Distance on a great circle

Average Error: 24.2 → 13.6

Time: 1.5min

Precision: binary64

Cost: 164288

Math

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

↓

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

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 (* phi1 0.5)))
        (t_1 (sin (* 0.5 phi2)))
        (t_2 (cos (* phi1 0.5)))
        (t_3 (* (cos phi1) (cos phi2)))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5 (cos (* 0.5 phi2))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (+ (pow (- (* t_5 t_0) (* t_1 t_2)) 2.0) (* t_3 (* t_4 t_4))))
      (sqrt
       (-
        (- 1.0 (pow (fma t_0 t_5 (* t_2 (- t_1))) 2.0))
        (*
         t_3
         (/
          (+
           (fma (cos lambda2) (cos lambda1) -1.0)
           (* (sin lambda2) (sin lambda1)))
          -2.0)))))))))

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((phi1 * 0.5));
	double t_1 = sin((0.5 * phi2));
	double t_2 = cos((phi1 * 0.5));
	double t_3 = cos(phi1) * cos(phi2);
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = cos((0.5 * phi2));
	return R * (2.0 * atan2(sqrt((pow(((t_5 * t_0) - (t_1 * t_2)), 2.0) + (t_3 * (t_4 * t_4)))), sqrt(((1.0 - pow(fma(t_0, t_5, (t_2 * -t_1)), 2.0)) - (t_3 * ((fma(cos(lambda2), cos(lambda1), -1.0) + (sin(lambda2) * sin(lambda1))) / -2.0))))));
}

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(phi1 * 0.5))
	t_1 = sin(Float64(0.5 * phi2))
	t_2 = cos(Float64(phi1 * 0.5))
	t_3 = Float64(cos(phi1) * cos(phi2))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = cos(Float64(0.5 * phi2))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((Float64(Float64(t_5 * t_0) - Float64(t_1 * t_2)) ^ 2.0) + Float64(t_3 * Float64(t_4 * t_4)))), sqrt(Float64(Float64(1.0 - (fma(t_0, t_5, Float64(t_2 * Float64(-t_1))) ^ 2.0)) - Float64(t_3 * Float64(Float64(fma(cos(lambda2), cos(lambda1), -1.0) + Float64(sin(lambda2) * sin(lambda1))) / -2.0)))))))
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[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[(t$95$5 * t$95$0), $MachinePrecision] - N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$3 * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Power[N[(t$95$0 * t$95$5 + N[(t$95$2 * (-t$95$1)), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 24.2

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

2. Simplified 24.2

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

   [Start] 24.2

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

   associate-*l* [=>] 24.2

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

3. Applied egg-rr 23.7

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

4. Applied egg-rr 14.1

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

5. Simplified 14.0

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

   [Start] 14.1	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   fma-neg [=>] 14.0	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\phi_2 \cdot 0.5\right), -\cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)\right)}}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   *-commutative [=>] 14.0	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \color{blue}{\left(0.5 \cdot \phi_1\right)}, \cos \left(\phi_2 \cdot 0.5\right), -\cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   *-commutative [=>] 14.0	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(0.5 \cdot \phi_1\right)} \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   distribute-rgt-neg-in [=>] 14.0	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)}\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Applied egg-rr 14.0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\frac{-\left(\cos 0 - \cos \left(\lambda_1 - \lambda_2\right)\right)}{-2}}}}\right) \]

7. Simplified 14.0

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

   [Start] 14.0	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{-\left(\cos 0 - \cos \left(\lambda_1 - \lambda_2\right)\right)}{-2}}}\right) \]
   neg-sub0 [=>] 14.0	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{0 - \left(\cos 0 - \cos \left(\lambda_1 - \lambda_2\right)\right)}}{-2}}}\right) \]
   cos-0 [=>] 14.0	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{0 - \left(\color{blue}{1} - \cos \left(\lambda_1 - \lambda_2\right)\right)}{-2}}}\right) \]
   associate--r- [=>] 14.0	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{\left(0 - 1\right) + \cos \left(\lambda_1 - \lambda_2\right)}}{-2}}}\right) \]
   metadata-eval [=>] 14.0	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{-1} + \cos \left(\lambda_1 - \lambda_2\right)}{-2}}}\right) \]

