Spherical law of cosines

Average Error: 16.1 → 3.7

Time: 49.9s

Precision: binary64

Cost: 77760

Math

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

↓

\[\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot R \]

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 phi2)
     (*
      (cos phi1)
      (+
       (log1p (expm1 (* (sin lambda1) (sin lambda2))))
       (* (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(phi2) * (cos(phi1) * (log1p(expm1((sin(lambda1) * sin(lambda2)))) + (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(phi2) * Float64(cos(phi1) * Float64(log1p(expm1(Float64(sin(lambda1) * sin(lambda2)))) + 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[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Log[1 + N[(Exp[N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 16.1

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

2. Simplified 16.1

   \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R} \]
   Proof
   (*.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): 2 points increase in error, 1 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): 6 points increase in error, 3 points decrease in error

3. Applied egg-rr 3.7

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

4. Applied egg-rr 3.7

   \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot R \]

5. Final simplification 3.7

   \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right) \cdot R \]

Alternatives

Alternative 1
Error	3.7
Cost	64960

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

Alternative 2
Error	3.7
Cost	64960

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

Alternative 3
Error	12.4
Cost	59352

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_2 := \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_1\right)\\ t_3 := \cos^{-1} t_0\\ t_4 := R \cdot \log \left(e^{t_3}\right)\\ \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+244}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -1.02 \cdot 10^{+93}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + t_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -2 \cdot 10^{+64}:\\ \;\;\;\;R \cdot t_3\\ \mathbf{elif}\;\phi_1 \leq -9.6 \cdot 10^{+32}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_2 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 4
Error	12.4
Cost	59096

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_2 := \sin \lambda_1 \cdot \sin \lambda_2 + t_1\\ t_3 := \cos^{-1} t_0\\ t_4 := R \cdot \log \left(e^{t_3}\right)\\ \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{+244}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -3.7 \cdot 10^{+90}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq -1.75 \cdot 10^{+64}:\\ \;\;\;\;R \cdot t_3\\ \mathbf{elif}\;\phi_1 \leq -2.9 \cdot 10^{+29}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.1 \cdot 10^{-10}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 5
Error	3.8
Cost	58688

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

Alternative 6
Error	12.4
Cost	52228

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_2 := \sin \lambda_1 \cdot \sin \lambda_2 + t_1\\ t_3 := R \cdot \cos^{-1} t_0\\ \mathbf{if}\;\phi_1 \leq -9.8 \cdot 10^{+244}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -4.6 \cdot 10^{+91}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq -1.75 \cdot 10^{+64}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_1 \leq -9 \cdot 10^{+30}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	12.4
Cost	46296

\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_1 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ t_2 := \sin \lambda_1 \cdot \sin \lambda_2 + t_0\\ \mathbf{if}\;\phi_1 \leq -1.56 \cdot 10^{+246}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -6.1 \cdot 10^{+91}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq -1.8 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -2.1 \cdot 10^{+30}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -5.4 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	12.4
Cost	46296

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_2 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ t_3 := \sin \lambda_1 \cdot \sin \lambda_2 + t_1\\ \mathbf{if}\;\phi_1 \leq -8.3 \cdot 10^{+244}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot t_0, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.12 \cdot 10^{+93}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_3\right)\\ \mathbf{elif}\;\phi_1 \leq -3.2 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -9.6 \cdot 10^{+32}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -2.2 \cdot 10^{-10}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	12.4
Cost	45904

\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \sin \lambda_1 \cdot \sin \lambda_2 + t_0\\ \mathbf{if}\;\phi_1 \leq -8.3 \cdot 10^{+244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1.12 \cdot 10^{+93}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq -6 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -9.6 \cdot 10^{+32}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.5 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	12.7
Cost	45904

\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \sin \lambda_1 \cdot \sin \lambda_2 + t_0\\ \mathbf{if}\;\phi_1 \leq -8.3 \cdot 10^{+244}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -4.1 \cdot 10^{+92}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq -2.6 \cdot 10^{+65}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{+30}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	12.5
Cost	40024

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ t_2 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.56 \cdot 10^{+246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.05 \cdot 10^{+93}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -4.6 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	16.6
Cost	39764

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ t_1 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ t_2 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.25 \cdot 10^{+246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -3 \cdot 10^{+92}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -1.05 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 0.00023:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	26.0
Cost	39236

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3200000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]

Alternative 14
Error	24.4
Cost	39236

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 15
Error	31.8
Cost	32580

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\cos \phi_1 \cdot t_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 16
Error	31.8
Cost	19780

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 8 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 17
Error	42.1
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 18
Error	40.6
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 19
Error	36.7
Cost	19648

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

Alternative 20
Error	47.4
Cost	13120

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

Reproduce

herbie shell --seed 2022329 
(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))