Average Error: 16.5 → 2.1
Time: 1.5min
Precision: binary64
Cost: 110404
\[ \begin{array}{c}[lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]
\[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
↓
\[\begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\lambda_2 \cdot R - \lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right)\\ \end{array} \]
(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
 (if (<=
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (* (cos (- lambda1 lambda2)) (* (cos phi1) (cos phi2)))))
      2e-6)
   (- (* lambda2 R) (* lambda1 R))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (*
       (cos phi1)
       (*
        (cos phi2)
        (fma
         (cos lambda2)
         (cos lambda1)
         (* (sin lambda2) (sin lambda1))))))))))
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) {
	double tmp;
	if (acos(((sin(phi1) * sin(phi2)) + (cos((lambda1 - lambda2)) * (cos(phi1) * cos(phi2))))) <= 2e-6) {
		tmp = (lambda2 * R) - (lambda1 * R);
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))))));
	}
	return tmp;
}

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)
	tmp = 0.0
	if (acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(Float64(lambda1 - lambda2)) * Float64(cos(phi1) * cos(phi2))))) <= 2e-6)
		tmp = Float64(Float64(lambda2 * R) - Float64(lambda1 * R));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))))));
	end
	return tmp
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_] := If[LessEqual[N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-6], N[(N[(lambda2 * R), $MachinePrecision] - N[(lambda1 * R), $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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
↓
\begin{array}{l}
\mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \leq 2 \cdot 10^{-6}:\\
\;\;\;\;\lambda_2 \cdot R - \lambda_1 \cdot R\\

\mathbf{else}:\\
\;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \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)\right)\\


\end{array}

Error

Derivation

  1. Split input into 2 regimes

  2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 1.99999999999999991e-6

    1. Initial program 55.5

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

    2. Simplified55.5

      \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R} \]
      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): 4 points increase in error, 3 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, 7 points decrease in error

    3. Taylor expanded in phi2 around 0 55.5

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

    4. Simplified55.5

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]
      Proof
      (*.f64 (cos.f64 (-.f64 lambda2 lambda1)) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (Rewrite<= unsub-neg_binary64 (+.f64 lambda2 (neg.f64 lambda1)))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (+.f64 lambda2 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 lambda1)))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 lambda1) lambda2))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= cos-neg_binary64 (cos.f64 (neg.f64 (+.f64 (*.f64 -1 lambda1) lambda2)))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (neg.f64 (Rewrite=> +-commutative_binary64 (+.f64 lambda2 (*.f64 -1 lambda1))))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (Rewrite=> distribute-neg-in_binary64 (+.f64 (neg.f64 lambda2) (neg.f64 (*.f64 -1 lambda1))))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (+.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 lambda2)) (neg.f64 (*.f64 -1 lambda1)))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (+.f64 (*.f64 -1 lambda2) (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 lambda1))))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (+.f64 (*.f64 -1 lambda2) (Rewrite=> remove-double-neg_binary64 lambda1))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (+.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 lambda2)) lambda1)) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (Rewrite=> +-commutative_binary64 (+.f64 lambda1 (neg.f64 lambda2)))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error
      (*.f64 (cos.f64 (Rewrite<= sub-neg_binary64 (-.f64 lambda1 lambda2))) (cos.f64 phi1)): 0 points increase in error, 0 points decrease in error

    5. Taylor expanded in phi1 around 0 55.6

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

    6. Applied egg-rr25.1

      \[\leadsto \color{blue}{\lambda_2 \cdot R + \left(-\lambda_1\right) \cdot R} \]

    if 1.99999999999999991e-6 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

    1. Initial program 14.2

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

    2. Simplified14.2

      \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R} \]
      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): 4 points increase in error, 3 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, 7 points decrease in error

    3. Applied egg-rr0.7

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

    4. Taylor expanded in phi1 around inf 0.7

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

    5. Simplified0.7

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right)} \cdot R \]
      Proof
      (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (fma.f64 (cos.f64 lambda2) (cos.f64 lambda1) (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)) (*.f64 (sin.f64 lambda2) (sin.f64 lambda1))))))): 7 points increase in error, 5 points decrease in error
      (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (cos.f64 phi1) (*.f64 (cos.f64 phi2) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (*.f64 (cos.f64 phi1) (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))) (cos.f64 phi2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cos.f64 phi1) (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1)))) (cos.f64 phi2)))): 7 points increase in error, 9 points decrease in error
      (fma.f64 (sin.f64 phi1) (sin.f64 phi2) (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))))))): 9 points increase in error, 14 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (cos.f64 phi2) (*.f64 (cos.f64 phi1) (+.f64 (*.f64 (sin.f64 lambda2) (sin.f64 lambda1)) (*.f64 (cos.f64 lambda2) (cos.f64 lambda1))))) (*.f64 (sin.f64 phi1) (sin.f64 phi2)))): 0 points increase in error, 0 points decrease in error
  3. Recombined 2 regimes into one program.

