Equirectangular approximation to distance on a great circle

Average Error: 39.1 → 0.1

Time: 2.0min

Precision: binary64

Cost: 39872

Math

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

↓

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

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
   (- phi2 phi1)
   (*
    (- lambda1 lambda2)
    (fma
     (cos (* 0.5 phi2))
     (cos (* 0.5 phi1))
     (* (- (sin (* 0.5 phi2))) (sin (* 0.5 phi1))))))))

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((phi2 - phi1), ((lambda1 - lambda2) * fma(cos((0.5 * phi2)), cos((0.5 * phi1)), (-sin((0.5 * phi2)) * sin((0.5 * phi1))))));
}

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(phi2 - phi1), Float64(Float64(lambda1 - lambda2) * fma(cos(Float64(0.5 * phi2)), cos(Float64(0.5 * phi1)), Float64(Float64(-sin(Float64(0.5 * phi2))) * sin(Float64(0.5 * phi1)))))))
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[(phi2 - phi1), $MachinePrecision] ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[((-N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]) * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.1

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

2. Simplified 3.9

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

3. Applied egg-rr 0.1

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

4. Applied egg-rr 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	33536

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

Alternative 2
Error	4.2
Cost	13832

\[\begin{array}{l} t_0 := R \cdot \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 39000000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2 - \phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	3.9
Cost	13696

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

Alternative 4
Error	13.4
Cost	13572

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6 \cdot 10^{+158}:\\ \;\;\;\;\left|\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right| \cdot \left(-R \cdot \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2 - \phi_1, \lambda_1 - \lambda_2\right)\\ \end{array} \]

Alternative 5
Error	13.4
Cost	13572

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.25 \cdot 10^{+160}:\\ \;\;\;\;\left(R \cdot \left|\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right|\right) \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2 - \phi_1, \lambda_1 - \lambda_2\right)\\ \end{array} \]

Alternative 6
Error	9.0
Cost	13568

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

Alternative 7
Error	19.3
Cost	7048

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.082:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 1.65 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot R - R \cdot \phi_2\\ \end{array} \]

Alternative 8
Error	39.6
Cost	6984

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{-129}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 1.9 \cdot 10^{-268}:\\ \;\;\;\;\left(\lambda_2 \cdot \left|1\right|\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.9 \cdot 10^{-194}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 3.4 \cdot 10^{-143}:\\ \;\;\;\;-R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 1.75 \cdot 10^{-40}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot R - R \cdot \phi_2\\ \end{array} \]

Alternative 9
Error	13.8
Cost	6912

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

Alternative 10
Error	39.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq 2.5 \cdot 10^{-193}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 1.9 \cdot 10^{-143}:\\ \;\;\;\;-R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 2.9 \cdot 10^{-41}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot R - R \cdot \phi_2\\ \end{array} \]

Alternative 11
Error	40.6
Cost	716

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq 1.95 \cdot 10^{-194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 1.9 \cdot 10^{-148}:\\ \;\;\;\;-R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 6.6 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot R\\ \end{array} \]

Alternative 12
Error	42.3
Cost	520

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{-83}:\\ \;\;\;\;-\phi_1 \cdot R\\ \mathbf{elif}\;\phi_1 \leq 0.0002:\\ \;\;\;\;-R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot R\\ \end{array} \]

Alternative 13
Error	42.5
Cost	456

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.65 \cdot 10^{+22}:\\ \;\;\;\;-R \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq 650000000:\\ \;\;\;\;\phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14
Error	48.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq 2.5 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot R\\ \end{array} \]

Alternative 15
Error	54.2
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023033 
(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))))))