Average Error: 17.2 → 4.0
Time: 1.2min
Precision: binary64
Cost: 71488
\[ \begin{array}{c}[phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \sin \lambda_2 \cdot \left(\cos \phi_1 \cdot \sin \lambda_1\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))
↓
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (fma
    (sin phi1)
    (sin phi2)
    (*
     (+
      (* (cos phi1) (* (cos lambda2) (cos lambda1)))
      (* (sin lambda2) (* (cos phi1) (sin lambda1))))
     (cos phi2))))
  R))
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
}
↓
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(fma(sin(phi1), sin(phi2), (((cos(phi1) * (cos(lambda2) * cos(lambda1))) + (sin(lambda2) * (cos(phi1) * sin(lambda1)))) * cos(phi2)))) * R;
}

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
end
↓
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(fma(sin(phi1), sin(phi2), Float64(Float64(Float64(cos(phi1) * Float64(cos(lambda2) * cos(lambda1))) + Float64(sin(lambda2) * Float64(cos(phi1) * sin(lambda1)))) * cos(phi2)))) * R)
end

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]
↓
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R
↓
\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \sin \lambda_2 \cdot \left(\cos \phi_1 \cdot \sin \lambda_1\right)\right) \cdot \cos \phi_2\right)\right) \cdot R

Error

Derivation

  1. Initial program 17.2

    \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

  2. Simplified17.2

    \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R} \]
    Proof
    (*.f64 (acos.f64 (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (cos.f64 (-.f64 lambda1 lambda2)))))) R): 0 points increase in error, 0 points decrease in error
    (*.f64 (acos.f64 (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi2) (cos.f64 phi1)) (cos.f64 (-.f64 lambda1 lambda2)))))) R): 0 points increase in error, 1 points decrease in error
    (*.f64 (acos.f64 (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 phi1) (cos.f64 phi2))) (cos.f64 (-.f64 lambda1 lambda2))))) R): 0 points increase in error, 0 points decrease in error
    (*.f64 (acos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))) R): 5 points increase in error, 4 points decrease in error

  3. Applied egg-rr4.0

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \color{blue}{\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1 + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right)}\right)\right) \cdot R \]

  4. Taylor expanded in phi1 around inf 4.0

    \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \sin \lambda_2 \cdot \left(\cos \phi_1 \cdot \sin \lambda_1\right)\right) \cdot \cos \phi_2\right)\right)} \cdot R \]

  5. Final simplification4.0

    \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \sin \lambda_2 \cdot \left(\cos \phi_1 \cdot \sin \lambda_1\right)\right) \cdot \cos \phi_2\right)\right) \cdot R \]

Alternatives

Alternative 1
Error4.0
Cost64960
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \]
Alternative 2
Error4.0
Cost64960
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right) \]
Alternative 3
Error11.3
Cost58824
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.7244957277850565 \cdot 10^{+29}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\log \left(1 + \mathsf{expm1}\left(t_0\right)\right) + t_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq 1.7504477548069122 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t_2, t_1, t_0\right)\right)\right)\\ \end{array} \]
Alternative 4
Error4.0
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \]
Alternative 5
Error11.3
Cost52552
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.7244957277850565 \cdot 10^{+29}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\log \left(1 + \mathsf{expm1}\left(t_0\right)\right) + t_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq 1.7504477548069122 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t_2, t_1, t_0\right)\right)\right)\\ \end{array} \]
Alternative 6
Error16.4
Cost52428
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \lambda_2 \cdot \cos \lambda_1\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot \sin \phi_2\\ t_4 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.3853542758470965 \cdot 10^{-154}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + t_0 \cdot t_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5342613052146748 \cdot 10^{-213}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_4 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.764340192315546 \cdot 10^{-39}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + t_0 \cdot \log \left(e^{t_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.189833140800419 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_4 + \left(t_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_2\right)\right)\right)\\ \end{array} \]
Alternative 7
Error16.4
Cost52428
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := \cos \lambda_2 \cdot \cos \lambda_1\\ t_4 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.3853542758470965 \cdot 10^{-154}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\log \left(1 + \mathsf{expm1}\left(t_0\right)\right) + t_2 \cdot t_4\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5342613052146748 \cdot 10^{-213}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_3\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.764340192315546 \cdot 10^{-39}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + t_2 \cdot \log \left(e^{t_4}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.189833140800419 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(t_3 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_4\right)\right)\right)\\ \end{array} \]
Alternative 8
Error16.4
Cost52296
\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)\\ t_3 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.3853542758470965 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 1.5342613052146748 \cdot 10^{-213}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.764340192315546 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 2.189833140800419 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + \left(t_0 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_1\right)\right)\right)\\ \end{array} \]
Alternative 9
Error16.4
Cost46032
\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)\\ t_3 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.3853542758470965 \cdot 10^{-154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 1.5342613052146748 \cdot 10^{-213}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 7.764340192315546 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 2.189833140800419 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_1\right)\right)\right)\\ \end{array} \]
Alternative 10
Error17.5
Cost45768
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 7.764340192315546 \cdot 10^{-39}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.189833140800419 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), t_0\right)\right)\\ \end{array} \]
Alternative 11
Error17.5
Cost45768
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 7.764340192315546 \cdot 10^{-39}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 2.189833140800419 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \end{array} \]
Alternative 12
Error17.5
Cost39496
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq 7.764340192315546 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.189833140800419 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 13
Error24.0
Cost39368
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{if}\;\phi_2 \leq -1326102924593198800:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 0.4206976665328204:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 14
Error22.1
Cost39368
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -3.519749827021072 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.0579427649226573 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_2 \cdot t_0\right)\\ \end{array} \]
Alternative 15
Error23.8
Cost39236
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 1.0579427649226573 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_2 \cdot t_0\right)\\ \end{array} \]
Alternative 16
Error28.0
Cost39108
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.635091269239073 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 17
Error30.2
Cost32964
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.635091269239073 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 18
Error36.5
Cost26948
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 4.3643240353646035 \cdot 10^{-69}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right) \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \]
Alternative 19
Error38.5
Cost26436
\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -158544303323.66702:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 20
Error36.5
Cost26436
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 4.3643240353646035 \cdot 10^{-69}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 21
Error48.0
Cost26308
\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.0483828761952882 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 22
Error49.2
Cost26308
\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 459030613.15081966:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 23
Error51.2
Cost19776
\[R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_2 - \lambda_1\right)\right) \]
Alternative 24
Error52.6
Cost13376
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right) \]

Error

Reproduce

herbie shell --seed 2022316 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))