Distance on a great circle

Average Error: 24.5 → 13.6

Time: 2.5min

Precision: binary64

Cost: 151552

\[ \begin{array}{c}[phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]

Math

\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

↓

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\cos \left(0.5 \cdot \phi_1\right)\right)\right)\right)}^{2} + t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1}}{\sqrt{1 - t_1}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt
     (+
      (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
      (*
       (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
       (sin (/ (- lambda1 lambda2) 2.0)))))
    (sqrt
     (-
      1.0
      (+
       (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
       (*
        (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
        (sin (/ (- lambda1 lambda2) 2.0))))))))))

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow
           (fma
            (cos (* 0.5 phi2))
            (sin (* 0.5 phi1))
            (* (sin (* 0.5 phi2)) (- (cos (* 0.5 phi1)))))
           2.0)
          (* t_0 (* (* (cos phi1) (cos phi2)) t_0)))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0))) * sin(((lambda1 - lambda2) / 2.0))))), sqrt((1.0 - (pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0))) * sin(((lambda1 - lambda2) / 2.0))))))));
}

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(fma(cos((0.5 * phi2)), sin((0.5 * phi1)), (sin((0.5 * phi2)) * -cos((0.5 * phi1)))), 2.0) + (t_0 * ((cos(phi1) * cos(phi2)) * t_0));
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))))), sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))) * sin(Float64(Float64(lambda1 - lambda2) / 2.0)))))))))
end

↓

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((fma(cos(Float64(0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(sin(Float64(0.5 * phi2)) * Float64(-cos(Float64(0.5 * phi1))))) ^ 2.0) + Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0)))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * (-N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 24.5

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Applied egg-rr 23.9

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied egg-rr 13.6

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Taylor expanded in phi1 around inf 13.6

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Simplified 13.6

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\cos \left(0.5 \cdot \phi_1\right)\right)\right)\right)}^{2}} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   Proof
   (pow.f64 (fma.f64 (cos.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (neg.f64 (cos.f64 (*.f64 1/2 phi1))))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (fma.f64 (cos.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1)))))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (cos.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1))) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1))))) 2): 21 points increase in error, 9 points decrease in error
   (pow.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 1/2 phi1)) (cos.f64 (*.f64 1/2 phi2)))) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1)))) 2): 0 points increase in error, 0 points decrease in error

6. Taylor expanded in phi1 around inf 13.6

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\cos \left(0.5 \cdot \phi_1\right)\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left(\color{blue}{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Simplified 13.6

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\cos \left(0.5 \cdot \phi_1\right)\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left(\color{blue}{{\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\cos \left(0.5 \cdot \phi_1\right)\right)\right)\right)}^{2}} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   Proof
   (pow.f64 (fma.f64 (cos.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (neg.f64 (cos.f64 (*.f64 1/2 phi1))))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (fma.f64 (cos.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1)) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1)))))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (cos.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1))) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1))))) 2): 21 points increase in error, 9 points decrease in error
   (pow.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 1/2 phi1)) (cos.f64 (*.f64 1/2 phi2)))) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1)))) 2): 0 points increase in error, 0 points decrease in error

8. Final simplification 13.6

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\cos \left(0.5 \cdot \phi_1\right)\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\cos \left(0.5 \cdot \phi_1\right)\right)\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

Alternatives

Alternative 1
Error	20.5
Cost	151432

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_3 := t_1 \cdot \left(t_0 \cdot t_1\right) + t_2\\ t_4 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_5 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{\sqrt{{\left(1 - \mathsf{fma}\left(t_0, t_4, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\right)}^{2}}}}\right)\\ \mathbf{if}\;t_1 \leq -0.01:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t_1 \leq 0.08:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + \cos \phi_2 \cdot t_4}}{\sqrt{1 - t_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 2
Error	20.9
Cost	145608

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_3 := t_1 + t_2\\ t_4 := \sqrt{1 - t_3}\\ t_5 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{if}\;t_0 \leq -0.06:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 + \left(\left(-0.5 + 0.5 \cdot \cos \left(2 \cdot t_5\right)\right) - t_1\right)}}\right)\\ \mathbf{elif}\;t_0 \leq 0.06:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \sqrt[3]{{\left({\sin t_5}^{2}\right)}^{3}}}}{t_4}\right)\\ \end{array} \]

Alternative 3
Error	20.9
Cost	145608

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_3 := t_1 + t_2\\ t_4 := \sqrt{1 - t_3}\\ t_5 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 + \left(\left(-0.5 + 0.5 \cdot \cos \left(2 \cdot t_5\right)\right) - t_1\right)}}\right)\\ \mathbf{elif}\;t_0 \leq 0.08:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{t_4}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \sqrt[3]{{\left({\sin t_5}^{2}\right)}^{3}}}}{t_4}\right)\\ \end{array} \]

Alternative 4
Error	13.6
Cost	145216

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \sin \left(0.5 \cdot \phi_2\right)\\ t_2 := \cos \left(0.5 \cdot \phi_1\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := t_3 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_3\right)\\ t_5 := \cos \left(0.5 \cdot \phi_2\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(t_5, t_0, t_1 \cdot \left(-t_2\right)\right)\right)}^{2} + t_4}}{\sqrt{1 - \left(t_4 + {\left(t_0 \cdot t_5 - t_1 \cdot t_2\right)}^{2}\right)}}\right) \end{array} \]

