Distance on a great circle

Average Error: 24.2 → 13.8

Time: 1.5min

Precision: binary64

Cost: 157952

\[ \begin{array}{c}[lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]

Math

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

↓

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

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) (cos phi2)))
        (t_1 (cos (* 0.5 phi1)))
        (t_2 (* (sin (* 0.5 phi1)) (cos (* 0.5 phi2))))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (fma t_1 (sin (* phi2 -0.5)) t_2) 2.0)
        (* (* t_0 (log1p (expm1 (sin (* 0.5 (- lambda1 lambda2)))))) t_3)))
      (sqrt
       (-
        1.0
        (+
         (pow (- t_2 (* t_1 (sin (* 0.5 phi2)))) 2.0)
         (* t_3 (* t_0 t_3))))))))))

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) * cos(phi2);
	double t_1 = cos((0.5 * phi1));
	double t_2 = sin((0.5 * phi1)) * cos((0.5 * phi2));
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt((pow(fma(t_1, sin((phi2 * -0.5)), t_2), 2.0) + ((t_0 * log1p(expm1(sin((0.5 * (lambda1 - lambda2)))))) * t_3))), sqrt((1.0 - (pow((t_2 - (t_1 * sin((0.5 * phi2)))), 2.0) + (t_3 * (t_0 * t_3)))))));
}

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(cos(phi1) * cos(phi2))
	t_1 = cos(Float64(0.5 * phi1))
	t_2 = Float64(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2)))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(t_1, sin(Float64(phi2 * -0.5)), t_2) ^ 2.0) + Float64(Float64(t_0 * log1p(expm1(sin(Float64(0.5 * Float64(lambda1 - lambda2)))))) * t_3))), sqrt(Float64(1.0 - Float64((Float64(t_2 - Float64(t_1 * sin(Float64(0.5 * phi2)))) ^ 2.0) + Float64(t_3 * Float64(t_0 * t_3))))))))
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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(t$95$1 * N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(t$95$0 * N[Log[1 + N[(Exp[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[(t$95$2 - N[(t$95$1 * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$3 * N[(t$95$0 * t$95$3), $MachinePrecision]), $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. Applied egg-rr 23.5

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

3. Applied egg-rr 13.8

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

4. Simplified 13.8

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   Proof
   (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (*.f64 -1/2 phi2)) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (Rewrite=> *-commutative_binary64 (*.f64 phi2 -1/2))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (*.f64 phi2 (Rewrite<= metadata-eval (neg.f64 1/2)))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (sin.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 phi2 1/2)))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (*.f64 (Rewrite=> sin-neg_binary64 (neg.f64 (sin.f64 (*.f64 phi2 1/2)))) (cos.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (sin.f64 (*.f64 phi2 1/2)) (cos.f64 (*.f64 1/2 phi1)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (neg.f64 (*.f64 (sin.f64 (*.f64 phi2 1/2)) (cos.f64 (Rewrite=> *-commutative_binary64 (*.f64 phi1 1/2)))))): 0 points increase in error, 0 points decrease in error
   (fma.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1)) (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2)))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (*.f64 1/2 phi1))) (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2))))): 10 points increase in error, 4 points decrease in error
   (-.f64 (*.f64 (cos.f64 (*.f64 phi2 1/2)) (sin.f64 (Rewrite=> *-commutative_binary64 (*.f64 phi1 1/2)))) (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 (*.f64 phi1 1/2)) (cos.f64 (*.f64 phi2 1/2)))) (*.f64 (cos.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2)))): 0 points increase in error, 0 points decrease in error

5. Taylor expanded in phi2 around inf 13.8

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

6. Simplified 13.8

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_1\right), \sin \left(\phi_2 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2}} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   Proof
   (pow.f64 (fma.f64 (cos.f64 (*.f64 1/2 phi1)) (sin.f64 (*.f64 phi2 -1/2)) (*.f64 (sin.f64 (*.f64 1/2 phi1)) (cos.f64 (*.f64 1/2 phi2)))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (fma.f64 (cos.f64 (*.f64 1/2 phi1)) (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/2 phi2))) (*.f64 (sin.f64 (*.f64 1/2 phi1)) (cos.f64 (*.f64 1/2 phi2)))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (fma.f64 (cos.f64 (*.f64 1/2 phi1)) (sin.f64 (*.f64 -1/2 phi2)) (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1))))) 2): 0 points increase in error, 0 points decrease in error
   (pow.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 (*.f64 1/2 phi1)) (sin.f64 (*.f64 -1/2 phi2))) (*.f64 (cos.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1))))) 2): 9 points increase in error, 8 points decrease in error

