Average Error: 16.6 → 3.8
Time: 1.3min
Precision: binary64
Cost: 71232
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
(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
    (* (cos phi2) (cos phi1))
    (fma (sin lambda2) (sin lambda1) (* (cos lambda2) (cos lambda1)))
    (* (sin phi1) (sin 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((cos(phi2) * cos(phi1)), fma(sin(lambda2), sin(lambda1), (cos(lambda2) * cos(lambda1))), (sin(phi1) * sin(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(Float64(cos(phi2) * cos(phi1)), fma(sin(lambda2), sin(lambda1), Float64(cos(lambda2) * cos(lambda1))), Float64(sin(phi1) * sin(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[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[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(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\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

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

  3. Taylor expanded in phi1 around 0 3.8

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

  4. Simplified3.8

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

  5. Final simplification3.8

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

Alternatives

Alternative 1
Error11.5
Cost59088
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \sin \lambda_2 \cdot \sin \lambda_1\\ t_3 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -6.59467577383806 \cdot 10^{+274}:\\ \;\;\;\;R \cdot {\left(\sqrt{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot t_1, t_3\right)\right)}\right)}^{2}\\ \mathbf{elif}\;\phi_1 \leq -8.596986855677736 \cdot 10^{+219}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + t_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -4.887307389258758 \cdot 10^{+25}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(t_1, t_0, t_3\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.104726322617007 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, t_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\\ \end{array} \]
Alternative 2
Error11.5
Cost58700
\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ t_2 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_3 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -6.59467577383806 \cdot 10^{+274}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_1 \cdot t_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -8.596986855677736 \cdot 10^{+219}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -4589286512.459332:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(t_2, t_1, t_3\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.104726322617007 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\\ \end{array} \]
Alternative 3
Error11.5
Cost58700
\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -6.59467577383806 \cdot 10^{+274}:\\ \;\;\;\;R \cdot {\left(\sqrt{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot t_1, t_2\right)\right)}\right)}^{2}\\ \mathbf{elif}\;\phi_1 \leq -8.596986855677736 \cdot 10^{+219}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -4589286512.459332:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(t_1, \cos \phi_2 \cdot \cos \phi_1, t_2\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.104726322617007 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\\ \end{array} \]
Alternative 4
Error3.8
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \]
Alternative 5
Error11.2
Cost52624
\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_1 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -6.59467577383806 \cdot 10^{+274}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -8.596986855677736 \cdot 10^{+219}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -1.5174202823642684 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 1.104726322617007 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\\ \end{array} \]
Alternative 6
Error10.5
Cost52228
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -7141439176.646912:\\ \;\;\;\;R \cdot \cos^{-1} \left({\left(\sqrt[3]{\sin \phi_1 \cdot \sin \phi_2}\right)}^{3} + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 5.566491248457847 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 7
Error10.5
Cost46024
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -7141439176.646912:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5.566491248457847 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error10.5
Cost45768
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -7141439176.646912:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5.566491248457847 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error17.2
Cost39632
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ t_1 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_2 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.368204658280704 \cdot 10^{+117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -1.4229866222232366 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.5174202823642684 \cdot 10^{-11}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 1.4450829827977262 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error26.7
Cost39500
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -4.0802968799897917 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq -9.070405085900488 \cdot 10^{+24}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -0.0035435174307683307:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
Alternative 11
Error10.5
Cost39496
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -7141439176.646912:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5.566491248457847 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 12
Error17.2
Cost39368
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -4589286512.459332:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.4450829827977262 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 13
Error31.4
Cost39108
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.658022492638147 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 14
Error32.0
Cost39108
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -4589286512.459332:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \]
Alternative 15
Error35.6
Cost32964
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.525710616834954 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0 + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 16
Error33.1
Cost32964
\[\begin{array}{l} t_0 := \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.525710616834954 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0\right)\\ \end{array} \]
Alternative 17
Error38.3
Cost32580
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -5.524808525466493 \cdot 10^{+138}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -4589286512.459332:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 18
Error42.8
Cost26440
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 1.6178342903602732 \cdot 10^{-47}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2830629669074637 \cdot 10^{+124}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
Alternative 19
Error38.7
Cost26436
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -4589286512.459332:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 20
Error46.0
Cost26308
\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3638.548816077099:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 21
Error44.4
Cost26308
\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 7.718490631243698 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 22
Error50.7
Cost19908
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.735136910404458 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
Alternative 23
Error50.1
Cost19908
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 7.718490631243698 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 24
Error47.3
Cost19904
\[R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \phi_1\right) \]
Alternative 25
Error52.5
Cost13376
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \phi_1\right) \]

Error

Reproduce

herbie shell --seed 2022228 
(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))