Distance on a great circle

Average Error: 23.6 → 13.1

Time: 2.1min

Precision: binary64

Cost: 164736

Math

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

↓

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

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 (cos (* phi1 0.5)))
        (t_1 (sin (* 0.5 phi2)))
        (t_2 (* (cos phi1) (cos phi2)))
        (t_3 (cos (* 0.5 phi2)))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5 (sin (* phi1 0.5))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (+ (* t_4 (* t_4 t_2)) (pow (- (* t_3 t_5) (* t_0 t_1)) 2.0)))
      (sqrt
       (-
        (- 1.0 (pow (fma t_3 t_5 (* t_0 (- t_1))) 2.0))
        (*
         t_2
         (pow
          (-
           (* (sin (* 0.5 lambda1)) (cos (* 0.5 lambda2)))
           (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
          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 = cos((phi1 * 0.5));
	double t_1 = sin((0.5 * phi2));
	double t_2 = cos(phi1) * cos(phi2);
	double t_3 = cos((0.5 * phi2));
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = sin((phi1 * 0.5));
	return R * (2.0 * atan2(sqrt(((t_4 * (t_4 * t_2)) + pow(((t_3 * t_5) - (t_0 * t_1)), 2.0))), sqrt(((1.0 - pow(fma(t_3, t_5, (t_0 * -t_1)), 2.0)) - (t_2 * pow(((sin((0.5 * lambda1)) * cos((0.5 * lambda2))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 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 = cos(Float64(phi1 * 0.5))
	t_1 = sin(Float64(0.5 * phi2))
	t_2 = Float64(cos(phi1) * cos(phi2))
	t_3 = cos(Float64(0.5 * phi2))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = sin(Float64(phi1 * 0.5))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_4 * Float64(t_4 * t_2)) + (Float64(Float64(t_3 * t_5) - Float64(t_0 * t_1)) ^ 2.0))), sqrt(Float64(Float64(1.0 - (fma(t_3, t_5, Float64(t_0 * Float64(-t_1))) ^ 2.0)) - Float64(t_2 * (Float64(Float64(sin(Float64(0.5 * lambda1)) * cos(Float64(0.5 * lambda2))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 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[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$4 * N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(t$95$3 * t$95$5), $MachinePrecision] - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Power[N[(t$95$3 * t$95$5 + N[(t$95$0 * (-t$95$1)), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[Power[N[(N[(N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 23.6

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

2. Applied egg-rr 23.0

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

3. Applied egg-rr 13.5

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

4. Applied egg-rr 13.5

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

5. Simplified 13.5

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

   [Start] 13.5	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 + \left(-\left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]
   sub-neg [<=] 13.5	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}\right) \]
   associate--r+ [=>] 13.5	\[ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}\right) \]

6. Applied egg-rr 13.1

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

7. Final simplification 13.1

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

Alternatives

Alternative 1
Error	13.5
Cost	144896

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

Alternative 2
Error	14.6
Cost	138697

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \sqrt{1 - \left(t_2 \cdot \left(t_2 \cdot t_0\right) + t_1\right)}\\ \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{-21} \lor \neg \left(\lambda_1 \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)}}{t_3}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_0 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{t_3}\right)\\ \end{array} \]

Alternative 3
Error	13.6
Cost	138697

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t_2 \cdot \left(t_2 \cdot t_0\right) + t_1\\ \mathbf{if}\;\lambda_2 \leq -2.9 \cdot 10^{-5} \lor \neg \left(\lambda_2 \leq 4.6 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_0 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - t_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 - \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)\right)}}\right)\\ \end{array} \]

Alternative 4
Error	18.2
Cost	138696

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := t_0 \cdot \left(t_0 \cdot t_1\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_5 := t_2 + t_4\\ t_6 := t_2 + t_3\\ t_7 := \sin \left(0.5 \cdot \lambda_1\right)\\ \mathbf{if}\;\lambda_2 \leq -6 \cdot 10^{-78}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_5}}{\sqrt{1 - t_6}}\right)\\ \mathbf{elif}\;\lambda_2 \leq 9.5 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_4 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {t_7}^{2}\right)}}{\sqrt{1 - t_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_6}}{\sqrt{\left|1 - \mathsf{fma}\left(t_1, {\left(t_7 \cdot \cos \left(0.5 \cdot \lambda_2\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, t_3\right)\right|}}\right)\\ \end{array} \]