  4. Final simplification2.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\lambda_2 \cdot R - \lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error2.1
Cost97860
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t_0 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\lambda_2 \cdot R - \lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \end{array} \]
Alternative 2
Error10.3
Cost58696
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.518655669388917 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.9271134508852888 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + {\left(\sqrt[3]{\sin \phi_1 \cdot \sin \phi_2}\right)}^{3}\right)\\ \end{array} \]
Alternative 3
Error10.3
Cost58696
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.518655669388917 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \log \left(1 + \mathsf{expm1}\left(\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.9271134508852888 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + {\left(\sqrt[3]{\sin \phi_1 \cdot \sin \phi_2}\right)}^{3}\right)\\ \end{array} \]
Alternative 4
Error10.3
Cost52360
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.518655669388917 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.9271134508852888 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + {\left(\sqrt[3]{\sin \phi_1 \cdot \sin \phi_2}\right)}^{3}\right)\\ \end{array} \]
Alternative 5
Error10.3
Cost52296
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.518655669388917 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.9271134508852888 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]
Alternative 6
Error10.3
Cost45768
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.518655669388917 \cdot 10^{-11}:\\ \;\;\;\;\cos^{-1} \left(t_1 + \cos \left(\lambda_1 - \lambda_2\right) \cdot t_0\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.9271134508852888 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t_0, \cos \left(\lambda_2 - \lambda_1\right), t_1\right)\right)\\ \end{array} \]
Alternative 7
Error10.3
Cost45768
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.518655669388917 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.9271134508852888 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t_0, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
Alternative 8
Error10.3
Cost45640
\[\begin{array}{l} t_0 := \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -1.518655669388917 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.9271134508852888 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error10.3
Cost39496
\[\begin{array}{l} t_0 := \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -1.518655669388917 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.9271134508852888 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error17.1
Cost39368
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{if}\;\phi_2 \leq -71005389823324184:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 9.196251035014615 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error15.1
Cost39368
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1.401269559717865 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 4.826221919811612 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
Alternative 12
Error15.1
Cost39368
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1.401269559717865 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 21828.65600782723:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
Alternative 13
Error20.5
Cost39236
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.115584907086277 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 14
Error27.0
Cost38980
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -71005389823324184:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1405763819360421.8:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot t_0 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 5.473368291690849 \cdot 10^{+108}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 15
Error27.0
Cost38980
\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -71005389823324184:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1, t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1405763819360421.8:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot t_1 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 5.473368291690849 \cdot 10^{+108}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 16
Error27.0
Cost33096
\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{if}\;\phi_2 \leq -71005389823324184:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1405763819360421.8:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot t_0 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 5.473368291690849 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 17
Error26.6
Cost32708
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 2.115584907086277 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \log \left(1 + \mathsf{expm1}\left(\cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 18
Error36.6
Cost19784
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.938748863857627 \cdot 10^{-101}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 17.978288611172104:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 19
Error29.6
Cost19780
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.009754333911024758:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
Alternative 20
Error26.6
Cost19780
\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 2.115584907086277 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]
Alternative 21
Error39.7
Cost19652
\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.401269559717865 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 22
Error36.6
Cost19652
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.115584907086277 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
Alternative 23
Error50.4
Cost13256
\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{if}\;\lambda_1 \leq -0.002717145980413742:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 1.248282664757156 \cdot 10^{-12}:\\ \;\;\;\;\lambda_2 \cdot R - \lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 24
Error47.3
Cost13124
\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.850880258250221 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
Alternative 25
Error47.3
Cost13120
\[R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \]
Alternative 26
Error58.9
Cost448
\[\lambda_2 \cdot R - \lambda_1 \cdot R \]
Alternative 27
Error59.2
Cost388
\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.1384283968035681 \cdot 10^{-247}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
Alternative 28
Error58.9
Cost320
\[R \cdot \left(\lambda_2 - \lambda_1\right) \]
Alternative 29
Error59.9
Cost192
\[\lambda_2 \cdot R \]

Error

Reproduce

herbie shell --seed 2022294 
(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))