Average Error: 16.9 → 3.9
Time: 51.8s
Precision: binary64
Cost: 71232
\[ \begin{array}{c}[phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
(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
    (cos phi2)
    (*
     (cos phi1)
     (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))
    (* (sin phi1) (sin phi2))))
  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(cos(phi2), (cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))), (sin(phi1) * sin(phi2)))) * 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(cos(phi2), Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1)))), Float64(sin(phi1) * sin(phi2)))) * 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[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $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(\cos \phi_2, \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R

Error

Derivation

  1. Initial program 16.9

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

  2. Applied egg-rr3.9

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

  3. Taylor expanded in phi1 around 0 3.9

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

  4. Simplified3.9

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

    [Start]3.9

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

    fma-def [=>]3.9

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

    fma-def [=>]3.9

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

  5. Final simplification3.9

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

Alternatives

Alternative 1
Error3.9
Cost64960
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right) \]
Alternative 2
Error11.2
Cost58828
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t_0\\ t_2 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -0.36:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.7 \cdot 10^{+16}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_2 + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.15 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \log \left(1 + \mathsf{expm1}\left(t_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.3 \cdot 10^{+114}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_2 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \frac{-\cos \phi_2}{\frac{-1}{t_0}}\right)\right)\\ \end{array} \]
Alternative 3
Error10.5
Cost58696
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.15 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.02 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \frac{-\cos \phi_2}{\frac{-1}{t_0}}\right)\right)\\ \end{array} \]
Alternative 4
Error3.9
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \]
Alternative 5
Error11.2
Cost52552
\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\\ t_1 := \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq -0.36:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.7 \cdot 10^{+16}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 \cdot t_0 + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 6.4 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{+114}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \frac{-\cos \phi_2}{\frac{-1}{t_1}}\right)\right)\\ \end{array} \]
Alternative 6
Error11.2
Cost52424
\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\\ t_1 := \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.00088:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 6.4 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, t_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{+114}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \frac{-\cos \phi_2}{\frac{-1}{t_1}}\right)\right)\\ \end{array} \]
Alternative 7
Error11.4
Cost46288
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -0.00088:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 5.6 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_2 \leq 5.5 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{+114}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \frac{-\cos \phi_2}{\frac{-1}{t_0}}\right)\right)\\ \end{array} \]
Alternative 8
Error11.2
Cost46288
\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\\ t_1 := \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.00088:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 6.4 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.88 \cdot 10^{+115}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \frac{-\cos \phi_2}{\frac{-1}{t_1}}\right)\right)\\ \end{array} \]
Alternative 9
Error10.6
Cost45769
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00088 \lor \neg \left(\phi_2 \leq 5.6 \cdot 10^{-19}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
Alternative 10
Error10.6
Cost45768
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.00088:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 5.6 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_0, t_1\right)\right)\\ \end{array} \]
Alternative 11
Error28.3
Cost39637
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_2 \leq 9.5 \cdot 10^{-299}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.1 \cdot 10^{-221}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{+114} \lor \neg \left(\phi_2 \leq 6.3 \cdot 10^{+168}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_1\right)\\ \end{array} \]
Alternative 12
Error17.0
Cost39500
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -0.00088:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{+114}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 13
Error10.6
Cost39497
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00088 \lor \neg \left(\phi_2 \leq 5.6 \cdot 10^{-19}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
Alternative 14
Error20.9
Cost39373
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{+113} \lor \neg \left(\phi_2 \leq 5.5 \cdot 10^{+168}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \end{array} \]
Alternative 15
Error17.1
Cost39368
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.00088:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \end{array} \]
Alternative 16
Error25.7
Cost39108
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq -2020000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 2.6:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \end{array} \]
Alternative 17
Error28.8
Cost33097
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+22} \lor \neg \left(\phi_1 \leq 315\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
Alternative 18
Error30.2
Cost32841
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00084 \lor \neg \left(\phi_1 \leq 0.11\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
Alternative 19
Error29.2
Cost32841
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0039 \lor \neg \left(\phi_1 \leq 0.047\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
Alternative 20
Error30.5
Cost32708
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.038:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
Alternative 21
Error36.0
Cost26436
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 1.7 \cdot 10^{-93}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 22
Error48.9
Cost26308
\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1.45 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 23
Error45.8
Cost26304
\[R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]
Alternative 24
Error48.7
Cost19908
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.022:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1 + \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right)\right)\\ \end{array} \]
Alternative 25
Error49.8
Cost19908
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.00014:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1 + \phi_2 \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2 + \phi_2 \cdot \phi_1\right)\\ \end{array} \]
Alternative 26
Error47.2
Cost19904
\[R \cdot \cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]
Alternative 27
Error52.3
Cost13376
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \phi_1\right) \]

Error

Reproduce

herbie shell --seed 2023011 
(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))