos(theta))));
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), ((pow(t_2, 3.0) + pow(t_3, 3.0)) / ((pow(t_2, 2.0) + (t_3 * (t_3 - t_2))) / cos(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)
	t_1 = Float64(cos(theta) * Float64(-sin(phi1)))
	t_2 = fma(cos(phi1), cos(delta), Float64(sin(delta) * t_1))
	t_3 = fma(t_1, sin(delta), Float64(sin(phi1) * Float64(sin(delta) * cos(theta))))
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(Float64((t_2 ^ 3.0) + (t_3 ^ 3.0)) / Float64(Float64((t_2 ^ 2.0) + Float64(t_3 * Float64(t_3 - t_2))) / cos(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_] := Block[{t$95$1 = N[(N[Cos[theta], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[Sin[delta], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Sin[delta], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] + N[Power[t$95$3, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[(t$95$3 * N[(t$95$3 - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 99.7%

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

2. Applied egg-rr 99.7%

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

3. Taylor expanded in delta around inf 99.7%

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

4. Simplified 99.9%

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

   [Start] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\cos delta + -1 \cdot \left(\cos delta \cdot {\sin \phi_1}^{2}\right)\right) + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   *-lft-identity [<=] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\color{blue}{1 \cdot \cos delta} + -1 \cdot \left(\cos delta \cdot {\sin \phi_1}^{2}\right)\right) + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   *-commutative [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(1 \cdot \cos delta + -1 \cdot \color{blue}{\left({\sin \phi_1}^{2} \cdot \cos delta\right)}\right) + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   associate-*r* [=>] 99.7	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(1 \cdot \cos delta + \color{blue}{\left(-1 \cdot {\sin \phi_1}^{2}\right) \cdot \cos delta}\right) + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   distribute-rgt-in [<=] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos delta \cdot \left(1 + -1 \cdot {\sin \phi_1}^{2}\right)} + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   mul-1-neg [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta \cdot \left(1 + \color{blue}{\left(-{\sin \phi_1}^{2}\right)}\right) + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   sub-neg [<=] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta \cdot \color{blue}{\left(1 - {\sin \phi_1}^{2}\right)} + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   *-commutative [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(1 - {\sin \phi_1}^{2}\right) \cdot \cos delta} + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   unpow2 [=>] 99.8	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(1 - \color{blue}{\sin \phi_1 \cdot \sin \phi_1}\right) \cdot \cos delta + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]
   1-sub-sin [=>] 99.9	\[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_1\right)} \cdot \cos delta + \left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(-\sin \phi_1\right)} \]

5. Taylor expanded in phi1 around inf 99.9%

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

6. Simplified 99.9%

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

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

7. Applied egg-rr 99.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{\cos \phi_1 \cdot \left({\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right)\right)}^{3} + {\left(\mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)}^{3}\right)}{{\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right)\right)}^{2} + \left(\mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) - \mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)}}} \]

8. Simplified 99.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\frac{{\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right)\right)}^{3} + {\left(\mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)\right)}^{3}}{\frac{{\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right)\right)}^{2} + \mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right) \cdot \left(\mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right) - \mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right)\right)}{\cos \phi_1}}}} \]
   Proof

   [Start] 99.9

   \[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\cos \phi_1 \cdot \left({\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right)\right)}^{3} + {\left(\mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)}^{3}\right)}{{\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right)\right)}^{2} + \left(\mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) - \mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)}} \]

   *-commutative [<=] 99.9

   \[ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{\color{blue}{\left({\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right)\right)}^{3} + {\left(\mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)}^{3}\right) \cdot \cos \phi_1}}{{\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right)\right)}^{2} + \left(\mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) - \mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\sin \phi_1 \cdot \left(-\cos theta\right)\right)\right) \cdot \mathsf{fma}\left(\sin \phi_1 \cdot \left(-\cos theta\right), \sin delta, \sin delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)}} \]

9. Final simplification 99.9%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\frac{{\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \left(-\sin \phi_1\right)\right)\right)\right)}^{3} + {\left(\mathsf{fma}\left(\cos theta \cdot \left(-\sin \phi_1\right), \sin delta, \sin \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right)\right)}^{3}}{\frac{{\left(\mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \left(-\sin \phi_1\right)\right)\right)\right)}^{2} + \mathsf{fma}\left(\cos theta \cdot \left(-\sin \phi_1\right), \sin delta, \sin \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \left(\mathsf{fma}\left(\cos theta \cdot \left(-\sin \phi_1\right), \sin delta, \sin \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) - \mathsf{fma}\left(\cos \phi_1, \cos delta, \sin delta \cdot \left(\cos theta \cdot \left(-\sin \phi_1\right)\right)\right)\right)}{\cos \phi_1}}} \]

Alternatives

Alternative 1
Accuracy	99.9%
Cost	65152

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

Alternative 2
Accuracy	94.5%
Cost	58624

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

Alternative 3
Accuracy	91.6%
Cost	45576

\[\begin{array}{l} t_1 := \sin theta \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -0.0037:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot t_1}{\cos \phi_1 \cdot \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin delta \cdot t_1\right)\right)}{\cos delta}\\ \end{array} \]

Alternative 4
Accuracy	92.0%
Cost	45504

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

Alternative 5
Accuracy	91.6%
Cost	32905

\[\begin{array}{l} \mathbf{if}\;delta \leq -0.0105 \lor \neg \left(delta \leq 10^{-21}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos \phi_1 \cdot \cos \phi_1}\\ \end{array} \]

Alternative 6
Accuracy	86.0%
Cost	32780

\[\begin{array}{l} t_1 := \sin theta \cdot \sin delta\\ \mathbf{if}\;\phi_1 \leq -2.4 \cdot 10^{+75}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.75 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1 \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)}{\cos delta}\\ \mathbf{elif}\;\phi_1 \leq 5 \cdot 10^{+123}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left|t_1\right|}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left|\sin delta\right|}{\cos delta}\\ \end{array} \]

Alternative 7
Accuracy	91.6%
Cost	32777

\[\begin{array}{l} \mathbf{if}\;delta \leq -0.0037 \lor \neg \left(delta \leq 1.1 \cdot 10^{-21}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \end{array} \]

Alternative 8
Accuracy	87.7%
Cost	26760

\[\begin{array}{l} \mathbf{if}\;delta \leq -1.2:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.65 \cdot 10^{-21}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 9
Accuracy	85.7%
Cost	26377

\[\begin{array}{l} \mathbf{if}\;theta \leq -8.2 \cdot 10^{-7} \lor \neg \left(theta \leq 1.15 \cdot 10^{+84}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]

Alternative 10
Accuracy	85.9%
Cost	25984

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

Alternative 11
Accuracy	79.5%
Cost	19849

\[\begin{array}{l} \mathbf{if}\;theta \leq -6.1 \cdot 10^{-80} \lor \neg \left(theta \leq 2.5 \cdot 10^{+22}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]

Alternative 12
Accuracy	76.4%
Cost	19848

\[\begin{array}{l} \mathbf{if}\;theta \leq -3200:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;theta \leq 9 \cdot 10^{+31}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 13
Accuracy	69.3%
Cost	13448

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{+97}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq -3.9 \cdot 10^{-107}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]

Alternative 14
Accuracy	69.9%
Cost	64

\[\lambda_1 \]

Reproduce

herbie shell --seed 2023129 
(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))))))))))