Spherical law of cosines

Average Error: 16.8 → 3.7

Time: 2.0min

Precision: binary64

Cost: 58816

Math

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

↓

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

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) (cos (- lambda2)))
      (* (sin lambda1) (sin (- 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) * cos(-lambda2)) - (sin(lambda1) * sin(-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) * cos(-lambda2)) - (sin(lambda1) * sin(-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) * Math.cos(-lambda2)) - (Math.sin(lambda1) * Math.sin(-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) * math.cos(-lambda2)) - (math.sin(lambda1) * math.sin(-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(Float64(cos(lambda1) * cos(Float64(-lambda2))) - Float64(sin(lambda1) * sin(Float64(-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) * cos(-lambda2)) - (sin(lambda1) * sin(-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[(N[Cos[lambda1], $MachinePrecision] * N[Cos[(-lambda2)], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 16.8

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

2. Applied egg-rr 3.7

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

3. Final simplification 3.7

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

Alternatives

Alternative 1
Error	12.0
Cost	52680

\[\begin{array}{l} t_0 := \cos \left(-\lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := \cos \lambda_1 \cdot t_0\\ t_4 := \cos^{-1} \left(\left(t_3 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\lambda_2 \leq -0.0095:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\lambda_2 \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\cos^{-1} \left(t_1 + t_2 \cdot \left(t_3 - \sin \lambda_1 \cdot \left(-\lambda_2\right)\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.38 \cdot 10^{+222}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t_1 + t_2 \cdot t_0\right) \cdot R\\ \end{array} \]

Alternative 2
Error	12.2
Cost	46156

\[\begin{array}{l} t_0 := \cos \left(-\lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ t_2 := \cos^{-1} \left(\left(\cos \lambda_1 \cdot t_0 - \sin \lambda_1 \cdot \sin \left(-\lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\lambda_2 \leq -1.35:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{elif}\;\lambda_2 \leq 1.38 \cdot 10^{+222}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) \cdot R\\ \end{array} \]

Alternative 3
Error	17.0
Cost	39432

\[\begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\lambda_2 \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 0.0125:\\ \;\;\;\;\cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.3
Cost	39368

\[\begin{array}{l} t_0 := \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\lambda_2 \leq -3.9 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	21.3
Cost	39368

\[\begin{array}{l} t_0 := \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) \cdot R\\ \mathbf{if}\;\lambda_2 \leq -3.9 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 5 \cdot 10^{-45}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.8
Cost	39232

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

Alternative 7
Error	16.8
Cost	39232

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

Alternative 8
Error	23.7
Cost	33096

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot t_0\right) \cdot R\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 - \lambda_2 \leq 0.0002:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	26.8
Cost	26376

\[\begin{array}{l} t_0 := \cos^{-1} \left(\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(-\lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\lambda_2 \leq -7 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 0.0125:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	26.7
Cost	26176

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

Alternative 11
Error	40.4
Cost	19784

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.75 \cdot 10^{-259}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \left(-\lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 7 \cdot 10^{-11}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \end{array} \]

Alternative 12
Error	33.3
Cost	19780

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -5000000:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \left(-\lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot R\\ \end{array} \]

Alternative 13
Error	41.7
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0185:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \cos \phi_2 \cdot R\\ \end{array} \]

Alternative 14
Error	40.1
Cost	19652

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-11}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right) \cdot R\\ \end{array} \]

Alternative 15
Error	47.5
Cost	13520

\[\begin{array}{l} t_0 := \cos^{-1} \cos \lambda_1 \cdot R\\ \mathbf{if}\;\lambda_1 \leq -0.00033:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 4.7 \cdot 10^{-262}:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \mathbf{elif}\;\lambda_1 \leq 2.9 \cdot 10^{-82}:\\ \;\;\;\;\cos^{-1} \cos \phi_2 \cdot R\\ \mathbf{elif}\;\lambda_1 \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	43.3
Cost	13384

\[\begin{array}{l} t_0 := \cos^{-1} \cos \lambda_1 \cdot R\\ \mathbf{if}\;\lambda_1 \leq -7800000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 16200000000000:\\ \;\;\;\;\cos^{-1} \cos \left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	51.8
Cost	13256

\[\begin{array}{l} t_0 := \cos^{-1} \cos \lambda_1 \cdot R\\ \mathbf{if}\;\lambda_1 \leq -2.4 \cdot 10^{-308}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 18
Error	47.3
Cost	13256

\[\begin{array}{l} t_0 := \cos^{-1} \cos \lambda_1 \cdot R\\ \mathbf{if}\;\lambda_1 \leq -0.00022:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 16200000000000:\\ \;\;\;\;\cos^{-1} \cos \phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 19
Error	60.9
Cost	192

\[\lambda_1 \cdot R \]

Reproduce

herbie shell --seed 2023099 
(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))