Spherical law of cosines

Average Error: 17.2 → 3.6

Time: 49.0s

Precision: binary64

Cost: 64960

Math

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

↓

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

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

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. Simplified 17.1

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

   [Start] 17.2	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   fma-def [=>] 17.1	\[ \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]
   *-commutative [=>] 17.1	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]
   associate-*l* [=>] 17.1	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R \]

3. Applied egg-rr 3.7

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

4. Simplified 3.6

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

   [Start] 3.7	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot R \]
   distribute-lft-out [=>] 3.6	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)\right) \cdot R \]

5. Final simplification 3.6

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

Alternatives

Alternative 1
Error	3.7
Cost	64960

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

Alternative 2
Error	11.0
Cost	58824

\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -6.2 \cdot 10^{+19}:\\ \;\;\;\;R \cdot \log \left(e^{t_0}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.75 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot t_0\\ \end{array} \]

Alternative 3
Error	3.7
Cost	58688

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

Alternative 4
Error	11.1
Cost	58436

\[\begin{array}{l} t_0 := \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \log \left(e^{t_0}\right)\\ \mathbf{elif}\;\phi_1 \leq 2 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot t_0\\ \end{array} \]

Alternative 5
Error	11.0
Cost	52553

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.2 \cdot 10^{+19} \lor \neg \left(\phi_1 \leq 1.75 \cdot 10^{-20}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 6
Error	16.1
Cost	45896

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.4 \cdot 10^{-222}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{-115}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	16.2
Cost	45768

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.3 \cdot 10^{-225}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_0, t_1\right)\right)\\ \end{array} \]

Alternative 8
Error	16.2
Cost	45768

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -8.5 \cdot 10^{-223}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-115}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	16.2
Cost	39625

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.1 \cdot 10^{-224} \lor \neg \left(\phi_2 \leq 3 \cdot 10^{-115}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]

Alternative 10
Error	28.7
Cost	39505

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right)\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -330000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 9 \cdot 10^{+197} \lor \neg \left(\phi_2 \leq 1.1 \cdot 10^{+227}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	28.7
Cost	39240

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.3 \cdot 10^{+207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -0.0076:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.135:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 + \left(\cos \phi_2 \cdot \left(\phi_1 \cdot \phi_1\right)\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	26.1
Cost	39236

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3400000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 13
Error	24.4
Cost	39236

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3400000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 14
Error	17.2
Cost	39232

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

Alternative 15
Error	29.7
Cost	33612

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{+208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -0.022:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot t_2 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0082:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_2 \cdot \left(\cos \phi_2 + \left(\cos \phi_2 \cdot \left(\phi_1 \cdot \phi_1\right)\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	29.6
Cost	33228

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{+205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -2.1 \cdot 10^{+24}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 58000000000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	29.8
Cost	33228

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ t_2 := t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.58 \cdot 10^{+206}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -5 \cdot 10^{+57}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq 7 \cdot 10^{+17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	29.8
Cost	32972

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{if}\;\phi_1 \leq -5.7 \cdot 10^{+205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -0.0017:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0008:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	39.1
Cost	26436

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.00037:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 20
Error	35.5
Cost	26436

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0031:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 21
Error	49.8
Cost	26308

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1.75 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 22
Error	46.4
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	51.5
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0036:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\phi_1 \cdot \phi_1\right) \cdot -0.5 + 1\right)\right)\\ \end{array} \]

Alternative 24
Error	50.9
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.75 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 25
Error	47.8
Cost	19904

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

Alternative 26
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 2023088 
(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))