7. Applied egg-rr 13.8

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_1\right), \sin \left(\phi_2 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

8. Final simplification 13.8

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_1\right), \sin \left(\phi_2 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

Alternatives

Alternative 1
Error	13.8
Cost	145152

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

Alternative 2
Error	13.8
Cost	138880

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

Alternative 3
Error	23.5
Cost	119040

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

Alternative 4
Error	23.5
Cost	119040

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

Alternative 5
Error	23.8
Cost	118016

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

Alternative 6
Error	24.2
Cost	105536

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

Alternative 7
Error	24.1
Cost	105408

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

Alternative 8
Error	24.1
Cost	92736

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

Alternative 9
Error	24.1
Cost	92544

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

Alternative 10
Error	36.0
Cost	92488

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sqrt{t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;t_1 \leq -0.12:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\frac{1 + t_0}{2}}}\right)\\ \mathbf{elif}\;t_1 \leq 0.21:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + 0.5 \cdot t_0\right)\right)}}\right)\\ \end{array} \]

Alternative 11
Error	36.0
Cost	92488

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ t_2 := 0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;t_0 \leq -0.12:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{\sqrt[3]{{t_2}^{3}}}}\right)\\ \mathbf{elif}\;t_0 \leq 0.21:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(t_2\right)\right)}}\right)\\ \end{array} \]

Alternative 12
Error	23.9
Cost	92228

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_2}}\right)\\ \end{array} \]

Alternative 13
Error	23.9
Cost	92228

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \sqrt{t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_1}}\right)\\ \end{array} \]

Alternative 14
Error	27.9
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{if}\;\phi_1 \leq -0.0015:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right)}}{\sqrt{{\cos \left(\phi_1 \cdot -0.5\right)}^{2} - \sin \left(0.5 \cdot \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \end{array} \]

Alternative 15
Error	24.7
Cost	92100

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_2}}\right)\\ \end{array} \]

Alternative 16
Error	36.0
Cost	86152

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{\frac{1 + \cos \left(\lambda_1 - \lambda_2\right)}{2}}}\right)\\ \mathbf{if}\;t_0 \leq -0.12:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_0 \leq 0.21:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	35.3
Cost	86084

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{if}\;\phi_1 \leq -0.000225:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + \left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right)}}{\sqrt{{\cos \left(\phi_1 \cdot -0.5\right)}^{2} - \sin \left(0.5 \cdot \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{\sqrt[3]{{\left(0.5 + 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}^{3}}}}\right)\\ \end{array} \]

Alternative 18
Error	41.5
Cost	66112

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

Alternative 19
Error	57.9
Cost	60096

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \mathsf{fma}\left(-0.0026041666666666665, \phi_1 \cdot \phi_1, 0.0625\right)\right)}{\sqrt{1 - \left(t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}\right) \end{array} \]

Alternative 20
Error	59.4
Cost	59968

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\phi_2 \cdot \left(0.125 \cdot \left(\phi_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}{\sqrt{1 - \left(t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}\right) \end{array} \]

Alternative 21
Error	59.4
Cost	59840

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \phi_1\right) \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.125\right)}{\sqrt{1 - \left(t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\right) \end{array} \]

Alternative 22
Error	59.4
Cost	59840

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1 \cdot \left(\left(\phi_2 \cdot \phi_2\right) \cdot 0.125\right)}{\sqrt{1 - \left(t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {t_1}^{2}\right)}}\right) \end{array} \]

Alternative 23
Error	60.2
Cost	53440

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\left(\phi_2 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot 0.0625\right)}{\sqrt{1 - \left(t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}\right) \end{array} \]

Reproduce

herbie shell --seed 2022339 
(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))))))))))