Alternative 5
Error	21.8
Cost	138696

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_3 := t_0 \cdot \left(t_1 \cdot t_0\right) + t_2\\ t_4 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_5 := \sqrt{1 - t_3}\\ \mathbf{if}\;\lambda_2 \leq -1.20752015092965 \cdot 10^{-214}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{\sqrt{{\left(1 - \mathsf{fma}\left(t_1, t_4, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\right)}^{2}}}}\right)\\ \mathbf{elif}\;\lambda_2 \leq 3.899859094326751 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + \cos \phi_1 \cdot t_4}}{t_5}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_1 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{t_5}\right)\\ \end{array} \]

Alternative 6
Error	20.8
Cost	138564

\[\begin{array}{l} t_0 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := \sqrt{1 - \left(t_1 \cdot \left(t_2 \cdot t_1\right) + t_0\right)}\\ \mathbf{if}\;\lambda_1 \leq -4.3856184526737666 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)}}{t_3}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{t_3}\right)\\ \end{array} \]

Alternative 7
Error	19.8
Cost	138564

\[\begin{array}{l} t_0 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := t_1 \cdot \left(t_2 \cdot t_1\right) + t_0\\ \mathbf{if}\;\lambda_1 \leq -2.0749004590653286 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)}}{\sqrt{1 - t_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 - \left(t_0 + t_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternative 8
Error	13.6
Cost	138560

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left(t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + t_1\right)}}\right) \end{array} \]

Alternative 9
Error	23.9
Cost	119296

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\cos \left(0.5 \cdot \phi_1\right)\right)\right)\right)}^{2} + t_1}}{\sqrt{1 + \left(\left(-0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - t_1\right)}}\right) \end{array} \]

Alternative 10
Error	23.9
Cost	112960

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 + \left(\left(-0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - t_1\right)}}\right) \end{array} \]

Alternative 11
Error	25.3
Cost	106696

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)\\ t_3 := t_2 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + \left(0.5 + -0.5 \cdot \cos \left(2 \cdot t_0\right)\right)}}{\sqrt{1 - t_3}}\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_1 \leq -1 \cdot 10^{-291}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 - {\sin t_0}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 12
Error	30.7
Cost	99592

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + \left(0.5 + -0.5 \cdot \cos \left(2 \cdot t_0\right)\right)}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \mathbf{if}\;t_1 \leq -0.01:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_1 \leq 0.071:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin t_0}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 13
Error	36.2
Cost	99144

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}}\right)\\ \mathbf{if}\;t_0 \leq -0.05:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_0 \leq 0.071:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	36.4
Cost	99080

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := \sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ t_3 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ \mathbf{if}\;t_0 \leq -0.05:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\frac{1 + \cos \left(\lambda_2 - \lambda_1\right)}{2}}}\right)\\ \mathbf{elif}\;t_0 \leq 0.071:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{1 - t_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \left(e^{\mathsf{log1p}\left(t_3\right)} + -1\right)}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 15
Error	36.4
Cost	99016

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := \sqrt{t_1 \cdot \left(t_0 \cdot t_1\right) + t_2}\\ \mathbf{if}\;t_1 \leq -0.05:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{\frac{1 + \cos \left(\lambda_2 - \lambda_1\right)}{2}}}\right)\\ \mathbf{elif}\;t_1 \leq 0.071:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_1 \cdot \left(t_0 \cdot \sqrt{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 16
Error	36.4
Cost	98952

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := \sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ t_3 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ \mathbf{if}\;t_0 \leq -0.05:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\frac{1 + \cos \left(\lambda_2 - \lambda_1\right)}{2}}}\right)\\ \mathbf{elif}\;t_0 \leq 0.071:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{1 - t_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \log \left(e^{t_3}\right)}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 17
Error	24.5
Cost	93120

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 + \left(\left(-0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - t_1\right)}}\right) \end{array} \]

Alternative 18
Error	24.2
Cost	92228

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq 6.084424075701019 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_2}}\right)\\ \end{array} \]

Alternative 19
Error	24.2
Cost	92228

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ t_2 := \sqrt{t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;\phi_2 \leq 6.084424075701019 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_1}}\right)\\ \end{array} \]

Alternative 20
Error	28.5
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\\ \mathbf{if}\;\phi_2 \leq 0.0001788308702225537:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot t_1 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \sin \left(\lambda_2 \cdot -0.5\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 21
Error	25.1
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{if}\;\phi_2 \leq 6.084424075701019 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 22
Error	36.4
Cost	86152

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := \sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ t_3 := 0.5 \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{if}\;t_0 \leq -0.05:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\frac{1 + \cos \left(\lambda_2 - \lambda_1\right)}{2}}}\right)\\ \mathbf{elif}\;t_0 \leq 0.071:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{1 - {\sin t_3}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \left(0.5 + -0.5 \cdot \cos \left(2 \cdot t_3\right)\right)}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 23
Error	43.6
Cost	66116

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;\lambda_1 \leq -5031.474308875478:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_2}}\right)\\ \end{array} \]

Alternative 24
Error	42.0
Cost	66112

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{\frac{1 + \cos \left(\lambda_2 - \lambda_1\right)}{2}}}\right) \end{array} \]

Alternative 25
Error	45.1
Cost	65984

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_2}}\right) \end{array} \]

Reproduce

herbie shell --seed 2022300 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))