Spherical law of cosines

Average Error: 16.6 → 3.9

Time: 1.2min

Precision: binary64

Cost: 84288

Math

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

↓

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \mathsf{fma}\left(t_0, \cos \lambda_1 \cdot \cos \lambda_2, t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right) \cdot R \end{array} \]

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
 (let* ((t_0 (* (cos phi2) (cos phi1))))
   (*
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (fma
       t_0
       (* (cos lambda1) (cos lambda2))
       (* t_0 (* (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) {
	double t_0 = cos(phi2) * cos(phi1);
	return acos(fma(sin(phi1), sin(phi2), fma(t_0, (cos(lambda1) * cos(lambda2)), (t_0 * (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)
	t_0 = Float64(cos(phi2) * cos(phi1))
	return Float64(acos(fma(sin(phi1), sin(phi2), fma(t_0, Float64(cos(lambda1) * cos(lambda2)), Float64(t_0 * 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_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $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

   \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R} \]
   Proof
   (*.f64 (acos.f64 (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (cos.f64 (-.f64 lambda1 lambda2)))))) R): 0 points increase in error, 0 points decrease in error
   (*.f64 (acos.f64 (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi2) (cos.f64 phi1)) (cos.f64 (-.f64 lambda1 lambda2)))))) R): 4 points increase in error, 2 points decrease in error
   (*.f64 (acos.f64 (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 phi1) (cos.f64 phi2))) (cos.f64 (-.f64 lambda1 lambda2))))) R): 0 points increase in error, 0 points decrease in error
   (*.f64 (acos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))) R): 3 points increase in error, 4 points decrease in error

3. Applied egg-rr 3.9

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

4. Final simplification 3.9

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

Alternatives

Alternative 1
Error	3.9
Cost	64960

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

Alternative 2
Error	3.9
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 3
Error	11.6
Cost	58824

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ t_2 := \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -0.00016355653694606447:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 9.490748390709758 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.6025750456240318 \cdot 10^{+99}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(t_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_1, t_0\right)\right)\\ \end{array} \]

Alternative 4
Error	11.7
Cost	58696

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -0.00016355653694606447:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.9104801520103361 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.6025750456240318 \cdot 10^{+99}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(t_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, t_0\right)\right)\\ \end{array} \]

Alternative 5
Error	3.9
Cost	58688

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

Alternative 6
Error	11.7
Cost	52556

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_2 \leq -0.00016355653694606447:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.9104801520103361 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.6025750456240318 \cdot 10^{+99}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, t_0\right)\right)\\ \end{array} \]

Alternative 7
Error	10.4
Cost	52424

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.00016355653694606447:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.9104801520103361 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sqrt[3]{{t_0}^{3}} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_1\right)\\ \end{array} \]

Alternative 8
Error	16.0
Cost	52360

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -2.6302203400713003 \cdot 10^{-80}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.0911530979295263 \cdot 10^{-143}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sqrt[3]{{\left(\sin \phi_1 \cdot \sin \phi_2\right)}^{3}} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \end{array} \]

Alternative 9
Error	16.0
Cost	52296

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -2.6302203400713003 \cdot 10^{-80}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.0911530979295263 \cdot 10^{-143}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0 + \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	16.0
Cost	52296

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -2.6302203400713003 \cdot 10^{-80}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.0911530979295263 \cdot 10^{-143}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0 + \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	16.0
Cost	45896

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.6302203400713003 \cdot 10^{-80}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.0911530979295263 \cdot 10^{-143}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]

Alternative 12
Error	16.0
Cost	45768

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -2.6302203400713003 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.0911530979295263 \cdot 10^{-143}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	16.0
Cost	45768

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.6302203400713003 \cdot 10^{-80}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.0911530979295263 \cdot 10^{-143}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]

Alternative 14
Error	16.0
Cost	39624

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -2.6302203400713003 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.0911530979295263 \cdot 10^{-143}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	23.0
Cost	39368

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{if}\;\phi_1 \leq -1.399854607501411 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 0.016762427276451514:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	27.6
Cost	39240

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -0.00016355653694606447:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 9.490748390709758 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	23.2
Cost	39236

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -15567.378987367014:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 18
Error	16.6
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 19
Error	29.0
Cost	33360

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ t_2 := t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.943533192873476 \cdot 10^{+110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -3.623605109198626 \cdot 10^{+52}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + t_2\right)\\ \mathbf{elif}\;\phi_1 \leq -2.616089606644368 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 0.016762427276451514:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	29.1
Cost	33096

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{if}\;\phi_1 \leq -2.616089606644368 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 0.016762427276451514:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	29.2
Cost	32840

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -2.7125478847829905 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 0.016762427276451514:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 22
Error	41.5
Cost	26948

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 6.873882961343751 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot t_0\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right)\\ \end{array} \]

Alternative 23
Error	41.5
Cost	26436

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 6.873882961343751 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 24
Error	48.9
Cost	26308

\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -15567.378987367014:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2 + t_0\right)\\ \end{array} \]

Alternative 25
Error	45.9
Cost	26304

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

Alternative 26
Error	50.6
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.399854607501411 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 27
Error	50.0
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -15567.378987367014:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 28
Error	47.2
Cost	19904

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

Alternative 29
Error	52.2
Cost	13376

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

Reproduce

herbie shell --seed 2022308 
(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))