Spherical law of cosines

Average Accuracy: 73.2% → 96.6%

Time: 1.1min

Precision: binary64

Cost: 110660

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

Math

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

↓

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t_0 + t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + t_1 \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

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 (<= (acos (+ t_0 (* t_1 (cos (- lambda1 lambda2))))) 4e-7)
     (* (- lambda2 lambda1) R)
     (*
      R
      (acos
       (+
        t_0
        (*
         t_1
         (+
          (expm1 (log1p (* (sin lambda1) (sin lambda2))))
          (* (cos lambda1) (cos lambda2))))))))))

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 (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 4e-7) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos((t_0 + (t_1 * (expm1(log1p((sin(lambda1) * sin(lambda2)))) + (cos(lambda1) * cos(lambda2))))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R;
}

↓

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi1) * Math.cos(phi2);
	double tmp;
	if (Math.acos((t_0 + (t_1 * Math.cos((lambda1 - lambda2))))) <= 4e-7) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * Math.acos((t_0 + (t_1 * (Math.expm1(Math.log1p((Math.sin(lambda1) * Math.sin(lambda2)))) + (Math.cos(lambda1) * Math.cos(lambda2))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R

↓

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	t_1 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if math.acos((t_0 + (t_1 * math.cos((lambda1 - lambda2))))) <= 4e-7:
		tmp = (lambda2 - lambda1) * R
	else:
		tmp = R * math.acos((t_0 + (t_1 * (math.expm1(math.log1p((math.sin(lambda1) * math.sin(lambda2)))) + (math.cos(lambda1) * math.cos(lambda2))))))
	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 (acos(Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2))))) <= 4e-7)
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(t_1 * Float64(expm1(log1p(Float64(sin(lambda1) * sin(lambda2)))) + Float64(cos(lambda1) * cos(lambda2)))))));
	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[ArcCos[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4e-7], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[(N[(Exp[N[Log[1 + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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))))) < 3.9999999999999998e-7

   1. Initial program 13.3%

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

   2. Simplified 13.3%

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

      [Start] 13.3	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
      fma-def [=>] 13.3	\[ \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]
      associate-*l* [=>] 13.3	\[ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R \]

   3. Taylor expanded in phi2 around 0 13.2%

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

   4. Taylor expanded in phi1 around 0 12.9%

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

   5. Simplified 12.9%

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

      [Start] 12.9	\[ \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \cdot R \]
      sub-neg [=>] 12.9	\[ \cos^{-1} \cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)} \cdot R \]
      remove-double-neg [<=] 12.9	\[ \cos^{-1} \cos \left(\color{blue}{\left(-\left(-\lambda_1\right)\right)} + \left(-\lambda_2\right)\right) \cdot R \]
      mul-1-neg [<=] 12.9	\[ \cos^{-1} \cos \left(\left(-\color{blue}{-1 \cdot \lambda_1}\right) + \left(-\lambda_2\right)\right) \cdot R \]
      distribute-neg-in [<=] 12.9	\[ \cos^{-1} \cos \color{blue}{\left(-\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)} \cdot R \]
      cos-neg [=>] 12.9	\[ \cos^{-1} \color{blue}{\cos \left(-1 \cdot \lambda_1 + \lambda_2\right)} \cdot R \]
      +-commutative [=>] 12.9	\[ \cos^{-1} \cos \color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot R \]
      mul-1-neg [=>] 12.9	\[ \cos^{-1} \cos \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right) \cdot R \]
      unsub-neg [=>] 12.9	\[ \cos^{-1} \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0 57.6%

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

   7. Simplified 57.6%

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

      [Start] 57.6	\[ \left(-1 \cdot \lambda_1 + \lambda_2\right) \cdot R \]
      mul-1-neg [=>] 57.6	\[ \left(\color{blue}{\left(-\lambda_1\right)} + \lambda_2\right) \cdot R \]
      +-commutative [=>] 57.6	\[ \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)} \cdot R \]
      sub-neg [<=] 57.6	\[ \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

   if 3.9999999999999998e-7 < (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 76.7%

