Spherical law of cosines

Average Error: 17.0 → 17.0

Time: 29.1s

Precision: binary64

Cost: 39232

Math

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

↓

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

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) (cos (- lambda1 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) * cos((lambda1 - lambda2)))))) * R;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
end function

↓

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2)))))) * r
end function

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.cos((lambda1 - 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.cos((lambda1 - 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(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
end

↓

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * cos((lambda1 - 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[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 17.0

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

2. Simplified 17.0

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

   [Start] 17.0	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   rational.json-simplify-2 [=>] 17.0	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) \cdot R \]
   rational.json-simplify-43 [=>] 17.0	\[ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot R \]

3. Final simplification 17.0

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

Alternatives

Alternative 1
Error	27.2
Cost	39368

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3 \cdot 10^{+96}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{elif}\;\lambda_1 \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right) \cdot R\\ \end{array} \]

Alternative 2
Error	23.6
Cost	39236

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 3.3 \cdot 10^{-6}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right) \cdot R\\ \end{array} \]

Alternative 3
Error	29.6
Cost	33096

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -34000:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 7600000000:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right) \cdot R\\ \end{array} \]

Alternative 4
Error	29.8
Cost	32840

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -8200:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.22:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \phi_1\right) \cdot R\\ \end{array} \]

Alternative 5
Error	35.5
Cost	26436

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -8200:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \end{array} \]

Alternative 6
Error	41.9
Cost	26312

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-16}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.35 \cdot 10^{-135}:\\ \;\;\;\;\cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2\right) \cdot R\\ \end{array} \]

Alternative 7
Error	42.5
Cost	20168

\[\begin{array}{l} t_0 := \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -8.1 \cdot 10^{-171}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-113}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	36.8
Cost	20168

\[\begin{array}{l} t_0 := \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -1.5 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 7.5 \cdot 10^{-88}:\\ \;\;\;\;\cos^{-1} \left(\phi_2 \cdot \phi_1 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	41.9
Cost	20040

\[\begin{array}{l} t_0 := \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -9.5 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 7.6 \cdot 10^{-168}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	42.9
Cost	19908

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.0031:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \lambda_2 \cdot \cos \phi_1\right) \cdot R\\ \end{array} \]

Alternative 11
Error	43.3
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.065:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \lambda_2\right) \cdot R\\ \end{array} \]

Alternative 12
Error	44.4
Cost	19520

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

Alternative 13
Error	53.0
Cost	12992

\[\cos^{-1} \cos \lambda_1 \cdot R \]

Alternative 14
Error	61.2
Cost	6592

\[\cos^{-1} 1 \cdot R \]

Alternative 15
Error	60.8
Cost	192

\[\lambda_1 \cdot R \]

Reproduce

herbie shell --seed 2023067 
(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))