Average Error: 16.4 → 3.8
Time: 1.1min
Precision: binary64
Cost: 71232
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right) \cdot R \]
(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 phi2)
    (sin phi1)
    (*
     (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))
     (* (cos phi2) (cos phi1)))))
  R))
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(phi2), sin(phi1), (fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))) * (cos(phi2) * cos(phi1))))) * R;
}

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(phi2), sin(phi1), Float64(fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))) * Float64(cos(phi2) * cos(phi1))))) * R)
end

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

\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_2, \sin \phi_1, \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)\right) \cdot R

Error

Derivation

  1. Initial program 16.4

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

  2. Applied egg-rr3.8

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

  3. Applied egg-rr3.8

    \[\leadsto \cos^{-1} \left(\color{blue}{\sqrt[3]{{\left(\sin \phi_1 \cdot \sin \phi_2\right)}^{3}}} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

  4. Taylor expanded in phi1 around 0 3.8

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

  5. Simplified3.8

    \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right)} \cdot R \]
    Proof
    (acos.f64 (fma.f64 (sin.f64 phi2) (sin.f64 phi1) (*.f64 (fma.f64 (cos.f64 lambda2) (cos.f64 lambda1) (*.f64 (sin.f64 lambda2) (sin.f64 lambda1))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (sin.f64 phi2) (sin.f64 phi1)) (*.f64 (fma.f64 (cos.f64 lambda2) (cos.f64 lambda1) (*.f64 (sin.f64 lambda2) (sin.f64 lambda1))) (*.f64 (cos.f64 phi2) (cos.f64 phi1)))))): 3 points increase in error, 5 points decrease in error
    (acos.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sin.f64 phi1) (sin.f64 phi2))) (*.f64 (fma.f64 (cos.f64 lambda2) (cos.f64 lambda1) (*.f64 (sin.f64 lambda2) (sin.f64 lambda1))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (fma.f64 (cos.f64 lambda2) (cos.f64 lambda1) (*.f64 (sin.f64 lambda2) (sin.f64 lambda1))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))) (*.f64 (sin.f64 phi1) (sin.f64 phi2))))): 0 points increase in error, 0 points decrease in error

  6. Final simplification3.8

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

Alternatives

Alternative 1
Error3.8
Cost64960
\[R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_1\right) + \sin \phi_2 \cdot \sin \phi_1\right) \]
Alternative 2
Error10.7
Cost58824
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -3923.6053227895122:\\ \;\;\;\;R \cdot \cos^{-1} \left(\log \left(1 + \mathsf{expm1}\left(\sin \phi_2 \cdot \sin \phi_1\right)\right) + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1087859110.54411:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 3
Error3.8
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right) \]
Alternative 4
Error10.7
Cost52552
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -3923.6053227895122:\\ \;\;\;\;R \cdot \cos^{-1} \left(\log \left(1 + \mathsf{expm1}\left(\sin \phi_2 \cdot \sin \phi_1\right)\right) + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1087859110.54411:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 5
Error15.5
Cost52296
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -7.59433999211927 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.095239535030533 \cdot 10^{-144}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \phi_1 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error15.5
Cost52296
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -7.59433999211927 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.095239535030533 \cdot 10^{-144}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_1 + \sin \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error15.5
Cost52296
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -7.59433999211927 \cdot 10^{-198}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sqrt[3]{{\left(\sin \phi_2 \cdot \sin \phi_1\right)}^{3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 3.095239535030533 \cdot 10^{-144}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_1 + \sin \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 8
Error15.5
Cost52296
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -7.59433999211927 \cdot 10^{-198}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\log \left(1 + \mathsf{expm1}\left(\sin \phi_2 \cdot \sin \phi_1\right)\right) + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.095239535030533 \cdot 10^{-144}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \cos \phi_1 + \sin \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 9
Error15.5
Cost46024
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -7.59433999211927 \cdot 10^{-198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.095239535030533 \cdot 10^{-144}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \phi_1 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error16.4
Cost45504
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right) \]
Alternative 11
Error23.1
Cost39368
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)\right)\\ \mathbf{if}\;\phi_2 \leq -3923.6053227895122:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 26223860418866.527:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 12
Error29.3
Cost39244
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -3923.6053227895122:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 1.659110507069364 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 7.956002351620146 \cdot 10^{+149}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ \end{array} \]
Alternative 13
Error27.8
Cost39240
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -3923.6053227895122:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 9.755531768440164 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 14
Error27.9
Cost39240
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -6.325929759800967 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.4402032423467717 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \]
Alternative 15
Error23.0
Cost39236
\[\begin{array}{l} t_0 := \sin \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_2 \leq 3.589699228275405 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \end{array} \]
Alternative 16
Error16.4
Cost39232
\[R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]
Alternative 17
Error30.0
Cost33104
\[\begin{array}{l} t_0 := \sin \phi_2 \cdot \phi_1\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.1633653951934317 \cdot 10^{+146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -3.977352586704644 \cdot 10^{+62}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -0.05829832698614223:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_1 \leq 5.78125110883513 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_2 \cdot t_1\right) \cdot \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 18
Error30.1
Cost33096
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + t_0\right)\\ \mathbf{if}\;\phi_1 \leq -1.764887932233001 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 5.78125110883513 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 19
Error29.3
Cost33096
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + t_0\right)\\ \mathbf{if}\;\phi_2 \leq -3923.6053227895122:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.659110507069364 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 7.956002351620146 \cdot 10^{+149}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 20
Error37.0
Cost26948
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.05829832698614223:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \phi_1 + \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right)\right)\\ \end{array} \]
Alternative 21
Error43.7
Cost26572
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \phi_1 + \cos \lambda_1 \cdot \cos \phi_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.044734356232553424:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -2.1955599710420832 \cdot 10^{-242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 8.891670849757005 \cdot 10^{-297}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 22
Error36.4
Cost26436
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.05829832698614223:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 23
Error48.1
Cost26308
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.1234870791901225 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \phi_1 + \cos \lambda_1 \cdot \cos \phi_2\right)\\ \end{array} \]
Alternative 24
Error53.6
Cost19780
\[\begin{array}{l} t_0 := \sin \phi_2 \cdot \phi_1\\ \mathbf{if}\;\lambda_1 \leq -4.296474584155235:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_2 + t_0\right)\\ \end{array} \]
Alternative 25
Error50.9
Cost19776
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \sin \phi_2 \cdot \phi_1\right) \]
Alternative 26
Error52.3
Cost13376
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_2 \cdot \phi_1\right) \]

Error

Reproduce

herbie shell --seed 2022311 
(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))