Equirectangular approximation to distance on a great circle

Average Error: 39.5 → 3.9

Time: 23.4s

Precision: binary64

Cost: 13696

Math

\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

↓

\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (sqrt
   (+
    (*
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))
     (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))))
    (* (- phi1 phi2) (- phi1 phi2))))))

↓

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))));
}

↓

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.sqrt(((((lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))));
}

↓

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.sqrt(((((lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))))

↓

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * sqrt(Float64(Float64(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0))) * Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

↓

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * sqrt(((((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))) * ((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0)))) + ((phi1 - phi2) * (phi1 - phi2))));
end

↓

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.5

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]

2. Simplified 3.9

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   Proof

   [Start] 39.5	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   hypot-def [=>] 3.9	\[ R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Final simplification 3.9

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right) \]

Alternatives

Alternative 1
Error	17.0
Cost	13964

\[\begin{array}{l} t_0 := \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\\ t_1 := R \cdot \mathsf{hypot}\left(\lambda_1 \cdot t_0, \phi_1 - \phi_2\right)\\ \mathbf{if}\;\lambda_2 \leq 1.05 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 3 \cdot 10^{+141}:\\ \;\;\;\;t_0 \cdot \left(R \cdot \left(\lambda_2 + \left(\lambda_1 - \lambda_1\right)\right) - R \cdot \lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 3.4 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot t_0\right)\\ \end{array} \]

Alternative 2
Error	44.3
Cost	13836

\[\begin{array}{l} t_0 := \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(R \cdot \left(\lambda_2 + \left(\lambda_1 - \lambda_1\right)\right) - R \cdot \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{+21}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -5.9 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -5.8 \cdot 10^{-114}:\\ \;\;\;\;R \cdot \sqrt{{\left(\lambda_1 - \lambda_2\right)}^{2} + \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;\phi_1 \leq -2.8 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -2.6 \cdot 10^{-303}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 1.26 \cdot 10^{-154}:\\ \;\;\;\;\left(R \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 3
Error	8.7
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -7 \cdot 10^{+92}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 4
Error	6.8
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -190000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 5
Error	43.1
Cost	8016

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -4.5 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq -6.5 \cdot 10^{-206}:\\ \;\;\;\;\cos \left(\phi_1 \cdot 0.5\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq -2.55 \cdot 10^{-228}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.6 \cdot 10^{-94}:\\ \;\;\;\;\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(R \cdot \left(\lambda_2 + \left(\lambda_1 - \lambda_1\right)\right) - R \cdot \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	45.2
Cost	7637

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq -1.9 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq -1.5 \cdot 10^{-53}:\\ \;\;\;\;R \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(-\lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 2.02 \cdot 10^{+126} \lor \neg \left(\lambda_2 \leq 6.5 \cdot 10^{+145}\right) \land \lambda_2 \leq 8.5 \cdot 10^{+176}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7
Error	45.2
Cost	7636

\[\begin{array}{l} t_0 := \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq -1.9 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq -1.5 \cdot 10^{-53}:\\ \;\;\;\;R \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(-\lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 9.6 \cdot 10^{+126}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.45 \cdot 10^{+146}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \mathbf{elif}\;\lambda_2 \leq 4.7 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot t_0\right)\\ \end{array} \]

Alternative 8
Error	47.7
Cost	7509

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ t_1 := \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -8.5 \cdot 10^{+233}:\\ \;\;\;\;R \cdot \left(t_1 \cdot \left(-\lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.85 \cdot 10^{-223}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_1 \leq 1.9 \cdot 10^{-138}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot t_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 4.8 \cdot 10^{-53} \lor \neg \left(\lambda_1 \leq 8 \cdot 10^{+51}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \]

Alternative 9
Error	43.1
Cost	7504

\[\begin{array}{l} t_0 := \cos \left(\phi_1 \cdot 0.5\right) \cdot \left(R \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq -3.5 \cdot 10^{-206}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq -5 \cdot 10^{-229}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.9 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	45.4
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.15 \cdot 10^{+176}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 11
Error	48.7
Cost	388

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.4 \cdot 10^{+40}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12
Error	46.9
Cost	320

\[R \cdot \left(\phi_2 - \phi_1\right) \]

Alternative 13
Error	54.0
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023067 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))