Spherical law of cosines

Average Error: 17.0 → 3.8

Time: 1.1min

Precision: binary64

Cost: 64960

Math

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

↓

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

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) (cos phi2))
     (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda2) (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(phi1) * cos(phi2)) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * 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(Float64(cos(phi1) * cos(phi2)) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * 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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $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. Simplified 17.0

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

3. Applied egg-rr 3.8

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

4. Final simplification 3.8

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

Alternatives

Alternative 1
Error	3.8
Cost	58688

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

Alternative 2
Error	10.9
Cost	58436

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3148555401365.3325:\\ \;\;\;\;R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(t_2, t_0, t_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.529952202849312 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_2, t_1\right)\right)\\ \end{array} \]

Alternative 3
Error	11.4
Cost	52552

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.607753673103088 \cdot 10^{+40}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 4.882370933655837 \cdot 10^{-29}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	10.9
Cost	52552

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -3148555401365.3325:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + t_0 \cdot t_1\right)\\ \mathbf{elif}\;\phi_2 \leq 3.529952202849312 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_1, t_2\right)\right)\\ \end{array} \]

Alternative 5
Error	16.3
Cost	46152

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -9.390396627828066 \cdot 10^{-256}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 2.496589322480471 \cdot 10^{-95}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, t_1\right)\right)\\ \end{array} \]

Alternative 6
Error	16.3
Cost	45896

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -9.390396627828066 \cdot 10^{-256}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.496589322480471 \cdot 10^{-95}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	16.3
Cost	45896

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -9.390396627828066 \cdot 10^{-256}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 2.496589322480471 \cdot 10^{-95}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, t_1\right)\right)\\ \end{array} \]

Alternative 8
Error	16.3
Cost	45768

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -9.390396627828066 \cdot 10^{-256}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.496589322480471 \cdot 10^{-95}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	16.3
Cost	39624

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -9.390396627828066 \cdot 10^{-256}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.496589322480471 \cdot 10^{-95}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	24.2
Cost	39368

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{if}\;\phi_1 \leq -3.5509407720656982:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 2.3717253196116622 \cdot 10^{-123}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 + \left(\cos \phi_2 \cdot \left(\phi_1 \cdot \phi_1\right)\right) \cdot \left(-0.5 + \left(\phi_1 \cdot \phi_1\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	23.6
Cost	39236

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -6.985255505077415 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 12
Error	23.6
Cost	39236

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -6.985255505077415 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 13
Error	17.0
Cost	39232

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

Alternative 14
Error	28.7
Cost	39108

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.5509407720656982:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 2.2065345907981463 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0 \cdot \left(\cos \phi_2 + \left(\cos \phi_2 \cdot \left(\phi_1 \cdot \phi_1\right)\right) \cdot \left(-0.5 + \left(\phi_1 \cdot \phi_1\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 15
Error	29.9
Cost	33996

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := R \cdot \cos^{-1} \left(t_2 + \sin \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq -3148555401365.3325:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_2 \leq 2.8192110658072435 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 \cdot t_1 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.411087625078105 \cdot 10^{+234}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + t_1 \cdot \left(\cos \phi_2 + \left(\cos \phi_2 \cdot \left(\phi_1 \cdot \phi_1\right)\right) \cdot \left(-0.5 + \left(\phi_1 \cdot \phi_1\right) \cdot 0.041666666666666664\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.519744578582305 \cdot 10^{+255}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot t_1\right)\\ \end{array} \]

Alternative 16
Error	29.9
Cost	33228

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := R \cdot \cos^{-1} \left(t_2 + \sin \phi_1 \cdot \sin \phi_2\right)\\ t_4 := t_2 \cdot t_1\\ \mathbf{if}\;\phi_2 \leq -3148555401365.3325:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_2 \leq 5.29047719721078 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_4 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.411087625078105 \cdot 10^{+234}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_4 + t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 6.519744578582305 \cdot 10^{+255}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot t_1\right)\\ \end{array} \]

Alternative 17
Error	29.9
Cost	33104

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := R \cdot \cos^{-1} \left(t_2 + \sin \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq -3148555401365.3325:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_2 \leq 2.8192110658072435 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 \cdot t_1 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.411087625078105 \cdot 10^{+234}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + t_1 \cdot \left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.519744578582305 \cdot 10^{+255}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot t_1\right)\\ \end{array} \]

Alternative 18
Error	29.8
Cost	32840

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.5509407720656982:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 2.2065345907981463 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 19
Error	42.7
Cost	26948

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 2.496589322480471 \cdot 10^{-95}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0 \cdot \left(\cos \phi_2 \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot -0.5\right)\right)\right)\\ \end{array} \]

Alternative 20
Error	39.6
Cost	26436

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -109914.66356005774:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 21
Error	49.5
Cost	26308

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -54227.68041796798:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 22
Error	51.2
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -109914.66356005774:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 23
Error	50.6
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -54227.68041796798:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 24
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 25
Error	52.8
Cost	13376

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

Reproduce

herbie shell --seed 2022241 
(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))