Average Error: 16.9 → 4.2
Time: 45.8s
Precision: binary64
Cost: 71232
\[ \begin{array}{c}[phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
(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 lambda1) (cos lambda2) (* (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(lambda1), cos(lambda2), (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(lambda1), cos(lambda2), 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[lambda1], $MachinePrecision] * N[Cos[lambda2], $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_1, \cos \lambda_2, \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-rr4.2

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

  3. Simplified4.2

    \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}\right) \cdot R \]
    Proof
    (*.f64 (fma.f64 (cos.f64 lambda1) (cos.f64 lambda2) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))) (*.f64 (cos.f64 phi2) (cos.f64 phi1))): 0 points increase in error, 0 points decrease in error
    (*.f64 (fma.f64 (cos.f64 lambda1) (cos.f64 lambda2) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))) (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (fma.f64 (cos.f64 lambda1) (cos.f64 lambda2) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (cos.f64 lambda1) (cos.f64 lambda2)) (*.f64 (sin.f64 lambda1) (sin.f64 lambda2))))): 9 points increase in error, 11 points decrease in error
    (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 (*.f64 (cos.f64 lambda1) (cos.f64 lambda2)) (*.f64 (cos.f64 phi1) (cos.f64 phi2))) (*.f64 (*.f64 (sin.f64 lambda1) (sin.f64 lambda2)) (*.f64 (cos.f64 phi1) (cos.f64 phi2))))): 12 points increase in error, 9 points decrease in error

  4. Taylor expanded in phi1 around 0 4.2

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

  5. Simplified4.2

    \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]
    Proof
    (acos.f64 (fma.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (fma.f64 (cos.f64 lambda1) (cos.f64 lambda2) (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)))) (*.f64 (sin.f64 phi1) (sin.f64 phi2)))): 0 points increase in error, 0 points decrease in error
    (acos.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (fma.f64 (cos.f64 lambda1) (cos.f64 lambda2) (*.f64 (sin.f64 lambda2) (sin.f64 lambda1))))) (*.f64 (sin.f64 phi1) (sin.f64 phi2))))): 2 points increase in error, 3 points decrease in error

  6. Final simplification4.2

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

Alternatives

Alternative 1
Error4.2
Cost64960
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right) \]
Alternative 2
Error4.2
Cost64960
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \]
Alternative 3
Error4.2
Cost64960
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \]
Alternative 4
Error11.1
Cost58696
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sqrt[3]{{t_1}^{3}}\right)\\ \end{array} \]
Alternative 5
Error4.2
Cost58688
\[R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \]
Alternative 6
Error11.1
Cost52360
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -3.4 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_0, t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \sqrt[3]{{\cos \left(\lambda_1 - \lambda_2\right)}^{3}}\right)\\ \end{array} \]
Alternative 7
Error11.1
Cost52360
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \sqrt[3]{{t_0}^{3}}\right)\\ \end{array} \]
Alternative 8
Error11.0
Cost46024
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right), t_0\right)\right)\\ \end{array} \]
Alternative 9
Error11.1
Cost45768
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -5.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot t_0, t_1\right)\right)\\ \end{array} \]
Alternative 10
Error11.1
Cost45768
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_2 \cdot \cos \phi_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right), t_0\right)\right)\\ \end{array} \]
Alternative 11
Error11.1
Cost45640
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 12
Error16.3
Cost39500
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ t_1 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.3 \cdot 10^{+131}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 13
Error16.6
Cost39500
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{+130}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
Alternative 14
Error11.1
Cost39496
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -6.2 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 15
Error21.4
Cost39236
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 16
Error17.6
Cost39236
\[\begin{array}{l} t_0 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 17
Error27.0
Cost33220
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_2 + \phi_1\right)}{2} + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 18
Error30.3
Cost32836
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \end{array} \]
Alternative 19
Error27.1
Cost32836
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 20
Error38.4
Cost32708
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \end{array} \]
Alternative 21
Error40.9
Cost32708
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1.25 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 22
Error30.9
Cost32708
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right)\\ \end{array} \]
Alternative 23
Error46.0
Cost26304
\[R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right) \]
Alternative 24
Error41.0
Cost26304
\[R \cdot \cos^{-1} \left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \]
Alternative 25
Error47.3
Cost19904
\[R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \phi_2 \cdot \phi_1\right) \]
Alternative 26
Error50.9
Cost19776
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right) \]
Alternative 27
Error52.2
Cost13376
\[R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_2 \cdot \phi_1\right) \]

Error

Reproduce

herbie shell --seed 2022325 
(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))