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

   2. Applied egg-rr 98.9%

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

      [Start] 76.7	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
      cos-diff [=>] 98.9	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]
      +-commutative [=>] 98.9	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

   3. Applied egg-rr 98.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]
      Proof

      [Start] 98.9	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]
      expm1-log1p-u [=>] 98.9	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

3. Recombined 2 regimes into one program.

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.6%
Cost	104132

\[\begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	96.6%
Cost	97860

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	96.6%
Cost	97860

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t_0 + t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 4 \cdot 10^{-7}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + t_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	83.4%
Cost	52552

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6.3 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0 \cdot \log \left(e^{t_1}\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.16:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot t_1\right)\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	83.3%
Cost	52360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6.3 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0 \cdot \log \left(e^{t_1}\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot t_1\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	83.5%
Cost	52164

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.3 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0 + \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	83.3%
Cost	52164

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6.3 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \log \left(e^{t_0}\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	83.5%
Cost	46024

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.3 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	83.4%
Cost	45768

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.3 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	73.9%
Cost	39764

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ t_1 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ t_2 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{+246}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{+198}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -1.15 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -0.00205:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 0.115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	74.6%
Cost	39632

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ t_2 := R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \lambda_1\right)\\ t_3 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\lambda_1 \leq -7.5 \cdot 10^{+201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq -7.2 \cdot 10^{+106}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_3\right)\\ \mathbf{elif}\;\lambda_1 \leq -2.5 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 1.9 \cdot 10^{+16}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_3\right)\\ \end{array} \]

Alternative 12
Accuracy	61.0%
Cost	39500

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{+156}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5\right) - R \cdot \sin^{-1} t_0\\ \mathbf{elif}\;\phi_1 \leq -3.3 \cdot 10^{+125}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -0.0046:\\ \;\;\;\;R \cdot \cos^{-1} t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	83.3%
Cost	39497

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.36 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 0.115\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 14
Accuracy	83.4%
Cost	39496

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -4.3 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \]

Alternative 15
Accuracy	56.0%
Cost	39376

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{+156}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5\right) - R \cdot \sin^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_1 \leq -8 \cdot 10^{+124}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -90000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \sqrt[3]{{t_1}^{3}}\right)\\ \mathbf{elif}\;\phi_1 \leq 0.42:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot t_1 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	68.9%
Cost	39368

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_2 \leq -0.00102:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 17
Accuracy	56.0%
Cost	33360

\[\begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{+156}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5\right) - R \cdot \sin^{-1} \left(\cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_1 \leq -3.25 \cdot 10^{+125}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -90000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \sqrt[3]{{t_1}^{3}}\right)\\ \mathbf{elif}\;\phi_1 \leq 900000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot t_1 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Accuracy	55.8%
Cost	33104

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot t_0\\ t_2 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{+156}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5\right) - R \cdot \sin^{-1} t_1\\ \mathbf{elif}\;\phi_1 \leq -5.7 \cdot 10^{+124}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -0.0047:\\ \;\;\;\;R \cdot \cos^{-1} t_1\\ \mathbf{elif}\;\phi_1 \leq 0.16:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Accuracy	50.5%
Cost	26500

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

Alternative 20
Accuracy	50.5%
Cost	26372

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 21
Accuracy	42.7%
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq -1.36 \cdot 10^{-188}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 22
Accuracy	48.0%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.00052:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 23
Accuracy	50.5%
Cost	19780

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \]

Alternative 24
Accuracy	37.7%
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.0025:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 25
Accuracy	42.7%
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 26
Accuracy	26.3%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.6 \cdot 10^{-108}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 5.9:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 27
Accuracy	26.3%
Cost	13256

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.6 \cdot 10^{-108}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 5.9:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]

Alternative 28
Accuracy	19.6%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \]

Alternative 29
Accuracy	7.7%
Cost	388

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -8.2 \cdot 10^{-157}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]

Alternative 30
Accuracy	7.7%
Cost	320

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

Alternative 31
Accuracy	6.0%
Cost	192

\[\lambda_2 \cdot R \]

Reproduce

herbie shell --seed 2023135 
(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))