Spherical law of cosines

Average Error: 73.4% → 96.5%

Time: 1.1min

Precision: binary64

Cost: 104132.00

\[ \begin{array}{c}[lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t_1 + \cos \left(\lambda_1 - \lambda_2\right) \cdot t_0\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot t_0\right)\\ \end{array} \]

FPCore

(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
 (let* ((t_0 (* (cos phi1) (cos phi2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= (acos (+ t_1 (* (cos (- lambda1 lambda2)) t_0))) 2e-6)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (+
        t_1
        (*
         (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2)))
         t_0)))))))

C

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 t_0 = cos(phi1) * cos(phi2);
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (acos((t_1 + (cos((lambda1 - lambda2)) * t_0))) <= 2e-6) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos((t_1 + (fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))) * t_0)));
	}
	return tmp;
}

Julia

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)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (acos(Float64(t_1 + Float64(cos(Float64(lambda1 - lambda2)) * t_0))) <= 2e-6)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))) * t_0))));
	end
	return tmp
end

Wolfram

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_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(t$95$1 + N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-6], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

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 13.1

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

   2. Simplified 13.1

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

      [Start] 13.1	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
      fma-def [=>] 13.1	\[ \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]
      *-commutative [=>] 13.1	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]
      associate-*l* [=>] 13.1	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R \]

   3. Taylor expanded in phi2 around 0 13.0

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

   4. Taylor expanded in phi1 around 0 12.8

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

   5. Simplified 12.8

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

      [Start] 12.8	\[ \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
      sub-neg [=>] 12.8	\[ \cos^{-1} \cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)} \cdot R \]
      remove-double-neg [<=] 12.8	\[ \cos^{-1} \cos \left(\color{blue}{\left(-\left(-\lambda_1\right)\right)} + \left(-\lambda_2\right)\right) \cdot R \]
      mul-1-neg [<=] 12.8	\[ \cos^{-1} \cos \left(\left(-\color{blue}{-1 \cdot \lambda_1}\right) + \left(-\lambda_2\right)\right) \cdot R \]
      distribute-neg-in [<=] 12.8	\[ \cos^{-1} \cos \color{blue}{\left(-\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)} \cdot R \]
      cos-neg [=>] 12.8	\[ \cos^{-1} \color{blue}{\cos \left(-1 \cdot \lambda_1 + \lambda_2\right)} \cdot R \]
      +-commutative [=>] 12.8	\[ \cos^{-1} \cos \color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot R \]
      mul-1-neg [=>] 12.8	\[ \cos^{-1} \cos \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right) \cdot R \]
      unsub-neg [=>] 12.8	\[ \cos^{-1} \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0 53.1

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

   7. Simplified 53.1

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

      [Start] 53.1	\[ \left(-1 \cdot \lambda_1 + \lambda_2\right) \cdot R \]
      +-commutative [=>] 53.1	\[ \color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot R \]
      mul-1-neg [=>] 53.1	\[ \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right) \cdot R \]
      sub-neg [<=] 53.1	\[ \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

   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 77.0

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

   2. Applied egg-rr 99.0

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

   3. Simplified 99.0

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

      [Start] 99.0	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]
      *-commutative [=>] 99.0	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]
      cos-neg [<=] 99.0	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \left(-\lambda_2\right)} \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]
      fma-def [=>] 99.0	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]
      cos-neg [=>] 99.0	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.5

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

Alternatives

Alternative 1
Error	96.5%
Cost	97860.00

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t_1 + \cos \left(\lambda_1 - \lambda_2\right) \cdot t_0\right) \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot t_0\right)\\ \end{array} \]

Alternative 2
Error	83.4%
Cost	58696.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \cos^{-1} \log \left(e^{t_0}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot {\left({\cos^{-1} t_0}^{3}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 3
Error	83.4%
Cost	58568.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \cos^{-1} \log \left(e^{t_0}\right)\\ \mathbf{elif}\;\phi_2 \leq 3.6 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\log \cos^{-1} t_0}\\ \end{array} \]

Alternative 4
Error	83.4%
Cost	58436.00

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \cos^{-1} \log \left(e^{t_0}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} t_0\\ \end{array} \]

Alternative 5
Error	83.4%
Cost	52164.00

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \cos^{-1} \log \left(e^{\sin \phi_1 \cdot \sin \phi_2 + t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)\\ \mathbf{elif}\;\phi_2 \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	84.0%
Cost	52164.00

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -5.9 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}\right)\\ \mathbf{elif}\;\phi_2 \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	84.0%
Cost	45769.00

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.8 \cdot 10^{-11} \lor \neg \left(\phi_2 \leq 1.95 \cdot 10^{-7}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]

Alternative 8
Error	84.0%
Cost	45768.00

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -5.1 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(t_1 + \cos \left(\lambda_1 - \lambda_2\right) \cdot t_0\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), t_0, t_1\right)\right)\\ \end{array} \]

Alternative 9
Error	84.0%
Cost	45641.00

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-10} \lor \neg \left(\phi_2 \leq 3.1 \cdot 10^{-9}\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]

Alternative 10
Error	75.9%
Cost	39632.00

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ t_2 := R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -1.26 \cdot 10^{+270}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq -1.08 \cdot 10^{+238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 2.95 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	75.9%
Cost	39632.00

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.6 \cdot 10^{+270}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -5.8 \cdot 10^{+237}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.65 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	84.0%
Cost	39497.00

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{-11} \lor \neg \left(\phi_2 \leq 1.4 \cdot 10^{-8}\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]

Alternative 13
Error	73.9%
Cost	39369.00

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.3 \cdot 10^{-5} \lor \neg \left(\lambda_2 \leq 0.024\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]

Alternative 14
Error	52.4%
Cost	39244.00

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -8 \cdot 10^{-147}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 8 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \end{array} \]

Alternative 15
Error	61.5%
Cost	39236.00

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 16
Error	52.4%
Cost	39112.00

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.55 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -5.2 \cdot 10^{-147}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 8 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 17
Error	52.4%
Cost	32972.00

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -9 \cdot 10^{-147}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\\ \mathbf{elif}\;\phi_1 \leq 8 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 18
Error	54.8%
Cost	32840.00

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.076:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 8 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 19
Error	49.8%
Cost	32580.00

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -18500000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 20
Error	49.8%
Cost	26500.00

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -18500000000:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5\right) - R \cdot \sin^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 21
Error	35.3%
Cost	19916.00

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+34}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -4.6 \cdot 10^{-39}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_1 \leq -9.2 \cdot 10^{-139}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 22
Error	48.4%
Cost	19780.00

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 23
Error	49.9%
Cost	19780.00

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -18500000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 24
Error	38.5%
Cost	19652.00

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.00029:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 25
Error	43.1%
Cost	19652.00

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 26
Error	29.3%
Cost	13380.00

\[\begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5000:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \]

Alternative 27
Error	19.6%
Cost	13124.00

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \]

Alternative 28
Error	26.7%
Cost	13124.00

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]

Alternative 29
Error	7.3%
Cost	388.00

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.6 \cdot 10^{-218}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]

Alternative 30
Error	7.4%
Cost	320.00

\[\left(\lambda_2 - \lambda_1\right) \cdot R \]

Alternative 31
Error	6.0%
Cost	192.00

\[\lambda_2 \cdot R \]

Reproduce

herbie shell --seed 2023097 
(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))