?

Average Error: 16.6 → 3.8
Time: 40.0s
Precision: binary64
Cost: 64960

?

\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \cdot R \]
(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
   (+
    (* (sin phi1) (sin phi2))
    (*
     (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2)))
     (* (cos phi2) (cos phi1)))))
  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(((sin(phi1) * sin(phi2)) + (fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))) * (cos(phi2) * cos(phi1))))) * 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(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))) * Float64(cos(phi2) * cos(phi1))))) * 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[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $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(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \cdot R

Error?

Derivation?

  1. Initial program 16.6

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

  2. Applied egg-rr3.8

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

  3. Simplified3.8

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

    [Start]3.8

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

    distribute-lft-out [=>]3.8

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

    +-commutative [<=]3.8

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

    *-commutative [=>]3.8

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

    +-commutative [=>]3.8

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

    *-commutative [=>]3.8

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

    cos-neg [<=]3.8

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

    fma-def [=>]3.8

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

    cos-neg [=>]3.8

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

    *-commutative [=>]3.8

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

  4. Final simplification3.8

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

Alternatives

Alternative 1
Error3.8
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \]
Alternative 2
Error10.4
Cost58436
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 5.3 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)\\ \end{array} \]
Alternative 3
Error10.4
Cost52296
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6.6 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(\left(-1 + \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\right) + 1\right)\\ \mathbf{elif}\;\phi_1 \leq 5.3 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \end{array} \]
Alternative 4
Error10.4
Cost52296
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(\left(-1 + \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\right) + 1\right)\\ \mathbf{elif}\;\phi_1 \leq 5.3 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)\\ \end{array} \]
Alternative 5
Error16.1
Cost46024
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.4 \cdot 10^{-96}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{-114}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \end{array} \]
Alternative 6
Error16.1
Cost46024
\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -3.4 \cdot 10^{-96}:\\ \;\;\;\;R \cdot t_0\\ \mathbf{elif}\;\phi_2 \leq 1.46 \cdot 10^{-114}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(-1 + t_0\right) + 1\right)\\ \end{array} \]
Alternative 7
Error16.6
Cost45504
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \]
Alternative 8
Error23.9
Cost39633
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)\right)\\ t_2 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.22 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -6.1 \cdot 10^{+29}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq -4.1 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 7.2 \cdot 10^{-71}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot t_2\right)\\ \end{array} \]
Alternative 9
Error23.5
Cost39236
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \end{array} \]
Alternative 10
Error16.6
Cost39232
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]
Alternative 11
Error31.8
Cost33220
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0 + \frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 12
Error38.4
Cost32841
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.1 \cdot 10^{-74} \lor \neg \left(\phi_2 \leq 1\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \end{array} \]
Alternative 13
Error34.8
Cost32840
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.1 \cdot 10^{-74}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_2\right)\\ \end{array} \]
Alternative 14
Error33.6
Cost32836
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 2.9 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_2\right)\\ \end{array} \]
Alternative 15
Error31.9
Cost32836
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 16
Error40.0
Cost32708
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 3500000000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \log \left(e^{t_0}\right)\right)\\ \end{array} \]
Alternative 17
Error40.5
Cost32708
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -7 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \cos \phi_1\right)\\ \end{array} \]
Alternative 18
Error42.9
Cost26436
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.2 \cdot 10^{+41}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0\right)\\ \end{array} \]
Alternative 19
Error50.2
Cost26308
\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 10^{-31}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 20
Error49.1
Cost26308
\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \cos \phi_1\right)\\ \end{array} \]
Alternative 21
Error46.2
Cost26304
\[R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \]
Alternative 22
Error52.1
Cost19780
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{+135}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
Alternative 23
Error53.4
Cost19780
\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -6.2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_2 + t_0\right)\\ \end{array} \]
Alternative 24
Error51.1
Cost19776
\[R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_2 - \lambda_1\right)\right) \]
Alternative 25
Error52.5
Cost13376
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right) \]

Error

Reproduce?

herbie shell --seed 2023059 
(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))