Equirectangular approximation to distance on a great circle

Average Accuracy: 38.6% → 93.7%

Time: 22.6s

Precision: binary64

Cost: 13700

\[ \begin{array}{c}[phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \end{array} \]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -6900:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

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
 (if (<= phi1 -6900.0)
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* 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) {
	double tmp;
	if (phi1 <= -6900.0) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

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) {
	double tmp;
	if (phi1 <= -6900.0) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

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):
	tmp = 0
	if phi1 <= -6900.0:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

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)
	tmp = 0.0
	if (phi1 <= -6900.0)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return tmp
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_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -6900.0)
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
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_] := If[LessEqual[phi1, -6900.0], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6900

   1. Initial program 27.3%

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

   2. Simplified 89.6%

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

      [Start] 27.3	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      hypot-def [=>] 89.6	\[ R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi2 around 0 89.6%

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

   if -6900 < phi1

   1. Initial program 46.5%

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

   2. Simplified 97.1%

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

      [Start] 46.5	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      hypot-def [=>] 97.1	\[ R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around 0 96.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6900:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	75.3%
Cost	13768

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.4 \cdot 10^{-118}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 5.3 \cdot 10^{+41}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 2
Accuracy	88.9%
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 380000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 3
Accuracy	94.0%
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 4
Accuracy	80.8%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.75 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 5
Accuracy	81.3%
Cost	13572

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 380000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 6
Accuracy	54.5%
Cost	7316

\[\begin{array}{l} t_0 := R \cdot \lambda_2 - R \cdot \lambda_1\\ t_1 := R \cdot \phi_2 - R \cdot \phi_1\\ t_2 := R \cdot \mathsf{hypot}\left(\lambda_1, \phi_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.45 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -2.9 \cdot 10^{-156}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.5 \cdot 10^{-183}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq -6 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 3.5 \cdot 10^{-214}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_1 \leq 7 \cdot 10^{-92}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	67.3%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.5 \cdot 10^{+166}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1, \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 8
Accuracy	78.7%
Cost	6912

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

Alternative 9
Accuracy	49.6%
Cost	1240

\[\begin{array}{l} t_0 := R \cdot \lambda_2 - R \cdot \lambda_1\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1.36 \cdot 10^{-163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -2.3 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -1.05 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 7.2 \cdot 10^{-277}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 5.3 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	49.6%
Cost	1240

\[\begin{array}{l} t_0 := R \cdot \lambda_2 - R \cdot \lambda_1\\ t_1 := R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -2.3 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.2 \cdot 10^{-183}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.04 \cdot 10^{-274}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 8.2 \cdot 10^{-277}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 5.6 \cdot 10^{-83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	44.3%
Cost	916

\[\begin{array}{l} t_0 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-259}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.9 \cdot 10^{-261}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 2.3 \cdot 10^{-209}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.06 \cdot 10^{-178}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 800000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12
Accuracy	50.4%
Cost	849

\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -6.6 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-265}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 4.6 \cdot 10^{-210} \lor \neg \left(\phi_2 \leq 9.2 \cdot 10^{-179}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \end{array} \]

Alternative 13
Accuracy	32.1%
Cost	588

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.5 \cdot 10^{-209}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.3 \cdot 10^{-109}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1100000:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14
Accuracy	32.2%
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1100000:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 15
Accuracy	10.6%
Cost	192

\[R \cdot \lambda_2 \]

Reproduce

herbie shell --seed 2023137 
(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))))))