Equirectangular approximation to distance on a great circle

Average Error: 38.9 → 3.5

Time: 18.6s

Precision: binary64

Cost: 26496

Math

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

↓

\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\right), \phi_1 - \phi_2\right) \]

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) (expm1 (log1p (cos (* (+ phi1 phi2) 0.5)))))
   (- 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) * expm1(log1p(cos(((phi1 + phi2) * 0.5))))), (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.expm1(Math.log1p(Math.cos(((phi1 + phi2) * 0.5))))), (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.expm1(math.log1p(math.cos(((phi1 + phi2) * 0.5))))), (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) * expm1(log1p(cos(Float64(Float64(phi1 + phi2) * 0.5))))), Float64(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[(Exp[N[Log[1 + N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.9

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

2. Simplified 3.5

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

   [Start] 38.9	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   hypot-def [=>] 3.5	\[ R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Applied egg-rr 3.5

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\right)}, \phi_1 - \phi_2\right) \]

4. Final simplification 3.5

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\right), \phi_1 - \phi_2\right) \]

Alternatives

Alternative 1
Error	15.5
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.02 \cdot 10^{-119}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.9 \cdot 10^{+155}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(-\cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 2
Error	12.5
Cost	13705

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.000405 \lor \neg \left(\phi_2 \leq 4 \cdot 10^{+156}\right):\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 3
Error	7.8
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0295:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(-\cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 4
Error	6.1
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 5
Error	3.5
Cost	13696

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

Alternative 6
Error	16.2
Cost	13572

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.2 \cdot 10^{-120}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 7
Error	24.3
Cost	6916

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.8 \cdot 10^{-29}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]

Alternative 8
Error	13.3
Cost	6912

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

Alternative 9
Error	44.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.25 \cdot 10^{-11} \lor \neg \left(\phi_1 \leq 5.5 \cdot 10^{-179}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2 - R \cdot \lambda_1\\ \end{array} \]

Alternative 10
Error	44.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 5.5 \cdot 10^{-179}:\\ \;\;\;\;R \cdot \lambda_2 - R \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]

Alternative 11
Error	48.0
Cost	652

\[\begin{array}{l} t_0 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 4.4 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-63}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12
Error	44.7
Cost	585

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{-12} \lor \neg \left(\phi_1 \leq 5.5 \cdot 10^{-179}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 13
Error	45.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 4.6 \cdot 10^{-179}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14
Error	50.4
Cost	388

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.15 \cdot 10^{-63}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 15
Error	54.0
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2022354 
(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))))))