Spherical law of cosines

Percentage Accurate: 73.7% → 94.0%

Time: 47.0s

Precision: binary64

Cost: 71488

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

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

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
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (*
     (* (cos phi1) (cos phi2))
     (+
      (expm1 (log1p (* (sin lambda1) (sin lambda2))))
      (* (cos lambda1) (cos lambda2))))))
  R))

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) {
	return acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * (expm1(log1p((sin(lambda1) * sin(lambda2)))) + (cos(lambda1) * cos(lambda2)))))) * R;
}

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) {
	return Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * (Math.expm1(Math.log1p((Math.sin(lambda1) * Math.sin(lambda2)))) + (Math.cos(lambda1) * Math.cos(lambda2)))))) * R;
}

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):
	return math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * (math.expm1(math.log1p((math.sin(lambda1) * math.sin(lambda2)))) + (math.cos(lambda1) * math.cos(lambda2)))))) * R

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)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * Float64(expm1(log1p(Float64(sin(lambda1) * sin(lambda2)))) + Float64(cos(lambda1) * cos(lambda2)))))) * R)
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_] := 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[(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] * R), $MachinePrecision]

Derivation

1. Initial program 71.9%

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

2. Applied egg-rr 92.6%

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

   [Start] 71.9	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   cos-diff [=>] 92.6	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \cdot R \]
   +-commutative [=>] 92.6	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \cdot R \]

3. Applied egg-rr 92.6%

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

   [Start] 92.6	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]
   expm1-log1p-u [=>] 92.6	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} + \cos \lambda_1 \cdot \cos \lambda_2\right)\right) \cdot R \]

4. Final simplification 92.6%

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

Alternatives

Alternative 1
Accuracy	81.9%
Cost	58700

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ t_3 := \cos \phi_1 \cdot \cos \phi_2\\ t_4 := \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_1 \leq -1.65 \cdot 10^{+152}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + t_3 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.3 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_4 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -4.3 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_3 \cdot t_1\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \cos \phi_2 \cdot \left(t_4 + t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), t_2\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	94.0%
Cost	58688

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

Alternative 3
Accuracy	94.0%
Cost	58688

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

Alternative 4
Accuracy	81.9%
Cost	52688

\[\begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ t_3 := \cos \phi_1 \cdot \cos \phi_2\\ t_4 := \cos \lambda_1 \cdot \cos \lambda_2\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{+152}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + t_3 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_1\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -1.6 \cdot 10^{+63}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_4 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq -1.92 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + t_3 \cdot t_1\right)\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_2 + \cos \phi_2 \cdot \left(t_4 + t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), t_2\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	83.6%
Cost	45896

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.9 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.55 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	83.6%
Cost	45769

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-10} \lor \neg \left(\phi_2 \leq 2.6 \cdot 10^{-9}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	83.6%
Cost	45768

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.3 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), t_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), t_0\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	83.6%
Cost	45768

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.1 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.7 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	73.2%
Cost	39632

\[\begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ t_2 := R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -9.5 \cdot 10^{+161}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq -1.15 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_1 \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Accuracy	83.6%
Cost	39497

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-9} \lor \neg \left(\phi_2 \leq 3.6 \cdot 10^{-9}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]

Alternative 11
Accuracy	73.3%
Cost	39369

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -6 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 3.7 \cdot 10^{-6}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	59.9%
Cost	39236

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]

Alternative 13
Accuracy	55.1%
Cost	32968

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00047:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 14
Accuracy	55.2%
Cost	32840

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -185:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.00092:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \phi_2 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 15
Accuracy	49.6%
Cost	32580

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -185:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \]

Alternative 16
Accuracy	49.6%
Cost	26372

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -185:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \]

Alternative 17
Accuracy	33.9%
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 18
Accuracy	36.2%
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 19
Accuracy	41.8%
Cost	19648

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

Alternative 20
Accuracy	25.5%
Cost	13120

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

Reproduce

herbie shell --seed 2023160 
(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))