Spherical law of cosines

Average Error: 16.6 → 3.9

Time: 1.1min

Precision: binary64

Cost: 77760

Math

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

↓

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

FPCore

(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 phi2)
     (fma
      (cos phi1)
      (* (cos lambda2) (cos lambda1))
      (* (sin lambda2) (* (cos phi1) (sin lambda1)))))))
  R))

C

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(phi2) * fma(cos(phi1), (cos(lambda2) * cos(lambda1)), (sin(lambda2) * (cos(phi1) * sin(lambda1))))))) * R;
}

Julia

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(cos(phi2) * fma(cos(phi1), Float64(cos(lambda2) * cos(lambda1)), Float64(sin(lambda2) * Float64(cos(phi1) * sin(lambda1))))))) * R)
end

Wolfram

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[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

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. Simplified 16.6

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

3. Applied egg-rr 3.9

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

4. Taylor expanded in phi1 around 0 3.9

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

5. Simplified 3.9

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

Alternatives

Alternative 1
Error	3.9
Cost	64960

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

Alternative 2
Error	10.4
Cost	58696

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.3607026683467793:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 3.386453956663696 \cdot 10^{-33}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;{\left({\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}^{3}\right)}^{0.3333333333333333} \cdot R\\ \end{array} \]

Alternative 3
Error	10.4
Cost	58696

\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -0.3607026683467793:\\ \;\;\;\;t_0 \cdot R\\ \mathbf{elif}\;\phi_1 \leq 3.386453956663696 \cdot 10^{-33}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;{\left({t_0}^{3}\right)}^{0.3333333333333333} \cdot R\\ \end{array} \]

Alternative 4
Error	3.9
Cost	58688

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

Alternative 5
Error	10.4
Cost	52552

\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -0.3607026683467793:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 3.386453956663696 \cdot 10^{-33}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.3
Cost	46024

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -8.672926346427996 \cdot 10^{-188}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 2.3927562724615613 \cdot 10^{-56}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R\\ \end{array} \]

Alternative 7
Error	10.5
Cost	46024

\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -5.4548644997183957 \cdot 10^{-8}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 3.386453956663696 \cdot 10^{-33}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	15.4
Cost	45768

\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -3.292689242123677 \cdot 10^{-202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 8.470776920393534 \cdot 10^{-169}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \phi_2 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	15.4
Cost	39624

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -3.292689242123677 \cdot 10^{-202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 8.470776920393534 \cdot 10^{-169}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \phi_2 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	27.3
Cost	39504

\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -9.872706369137863 \cdot 10^{+226}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.704948877871402 \cdot 10^{+141}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -0.006987233783562096:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 4.641937990596982 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	23.3
Cost	39368

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.034292240347147965:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.551396719605452 \cdot 10^{-34}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	16.9
Cost	39368

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot R\\ \mathbf{if}\;\lambda_2 \leq -0.00015194537818668068:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 74157.60373199143:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	26.9
Cost	39240

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -56839.816271683245:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 7.097017684799589 \cdot 10^{-5}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	16.6
Cost	39232

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

Alternative 15
Error	30.1
Cost	33096

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.773114741715905 \cdot 10^{+84}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 7.097017684799589 \cdot 10^{-5}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right) \cdot R\\ \end{array} \]

Alternative 16
Error	29.2
Cost	32840

\[\begin{array}{l} t_0 := \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -0.3607026683467793:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 0.1802658131756578:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	36.3
Cost	27080

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -0.034292240347147965:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 7.795962036197749 \cdot 10^{-56}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \phi_1 \cdot t_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \left(\cos \phi_2 \cdot t_1\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right) \cdot R\\ \end{array} \]

Alternative 18
Error	39.7
Cost	26568

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ t_1 := \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -2.5945394446489263 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.551396719605452 \cdot 10^{-34}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	36.3
Cost	26568

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ t_1 := \cos^{-1} \left(t_0 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.034292240347147965:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 7.795962036197749 \cdot 10^{-56}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	43.6
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.5265610430811174 \cdot 10^{+26}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq 6.187207818773856 \cdot 10^{-29}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \lambda_2 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \end{array} \]

Alternative 21
Error	41.6
Cost	26440

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ t_1 := \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{if}\;\lambda_1 \leq -2.5265610430811174 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 2.1637307840524583 \cdot 10^{-45}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 22
Error	49.6
Cost	20040

\[\begin{array}{l} t_0 := \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\lambda_1 \leq -2.5265610430811174 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 9.177905557441422 \cdot 10^{-107}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_2 \cdot \cos \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 23
Error	47.9
Cost	20040

\[\begin{array}{l} t_0 := \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{if}\;\lambda_1 \leq -2.5265610430811174 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 2.1637307840524583 \cdot 10^{-45}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_2 \cdot \cos \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 24
Error	47.3
Cost	19904

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

Alternative 25
Error	52.1
Cost	13376

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

Reproduce

herbie shell --seed 2022320 
(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))