8. Applied egg-rr 13.6

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

9. Simplified 13.6

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

   [Start] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \left(\sin \lambda_1 \cdot \sin \left(-\lambda_2\right) - -1\right)}{-2}}}\right) \]
   associate--r- [=>] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{\left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) + -1}}{-2}}}\right) \]
   +-commutative [<=] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{-1 + \left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right)}}{-2}}}\right) \]
   associate--l+ [<=] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{\left(-1 + \cos \lambda_1 \cdot \cos \left(-\lambda_2\right)\right) - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)}}{-2}}}\right) \]
   *-commutative [=>] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\left(-1 + \cos \lambda_1 \cdot \cos \left(-\lambda_2\right)\right) - \color{blue}{\sin \left(-\lambda_2\right) \cdot \sin \lambda_1}}{-2}}}\right) \]
   sin-neg [=>] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\left(-1 + \cos \lambda_1 \cdot \cos \left(-\lambda_2\right)\right) - \color{blue}{\left(-\sin \lambda_2\right)} \cdot \sin \lambda_1}{-2}}}\right) \]
   cancel-sign-sub [=>] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{\left(-1 + \cos \lambda_1 \cdot \cos \left(-\lambda_2\right)\right) + \sin \lambda_2 \cdot \sin \lambda_1}}{-2}}}\right) \]
   +-commutative [=>] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{\left(\cos \lambda_1 \cdot \cos \left(-\lambda_2\right) + -1\right)} + \sin \lambda_2 \cdot \sin \lambda_1}{-2}}}\right) \]
   *-commutative [=>] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\left(\color{blue}{\cos \left(-\lambda_2\right) \cdot \cos \lambda_1} + -1\right) + \sin \lambda_2 \cdot \sin \lambda_1}{-2}}}\right) \]
   fma-def [=>] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(\cos \left(-\lambda_2\right), \cos \lambda_1, -1\right)} + \sin \lambda_2 \cdot \sin \lambda_1}{-2}}}\right) \]
   cos-neg [=>] 13.6	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(\phi_2 \cdot 0.5\right)\right)\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, -1\right) + \sin \lambda_2 \cdot \sin \lambda_1}{-2}}}\right) \]

10. Final simplification 13.6

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

Alternatives

Alternative 1
Error	13.6
Cost	158016

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

Alternative 2
Error	14.0
Cost	151360

\[\begin{array}{l} t_0 := \sin \left(\phi_1 \cdot 0.5\right)\\ t_1 := \sin \left(0.5 \cdot \phi_2\right)\\ t_2 := \cos \left(\phi_1 \cdot 0.5\right)\\ t_3 := \cos \phi_1 \cdot \cos \phi_2\\ t_4 := \cos \left(0.5 \cdot \phi_2\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(t_4 \cdot t_0 - t_1 \cdot t_2\right)}^{2} + t_3 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)}}{\sqrt{\left(1 - {\left(\mathsf{fma}\left(t_0, t_4, t_2 \cdot \left(-t_1\right)\right)\right)}^{2}\right) + t_3 \cdot \frac{1 - \cos \left(\lambda_1 - \lambda_2\right)}{-2}}}\right) \end{array} \]

Alternative 3
Error	14.0
Cost	138560

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

Alternative 4
Error	14.0
Cost	132224

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

Alternative 5
Error	23.4
Cost	131840

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

Alternative 6
Error	23.7
Cost	112384

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

Alternative 7
Error	23.7
Cost	112384

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

Alternative 8
Error	23.9
Cost	105216

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

Alternative 9
Error	24.2
Cost	92544

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

Alternative 10
Error	24.2
Cost	92544

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

Alternative 11
Error	24.3
Cost	92356

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_4 := \cos \phi_1 \cdot t_3\\ t_5 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_6 := t_5 \cdot \left(t_0 \cdot t_5\right)\\ \mathbf{if}\;\phi_1 \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_4}}{\sqrt{1 - \left(t_2 + t_6\right)}}\right)\\ \mathbf{elif}\;\phi_1 \leq 2.9 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(t_5 \cdot t_5\right) + t_2}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot t_3\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_6}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - t_4}}\right)\\ \end{array} \]

Alternative 12
Error	24.3
Cost	92233

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -2.95 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 2.9 \cdot 10^{-28}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + t_1 \cdot \left(t_2 \cdot t_1\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 \cdot \left(t_1 \cdot t_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot t_0\right)}}\right)\\ \end{array} \]

Alternative 13
Error	33.3
Cost	85824

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

Alternative 14
Error	42.1
Cost	66368

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

Alternative 15
Error	42.2
Cost	66249

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;\lambda_2 \leq -0.033 \lor \neg \left(\lambda_2 \leq 3.3 \cdot 10^{-7}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_1}}\right)\\ \end{array} \]

Alternative 16
Error	42.9
Cost	66249

\[\begin{array}{l} t_0 := \sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_1 \cdot t_1\right)\\ \mathbf{if}\;\phi_1 \leq -9.2 \cdot 10^{-18} \lor \neg \left(\phi_1 \leq 1.4 \cdot 10^{-27}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{t_0}\right)\\ \end{array} \]

Alternative 17
Error	42.1
Cost	66112

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

Alternative 18
Error	45.1
Cost	65984

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

Alternative 19
Error	55.7
Cost	65536

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

Reproduce

herbie shell --seed 2023039 
(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))))))))))