Distance on a great circle

Average Error: 24.5 → 13.8

Time: 1.8min

Precision: binary64

Cost: 151424

Math

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

↓

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

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

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

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 = Float64(Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(phi1 * 0.5))) - Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(phi1 * 0.5)))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda2 - lambda1) / 2.0))
	return Float64(atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(t_1 * t_1)), t_0)), sqrt(Float64(fma(cos(phi1), Float64(cos(phi2) * Float64(t_1 * sin(Float64(Float64(lambda2 - lambda1) * -0.5)))), 1.0) - t_0))) * Float64(2.0 * R))
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[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 * N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $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. Simplified 24.5

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_2 - \lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right) \cdot \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\right), 1\right) - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}} \cdot \left(R \cdot 2\right)} \]
   Proof
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (Rewrite<= sqr-neg_binary64 (*.f64 (neg.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2))) (neg.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (Rewrite<= sin-neg_binary64 (sin.f64 (neg.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) (neg.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (-.f64 lambda2 lambda1)) 2))) (neg.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 lambda2 lambda1))) 2)) (neg.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 lambda2) lambda1)) 2)) (neg.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 lambda2)) lambda1) 2)) (neg.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 lambda1 (neg.f64 lambda2))) 2)) (neg.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 lambda1 lambda2)) 2)) (neg.f64 (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (Rewrite<= sin-neg_binary64 (sin.f64 (neg.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 (-.f64 lambda2 lambda1)) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 lambda2 lambda1))) 2)))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 lambda2) lambda1)) 2)))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 lambda2)) lambda1) 2)))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 lambda1 (neg.f64 lambda2))) 2)))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 lambda1 lambda2)) 2)))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 6 points increase in error, 2 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 1 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 5 points increase in error, 6 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 -1/2 (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) (-.f64 lambda2 lambda1))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (Rewrite<= associate-/r/_binary64 (/.f64 -1 (/.f64 2 (-.f64 lambda2 lambda1))))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 25 points increase in error, 6 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 (-.f64 lambda2 lambda1)) 2))) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 6 points increase in error, 25 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 lambda2 lambda1))) 2)) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 lambda2 lambda1))) 2)) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 lambda2) lambda1)) 2)) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 lambda2)) lambda1) 2)) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 lambda1 (neg.f64 lambda2))) 2)) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 lambda1 lambda2)) 2)) (sin.f64 (/.f64 (-.f64 lambda2 lambda1) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 lambda2 (neg.f64 lambda1))) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 lambda1) lambda2)) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 lambda1)) lambda2) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 lambda1 lambda2))) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (/.f64 (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 lambda1 lambda2))) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (sin.f64 (Rewrite=> distribute-frac-neg_binary64 (neg.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (*.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) (Rewrite=> sin-neg_binary64 (neg.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi2) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (neg.f64 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 3 points increase in error, 1 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (fma.f64 (cos.f64 phi1) (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 (cos.f64 phi2) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 phi1) (neg.f64 (*.f64 (*.f64 (cos.f64 phi2) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) 1)) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 4 points increase in error, 1 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (cos.f64 phi1) (*.f64 (*.f64 (cos.f64 phi2) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (+.f64 (neg.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 2 points increase in error, 2 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (+.f64 (neg.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))) 1) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 1 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (Rewrite=> +-commutative_binary64 (+.f64 1 (neg.f64 (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 (Rewrite<= sub-neg_binary64 (-.f64 1 (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (Rewrite<= associate--r+_binary64 (-.f64 1 (+.f64 (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2)))))) (*.f64 R 2)): 12 points increase in error, 11 points decrease in error
   (*.f64 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 1 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))))))) (*.f64 R 2)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 R 2) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 1 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*r*_binary64 (*.f64 R (*.f64 2 (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))))) (sqrt.f64 (-.f64 1 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) 2)) 2) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)))))))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 23.9

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

4. Applied egg-rr 13.8

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

5. Final simplification 13.8

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

Alternatives

Alternative 1
Error	13.8
Cost	145152

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

Alternative 2
Error	20.3
Cost	145028

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\lambda_2 \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_0 \cdot t_0\right), t_2\right)}}{\sqrt{\left(1 + \cos \phi_2 \cdot \left(\left(\cos \phi_1 \cdot \sin \left(\lambda_1 \cdot 0.5\right)\right) \cdot \sin \left(\lambda_1 \cdot -0.5\right)\right)\right) - t_2}}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_1 \cdot \left(\sin \left(\lambda_2 \cdot -0.5\right) \cdot t_3\right)}}{\sqrt{\left(1 - t_2\right) - t_1 \cdot \left(t_3 \cdot t_3\right)}}\right)\\ \end{array} \]

