Spherical law of cosines

Average Error: 16.9 → 3.7

Time: 42.8s

Precision: binary64

Cost: 71488

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

Math

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

↓

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

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 phi1)
     (+
      (* (sin lambda2) (* (cos phi2) (sin lambda1)))
      (* (cos phi2) (* (cos lambda1) (cos lambda2)))))))
  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(phi1) * ((sin(lambda2) * (cos(phi2) * sin(lambda1))) + (cos(phi2) * (cos(lambda1) * cos(lambda2))))))) * 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(phi1) * Float64(Float64(sin(lambda2) * Float64(cos(phi2) * sin(lambda1))) + Float64(cos(phi2) * Float64(cos(lambda1) * cos(lambda2))))))) * 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[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 16.9

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

2. Simplified 16.9

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

   [Start] 16.9	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   fma-def [=>] 16.9	\[ \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]
   associate-*l* [=>] 16.9	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R \]

3. Applied egg-rr 3.6

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

4. Taylor expanded in lambda1 around inf 3.6

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

5. Simplified 3.7

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

   [Start] 3.6	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)\right)\right) \cdot R \]
   *-commutative [=>] 3.6	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)} + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)\right)\right) \cdot R \]
   associate-*l* [<=] 3.7	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \phi_2 \cdot \sin \lambda_1\right) \cdot \sin \lambda_2} + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)\right)\right) \cdot R \]
   *-commutative [=>] 3.7	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\color{blue}{\sin \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)} + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2\right)\right)\right) \cdot R \]

6. Final simplification 3.7

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

Alternatives

Alternative 1
Error	3.6
Cost	71232

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

Alternative 2
Error	3.6
Cost	64960

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

Alternative 3
Error	11.0
Cost	58696

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -7.5 \cdot 10^{-70}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} t_0}\right)\\ \end{array} \]

Alternative 4
Error	10.9
Cost	58696

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -7.5 \cdot 10^{-70}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.25 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot {\left({\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 5
Error	3.7
Cost	58688

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

Alternative 6
Error	10.3
Cost	52552

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -25500:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00021:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot t_1 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_1 \cdot \sqrt[3]{{t_0}^{3}}\right)\\ \end{array} \]

Alternative 7
Error	15.6
Cost	52297

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.22 \cdot 10^{-198} \lor \neg \left(\phi_2 \leq 5.5 \cdot 10^{-175}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 8
Error	15.7
Cost	52296

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -2.25 \cdot 10^{-203}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-180}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	15.6
Cost	46025

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.56 \cdot 10^{-201} \lor \neg \left(\phi_2 \leq 3.5 \cdot 10^{-175}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]

Alternative 10
Error	16.9
Cost	45504

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

Alternative 11
Error	23.6
Cost	39633

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{+136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -7.2 \cdot 10^{+49}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -0.0022 \lor \neg \left(\phi_1 \leq 1.06 \cdot 10^{-19}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 + \left(\cos \phi_2 \cdot \left(\phi_1 \cdot \phi_1\right)\right) \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 12
Error	17.2
Cost	39236

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_2 \cdot t_0\right)\\ \end{array} \]

Alternative 13
Error	16.9
Cost	39232

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

Alternative 14
Error	26.2
Cost	39108

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -1:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 760000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0 \cdot \left(1 + -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right)\\ \end{array} \]

Alternative 15
Error	28.4
Cost	33220

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right) \cdot t_0\\ \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{+16}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0 \cdot \left(1 + -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 16
Error	29.4
Cost	32964

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 17
Error	35.8
Cost	32836

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 3.7 \cdot 10^{-16}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_2 \cdot t_0\right)\\ \end{array} \]

Alternative 18
Error	35.7
Cost	32832

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

Alternative 19
Error	36.1
Cost	26436

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 68000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 20
Error	47.8
Cost	26308

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 6 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_2 + t_0\right)\\ \end{array} \]

Alternative 21
Error	46.2
Cost	26308

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 22
Error	46.1
Cost	26304

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

Alternative 23
Error	51.2
Cost	19780

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_2 + t_0\right)\\ \end{array} \]

Alternative 24
Error	51.1
Cost	19776

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

Alternative 25
Error	52.5
Cost	13376

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

Reproduce

herbie shell --seed 2023027 
(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))