Alternative 5
Error	13.6
Cost	138696

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := {\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_3 := t_2 + t_1 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\\ t_4 := t_0 \cdot \left(t_0 \cdot t_1\right) + t_2\\ t_5 := \sqrt{t_4}\\ \mathbf{if}\;\lambda_2 \leq -2.8 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_5}{\sqrt{1 - t_3}}\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_5}{\sqrt{1 - \left(t_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3}}{\sqrt{1 - t_4}}\right)\\ \end{array} \]

Alternative 6
Error	13.5
Cost	138560

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

Alternative 7
Error	23.1
Cost	131392

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

Alternative 8
Error	23.0
Cost	119040

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

Alternative 9
Error	23.3
Cost	111552

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

Alternative 10
Error	23.6
Cost	99200

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

Alternative 11
Error	35.7
Cost	98952

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\ t_3 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot t_2 + t_1}}{\sqrt{1 - {\log \left(e^{t_3}\right)}^{2}}}\right)\\ \mathbf{elif}\;t_0 \leq 0.24:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_3 \cdot t_3\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)}}{\sqrt{1 - {t_3}^{2}}}\right)\\ \end{array} \]

Alternative 12
Error	23.6
Cost	98880

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

Alternative 13
Error	23.4
Cost	92360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_3 := \cos \phi_2 \cdot t_2\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_5 := t_4 \cdot \left(t_4 \cdot t_0\right)\\ \mathbf{if}\;\phi_2 \leq -9 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_3}}{\sqrt{\left(1 - t_1\right) - t_0 \cdot \left(t_4 \cdot t_4\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 0.000145:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_5 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_5 + t_1}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_3}}\right)\\ \end{array} \]

Alternative 14
Error	29.5
Cost	92228

\[\begin{array}{l} t_0 := {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := t_1 \cdot \left(t_3 \cdot t_3\right)\\ t_5 := \sin \left(\frac{\phi_1 - \phi_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -0.0001:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2}}{\sqrt{\left(1 - t_0\right) - t_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 3.2 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t_3 \cdot \sqrt{t_1}, t_5\right)}{\sqrt{\left(1 + \left(-0.5 - -0.5 \cdot \cos \left(\phi_1 - \phi_2\right)\right)\right) - t_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3 \cdot \left(t_3 \cdot t_1\right) + {t_5}^{2}}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - t_2\right)}}\right)\\ \end{array} \]

Alternative 15
Error	29.5
Cost	86089

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-8} \lor \neg \left(\phi_1 \leq 0.0132\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_1 \cdot t_1\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{0.5 + \left(0.5 \cdot \cos \left(-\phi_2\right) - \cos \phi_2 \cdot {t_1}^{2}\right)}}\right)\\ \end{array} \]

Alternative 16
Error	35.6
Cost	79368

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ t_2 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.0001:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {t_2}^{2}}}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 200:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_2 \cdot t_2\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\left(\sin \left(\phi_1 \cdot 0.5\right) + \phi_2 \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot -0.5\right)\right)}^{2}}}{\sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \end{array} \]

Alternative 17
Error	35.6
Cost	79368

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t_1 \cdot \left(t_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{-0.5 + e^{\mathsf{log1p}\left(0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 200:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_0 \cdot t_0\right), {\sin \left(-0.5 \cdot \left(\phi_2 - \phi_1\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + {\left(\sin \left(\phi_1 \cdot 0.5\right) + \phi_2 \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot -0.5\right)\right)}^{2}}}{\sqrt{0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \end{array} \]

Alternative 18
Error	41.5
Cost	66249

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;\lambda_1 \leq -0.0026 \lor \neg \left(\lambda_1 \leq 0.032\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{0.5 + 0.5 \cdot \cos \lambda_2}}\right)\\ \end{array} \]

Alternative 19
Error	41.4
Cost	66112

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

Alternative 20
Error	44.7
Cost	65984

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

Reproduce

herbie shell --seed 2023011 
(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))))))))))