Spherical law of cosines

Average Error: 16.3 → 2.2

Time: 57.0s

Precision: binary64

Cost: 110596

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

Math

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

↓

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;t_0 + t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 0.9998:\\ \;\;\;\;\cos^{-1} \left(t_0 + t_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

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 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (+ t_0 (* t_1 (cos (- lambda1 lambda2)))) 0.9998)
     (*
      (acos
       (+
        t_0
        (*
         t_1
         (fma
          (cos lambda2)
          (cos lambda1)
          (cbrt (pow (* (sin lambda1) (sin lambda2)) 3.0))))))
      R)
     (* R (- lambda2 lambda1)))))

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 = sin(phi1) * sin(phi2);
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 0.9998) {
		tmp = acos((t_0 + (t_1 * fma(cos(lambda2), cos(lambda1), cbrt(pow((sin(lambda1) * sin(lambda2)), 3.0)))))) * R;
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	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(sin(phi1) * sin(phi2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2)))) <= 0.9998)
		tmp = Float64(acos(Float64(t_0 + Float64(t_1 * fma(cos(lambda2), cos(lambda1), cbrt((Float64(sin(lambda1) * sin(lambda2)) ^ 3.0)))))) * R);
	else
		tmp = Float64(R * Float64(lambda2 - lambda1));
	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[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.9998], N[(N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[Power[N[Power[N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))) < 0.99980000000000002

   1. Initial program 14.0

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

   2. Applied egg-rr 0.6

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

   3. Simplified 0.6

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

      [Start] 0.6	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]
      *-commutative [=>] 0.6	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]
      cos-neg [<=] 0.6	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \left(-\lambda_2\right)} \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]
      fma-def [=>] 0.6	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(-\lambda_2\right), \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]
      cos-neg [=>] 0.6	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \cdot R \]

   4. Applied egg-rr 0.6

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}}\right)\right) \cdot R \]

   if 0.99980000000000002 < (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))

   1. Initial program 51.8

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

   2. Simplified 51.8

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

      [Start] 51.8	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
      fma-def [=>] 51.8	\[ \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]
      *-commutative [=>] 51.8	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot R \]
      associate-*l* [=>] 51.8	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R \]

   3. Taylor expanded in phi2 around 0 52.8

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

   4. Taylor expanded in phi1 around 0 53.8

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

   5. Simplified 53.8

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

      [Start] 53.8	\[ \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
      sub-neg [=>] 53.8	\[ \cos^{-1} \cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)} \cdot R \]
      remove-double-neg [<=] 53.8	\[ \cos^{-1} \cos \left(\color{blue}{\left(-\left(-\lambda_1\right)\right)} + \left(-\lambda_2\right)\right) \cdot R \]
      mul-1-neg [<=] 53.8	\[ \cos^{-1} \cos \left(\left(-\color{blue}{-1 \cdot \lambda_1}\right) + \left(-\lambda_2\right)\right) \cdot R \]
      distribute-neg-in [<=] 53.8	\[ \cos^{-1} \cos \color{blue}{\left(-\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)} \cdot R \]
      cos-neg [=>] 53.8	\[ \cos^{-1} \color{blue}{\cos \left(-1 \cdot \lambda_1 + \lambda_2\right)} \cdot R \]
      +-commutative [=>] 53.8	\[ \cos^{-1} \cos \color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot R \]
      mul-1-neg [=>] 53.8	\[ \cos^{-1} \cos \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right) \cdot R \]
      unsub-neg [=>] 53.8	\[ \cos^{-1} \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0 27.9

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

   7. Simplified 27.9

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

      [Start] 27.9	\[ \left(-1 \cdot \lambda_1 + \lambda_2\right) \cdot R \]
      mul-1-neg [=>] 27.9	\[ \left(\color{blue}{\left(-\lambda_1\right)} + \lambda_2\right) \cdot R \]
      +-commutative [=>] 27.9	\[ \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)} \cdot R \]
      sub-neg [<=] 27.9	\[ \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 0.9998:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sqrt[3]{{\left(\sin \lambda_1 \cdot \sin \lambda_2\right)}^{3}}\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	2.2
Cost	97732

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;t_0 + t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 0.9998:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + t_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 2
Error	2.2
Cost	91460

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;t_0 + t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 0.9998:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + t_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 3
Error	10.2
Cost	58564

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -9 \cdot 10^{-47}:\\ \;\;\;\;R \cdot {\left({\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;\phi_2 \leq 6 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0 + \log \left(1 + \mathsf{expm1}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	10.2
Cost	52424

\[\begin{array}{l} t_0 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -9 \cdot 10^{-47}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 1.2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \log \left(1 + \mathsf{expm1}\left(t_1\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	10.2
Cost	52296

\[\begin{array}{l} t_0 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -9 \cdot 10^{-47}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 2.9 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	10.2
Cost	45641

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -9 \cdot 10^{-47} \lor \neg \left(\phi_2 \leq 1.65 \cdot 10^{-8}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]

Alternative 7
Error	16.8
Cost	39764

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ t_2 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\\ t_3 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_2\right)\\ \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{+260}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -6.3 \cdot 10^{+211}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_1 \leq -1.2 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1.6 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 8
Error	16.7
Cost	39764

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ t_2 := R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \lambda_1\right)\\ t_3 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\\ t_4 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_3\right)\\ \mathbf{if}\;\phi_1 \leq -2.35 \cdot 10^{+260}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -4.5 \cdot 10^{+211}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_1 \leq -5 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -0.00064:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 9
Error	10.2
Cost	39497

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -9 \cdot 10^{-47} \lor \neg \left(\phi_2 \leq 1.05 \cdot 10^{-8}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]

Alternative 10
Error	16.2
Cost	39369

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 3.7 \cdot 10^{-8}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]

Alternative 11
Error	21.1
Cost	39368

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.6 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\log \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]

Alternative 12
Error	20.1
Cost	39368

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_1 \leq -1.02 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 13
Error	29.5
Cost	33097

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -9.5 \cdot 10^{+74} \lor \neg \left(\phi_1 \leq 2.4 \cdot 10^{-14}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 14
Error	28.9
Cost	32972

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -0.000105:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \phi_1 \cdot t_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	31.9
Cost	32844

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot t_0\\ \mathbf{if}\;\phi_2 \leq 1.8 \cdot 10^{-81}:\\ \;\;\;\;R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} t_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot e^{\log \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)}\\ \end{array} \]

Alternative 16
Error	31.2
Cost	32580

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 17
Error	31.2
Cost	26372

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.0029:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 18
Error	40.4
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.9 \cdot 10^{-251}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]

Alternative 19
Error	33.1
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]

Alternative 20
Error	31.2
Cost	19780

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.34 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 21
Error	39.5
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.9:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 22
Error	36.6
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 7 \cdot 10^{-27}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 23
Error	47.4
Cost	13388

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{if}\;\lambda_2 \leq 2.05 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.9:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]

Alternative 24
Error	47.3
Cost	13385

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.05 \cdot 10^{-116} \lor \neg \left(\lambda_2 \leq 2.3 \cdot 10^{-8}\right):\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 25
Error	51.6
Cost	13124

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

Alternative 26
Error	59.2
Cost	388

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -8.2 \cdot 10^{-207}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]

Alternative 27
Error	59.1
Cost	320

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

Alternative 28
Error	60.0
Cost	192

\[\lambda_2 \cdot R \]

Reproduce

herbie shell --seed 2023083 
(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))