Alternative 3
Error	20.1
Cost	145028

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\\ t_1 := {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\\ t_2 := \sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_0 \cdot t_0\right), t_1\right)}\\ \mathbf{if}\;\lambda_2 \leq 6.9 \cdot 10^{-6}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\left(1 + \cos \phi_2 \cdot \left(\left(\cos \phi_1 \cdot \sin \left(\lambda_1 \cdot 0.5\right)\right) \cdot \sin \left(\lambda_1 \cdot -0.5\right)\right)\right) - t_1}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\left(1 + \sin \left(\lambda_2 \cdot -0.5\right) \cdot \left(\sin \left(\lambda_2 \cdot 0.5\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right) - t_1}}\\ \end{array} \]

Alternative 4
Error	23.1
Cost	138884

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_2 - \lambda_1}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)}^{2}\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\lambda_1 \leq -3.45 \cdot 10^{-74}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_0 \cdot t_0\right), t_2\right)}}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_0 \cdot \sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right), 1\right) - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_1 \cdot \left(\sin \left(\lambda_2 \cdot -0.5\right) \cdot t_3\right)}}{\sqrt{\left(1 - t_2\right) - t_1 \cdot \left(t_3 \cdot t_3\right)}}\right)\\ \end{array} \]

Alternative 5
Error	23.9
Cost	131584

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

Alternative 6
Error	23.9
Cost	119040

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

Alternative 7
Error	23.9
Cost	119040

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

Alternative 8
Error	24.5
Cost	92864

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

Alternative 9
Error	28.5
Cost	92548

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq 2.05 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_1 \cdot \left(t_2 \cdot t_1\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 \cdot \left(t_1 \cdot t_1\right) + t_0}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot \left(0.5 - \frac{\cos \left(2 \cdot \mathsf{fma}\left(0.5, \lambda_2, \lambda_1 \cdot -0.5\right)\right)}{2}\right)}}\right)\\ \end{array} \]

Alternative 10
Error	24.5
Cost	92544

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

Alternative 11
Error	31.6
Cost	92228

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{if}\;\phi_1 \leq -0.195:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{\left(1 - t_1\right) - t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_1}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 12
Error	28.5
Cost	92228

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ t_2 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq 2.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_2}}\right)\\ \end{array} \]

Alternative 13
Error	28.5
Cost	92228

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_1 \cdot \left(t_2 \cdot t_1\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 \cdot \left(t_1 \cdot t_1\right) + t_0}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 14
Error	33.4
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{if}\;\phi_1 \leq -0.18:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 15
Error	38.0
Cost	86160

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := {\cos \left(\phi_2 \cdot -0.5\right)}^{2}\\ t_3 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)\\ t_4 := \sqrt{t_3 + t_1}\\ t_5 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{t_2 - \cos \phi_2 \cdot \left(0.5 - \frac{\cos \lambda_1}{2}\right)}}\right)\\ \mathbf{if}\;\phi_1 \leq -4.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{\left(1 - t_1\right) - t_3}}\right)\\ \mathbf{elif}\;\phi_1 \leq -8.2 \cdot 10^{-140}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{1 - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}\right)\\ \mathbf{elif}\;\phi_1 \leq -1.85 \cdot 10^{-299}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-223}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{t_2 + \cos \phi_2 \cdot \left(-0.5 + \frac{\cos \lambda_2}{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

Alternative 16
Error	38.1
Cost	85896

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)\\ t_3 := \sqrt{t_2 + t_1}\\ \mathbf{if}\;\phi_1 \leq -0.00017:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{\left(1 - t_1\right) - t_2}}\right)\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{-140}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{1 - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot \left(0.5 - \frac{\cos \lambda_1}{2}\right)}}\right)\\ \end{array} \]

Alternative 17
Error	45.0
Cost	72840

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t_0 \cdot \left(t_1 \cdot t_1\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-163}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_3}}{\sqrt{1 - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{-218}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{\left(1 - t_3\right) - t_0 \cdot \left(\sin \left(\lambda_2 \cdot -0.5\right) \cdot t_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{t_4}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_2}}\right)\\ \end{array} \]

Alternative 18
Error	40.5
Cost	72708

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{if}\;\phi_1 \leq -0.00014:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{\left(1 - t_1\right) - t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_1}}{\sqrt{1 - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 19
Error	48.3
Cost	72320

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

Alternative 20
Error	54.5
Cost	59968

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

Alternative 21
Error	55.4
Cost	59716

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)}\\ \mathbf{if}\;\phi_2 \leq 4.9 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sin \left(\phi_1 \cdot 0.5\right)}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sin \left(\phi_2 \cdot -0.5\right)}{t_1}\right)\\ \end{array} \]

Alternative 22
Error	53.5
Cost	59712

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

Alternative 23
Error	55.4
Cost	59588

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t_0 \cdot t_0\right)\\ \mathbf{if}\;\phi_1 \leq -6.2:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sin \left(\phi_2 \cdot -0.5\right)}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_1}}\right)\\ \end{array} \]

Alternative 24
Error	53.6
Cost	59584

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

Alternative 25
Error	53.6
Cost	59584

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

Alternative 26
Error	55.0
Cost	52608

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

Alternative 27
Error	56.5
Cost	39616

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

Reproduce

herbie shell --seed 2022331 
(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))))))))))