Spherical law of cosines

Average Error: 17.0 → 3.7

Time: 51.4s

Precision: binary64

Cost: 71232

Math

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

↓

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

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

Derivation

1. Initial program 17.0

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

2. Applied egg-rr 3.7

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

3. Simplified 3.7

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

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

4. Taylor expanded in phi1 around 0 3.7

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

5. Simplified 3.7

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

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

6. Final simplification 3.7

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

Alternatives

Alternative 1
Error	3.7
Cost	64960

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

Alternative 2
Error	3.7
Cost	64960

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

Alternative 3
Error	10.5
Cost	58696

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.45 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(e^{\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)} + -1\right) + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	3.7
Cost	58688

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

Alternative 5
Error	10.9
Cost	58436

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3750000000:\\ \;\;\;\;R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	10.9
Cost	52553

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -6000000 \lor \neg \left(\phi_1 \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 7
Error	10.9
Cost	52425

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.105 \lor \neg \left(\phi_1 \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]

Alternative 8
Error	16.0
Cost	46025

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.9 \cdot 10^{-206} \lor \neg \left(\phi_2 \leq 2.35 \cdot 10^{-191}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]

Alternative 9
Error	10.9
Cost	46025

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.105 \lor \neg \left(\phi_1 \leq 6.5 \cdot 10^{-21}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]

Alternative 10
Error	17.0
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	24.4
Cost	39500

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1.5 \cdot 10^{+182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0013:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 + \left(\cos \phi_2 \cdot \left(\phi_1 \cdot \phi_1\right)\right) \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	23.2
Cost	39236

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -320:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \end{array} \]

Alternative 13
Error	17.0
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 14
Error	31.6
Cost	39108

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -6000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 15
Error	31.8
Cost	39108

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \]

Alternative 16
Error	39.9
Cost	33868

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ t_2 := t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+86}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -5000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq -0.4:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \left(1 + -0.5 \cdot \left(\lambda_2 \cdot \lambda_2\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	37.5
Cost	32964

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq 8200000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \lambda_2 \cdot \sin \lambda_1 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \phi_2 \cdot \phi_1\right)\right)\\ \end{array} \]

Alternative 18
Error	33.1
Cost	32964

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{+77}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 19
Error	36.5
Cost	32832

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

Alternative 20
Error	42.0
Cost	26436

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 2.9 \cdot 10^{-123}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 21
Error	50.1
Cost	26308

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.105:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 22
Error	49.2
Cost	26308

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1850000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \cos \phi_1\right)\\ \end{array} \]

Alternative 23
Error	46.6
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 24
Error	50.8
Cost	20420

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot -0.5\right)\right)\\ \end{array} \]

Alternative 25
Error	50.2
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.5 \cdot 10^{+30}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 26
Error	53.1
Cost	19780

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1850000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_2 + t_0\right)\\ \end{array} \]

Alternative 27
Error	51.0
Cost	19776

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

Alternative 28
Error	52.2
Cost	13376

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

Reproduce

herbie shell --seed 2023041 
(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))