Average Error: 17.6 → 4.0
Time: 1.2min
Precision: binary64
Cost: 71232
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
(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 (sin lambda1) (sin lambda2) (* (cos lambda2) (cos 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(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(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(Float64(cos(phi2) * cos(phi1)), fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(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[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $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 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R

Error

Derivation

  1. Initial program 17.6

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

  2. Applied egg-rr4.0

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

  3. Taylor expanded in phi1 around 0 4.0

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

  4. Simplified4.0

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

  5. Final simplification4.0

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

Alternatives

Alternative 1
Error4.0
Cost64960
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \]
Alternative 2
Error4.0
Cost64960
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \]
Alternative 3
Error10.9
Cost58824
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -0.001771095156682198:\\ \;\;\;\;R \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3785298.970286832:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\\ \end{array} \]
Alternative 4
Error4.0
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \]
Alternative 5
Error10.8
Cost58568
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.001771095156682198:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3785298.970286832:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_0, t_1\right)\right)}\right)\\ \end{array} \]
Alternative 6
Error10.8
Cost58568
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.001771095156682198:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3785298.970286832:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_0, t_1\right)\right)\right)\right)\\ \end{array} \]
Alternative 7
Error10.9
Cost58568
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -0.001771095156682198:\\ \;\;\;\;R \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3785298.970286832:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\\ \end{array} \]
Alternative 8
Error10.8
Cost52552
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.001771095156682198:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3785298.970286832:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_0, t_1\right)\right)\\ \end{array} \]
Alternative 9
Error11.0
Cost46024
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.001771095156682198:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.6452787870445573 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, t_0\right)\right)\\ \end{array} \]
Alternative 10
Error11.1
Cost45768
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.0489324248723593 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.6452787870445573 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, t_0\right)\right)\\ \end{array} \]
Alternative 11
Error11.1
Cost45640
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.0489324248723593 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.6452787870445573 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \end{array} \]
Alternative 12
Error17.4
Cost39896
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ t_1 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\\ t_2 := R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{if}\;\phi_2 \leq -3.653964539340922 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.3675086527382384 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.959794502762625 \cdot 10^{+127}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 4.8504165826405776 \cdot 10^{+191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.0099808415093212 \cdot 10^{+228}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 7.111005260934945 \cdot 10^{+295}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 13
Error17.3
Cost39632
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ t_2 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\lambda_2 \leq -3.324560186503782:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.002215551339273201:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.437220474107897 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 2.6616776691247708 \cdot 10^{+225}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 14
Error17.3
Cost39632
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\lambda_2 \leq -3.324560186503782:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.002215551339273201:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.437220474107897 \cdot 10^{+154}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.6616776691247708 \cdot 10^{+225}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
Alternative 15
Error11.1
Cost39496
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -2.0489324248723593 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.6452787870445573 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 16
Error11.1
Cost39496
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.0489324248723593 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.6452787870445573 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \end{array} \]
Alternative 17
Error25.9
Cost39372
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq 3.3675086527382384 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 9.411803137543734 \cdot 10^{+208}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 7.111005260934945 \cdot 10^{+295}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 18
Error20.8
Cost39368
\[\begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_1 \leq -7325599.827869333:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 1.2193725882016935 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \end{array} \]
Alternative 19
Error30.0
Cost39244
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ t_2 := t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.2150024426448447 \cdot 10^{+253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -7325599.827869333:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + t_2\right)\\ \mathbf{elif}\;\phi_1 \leq 1.2193725882016935 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 20
Error28.6
Cost39112
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq -7325599.827869333:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.2193725882016935 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ \end{array} \]
Alternative 21
Error32.9
Cost32964
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -7325599.827869333:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + t_0 \cdot \left(\cos \phi_1 \cdot \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
Alternative 22
Error32.9
Cost32964
\[\begin{array}{l} t_0 := \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -7325599.827869333:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
Alternative 23
Error40.7
Cost32836
\[\begin{array}{l} t_0 := \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_2 \leq 0.002215551339273201:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \end{array} \]
Alternative 24
Error41.3
Cost26948
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\phi_1 \leq -2.3967311060527453 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \left(\cos \phi_1 \cdot \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 25
Error40.1
Cost26436
\[\begin{array}{l} t_0 := \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\phi_2 \leq 3.3675086527382384 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 26
Error41.4
Cost26436
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\phi_1 \leq -2.3967311060527453 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 27
Error49.9
Cost26308
\[\begin{array}{l} t_0 := \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -2.9452362796445526 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 28
Error49.2
Cost26308
\[\begin{array}{l} t_0 := \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -2.9452362796445526 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 29
Error53.6
Cost19780
\[\begin{array}{l} t_0 := \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -2.9452362796445526 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_2 + t_0\right)\\ \end{array} \]
Alternative 30
Error51.2
Cost19776
\[R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)\right) \]
Alternative 31
Error52.5
Cost13376
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \phi_1\right) \]

Error

Reproduce

herbie shell --seed 2022293 
(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))