Average Error: 16.9 → 3.8
Time: 1.6min
Precision: binary64
Cost: 84032
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)}\right) \cdot R \]
(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
 (*
  (log
   (exp
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (*
       (cos phi2)
       (*
        (cos phi1)
        (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2)))))))))
  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 log(exp(acos(fma(sin(phi1), sin(phi2), (cos(phi2) * (cos(phi1) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2))))))))) * 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(log(exp(acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * Float64(cos(phi1) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2))))))))) * 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[Log[N[Exp[N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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
↓
\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\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. Simplified16.9

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

  3. Applied egg-rr16.9

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

  4. Applied egg-rr3.8

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

  5. Final simplification3.8

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

Alternatives

Alternative 1
Error3.8
Cost71232
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
Alternative 2
Error3.8
Cost64960
\[R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right) \]
Alternative 3
Error3.8
Cost64960
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \]
Alternative 4
Error10.7
Cost58696
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq -1161800693789283:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00037065977832448897:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot {\left({\cos^{-1} \left(\mathsf{fma}\left(t_1, t_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]
Alternative 5
Error3.8
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \]
Alternative 6
Error11.7
Cost52688
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.2082743525283767 \cdot 10^{+116}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_2\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -13364422.449539104:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -1.2891045211223944 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t_1, t_2, t_3\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00037065977832448897:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + \cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_1 \cdot t_2\right)\right)\right)\right)\\ \end{array} \]
Alternative 7
Error10.7
Cost52552
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq -1161800693789283:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00037065977832448897:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_1 \cdot t_0\right)\right)\right)\right)\\ \end{array} \]
Alternative 8
Error11.7
Cost52492
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq -1.2082743525283767 \cdot 10^{+116}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -13364422.449539104:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.2891045211223944 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t_2, t_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00037065977832448897:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(t_0 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_2 \cdot t_1\right)\right)\right)\right)\\ \end{array} \]
Alternative 9
Error11.7
Cost52492
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.2082743525283767 \cdot 10^{+116}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_2\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -13364422.449539104:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_3 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -1.2891045211223944 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t_1, t_2, t_3\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00037065977832448897:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_1 \cdot t_2\right)\right)\right)\right)\\ \end{array} \]
Alternative 10
Error11.8
Cost46288
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.2082743525283767 \cdot 10^{+116}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -13364422.449539104:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.2891045211223944 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00037065977832448897:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(t_0 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_1\right)\\ \end{array} \]
Alternative 11
Error11.8
Cost46288
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.2082743525283767 \cdot 10^{+116}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -13364422.449539104:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.2891045211223944 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00037065977832448897:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(t_0 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(3 \cdot \left(0.3333333333333333 \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_1\right)\right)\right)\right)\\ \end{array} \]
Alternative 12
Error12.0
Cost45640
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ t_2 := R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1.2082743525283767 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -13364422.449539104:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.6490259201387514 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 2.352903685598969 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(t_0 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 13
Error12.0
Cost45640
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.2082743525283767 \cdot 10^{+116}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -13364422.449539104:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.6490259201387514 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 2.352903685598969 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(t_0 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_1\right)\\ \end{array} \]
Alternative 14
Error12.0
Cost39760
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\\ \mathbf{if}\;\phi_1 \leq -1.2082743525283767 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -13364422.449539104:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_2\right)\\ \mathbf{elif}\;\phi_1 \leq -1.6490259201387514 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 2.352903685598969 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 15
Error17.2
Cost39632
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_2 := R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{if}\;\phi_2 \leq -2.5795563652859377 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 5.5158265213743536 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_2 \leq 4.059721371960454 \cdot 10^{+254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 3.133557406109485 \cdot 10^{+302}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 16
Error16.9
Cost39500
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ t_1 := \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\\ \mathbf{if}\;\phi_1 \leq -1.1322134743168485 \cdot 10^{+202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -0.0003088993329805857:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_1 \leq 7.388132624794298 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 17
Error25.4
Cost39236
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.8462986057918003 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 18
Error23.9
Cost39236
\[\begin{array}{l} t_0 := \cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\\ \mathbf{if}\;\phi_2 \leq 5.5158265213743536 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 19
Error31.5
Cost39108
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1161800693789283:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
Alternative 20
Error31.7
Cost39108
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.0003088993329805857:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t_0, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \]
Alternative 21
Error36.2
Cost33164
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.158507859769614 \cdot 10^{+195}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.9166791020619406 \cdot 10^{+94}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_1 \leq -0.013378565766802002:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\phi_1, \phi_2, \cos \phi_1 \cdot t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot t_0\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right)\\ \end{array} \]
Alternative 22
Error35.9
Cost32964
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.013378565766802002:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right)\\ \end{array} \]
Alternative 23
Error32.9
Cost32964
\[\begin{array}{l} t_0 := \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1161800693789283:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0\right)\\ \end{array} \]
Alternative 24
Error36.2
Cost32840
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.158507859769614 \cdot 10^{+195}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.9166791020619406 \cdot 10^{+94}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_1 \leq -0.013378565766802002:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0 + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot t_0\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right)\\ \end{array} \]
Alternative 25
Error35.9
Cost32708
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.013378565766802002:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right)\\ \end{array} \]
Alternative 26
Error42.0
Cost26948
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 1.68903798352085 \cdot 10^{-52}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0 + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot t_0\right) \cdot \left(-0.5 \cdot \left(\phi_1 \cdot \phi_1\right) + 1\right)\right)\\ \end{array} \]
Alternative 27
Error42.0
Cost26436
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 1.68903798352085 \cdot 10^{-52}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0 + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 28
Error49.2
Cost26308
\[\begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 2.9705120672662974 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 29
Error46.2
Cost26304
\[R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \]
Alternative 30
Error51.1
Cost19908
\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2729.8956173312417:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2 + \phi_1 \cdot \phi_2\right)\\ \end{array} \]
Alternative 31
Error50.5
Cost19908
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.292261328241503 \cdot 10^{-19}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1 + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2 + \phi_1 \cdot \phi_2\right)\\ \end{array} \]
Alternative 32
Error47.5
Cost19904
\[R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right) \]
Alternative 33
Error52.2
Cost13376
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right) \]

Error

Reproduce

herbie shell --seed 2022298 
(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))