Equirectangular approximation to distance on a great circle

Average Error: 60.57% → 6.71%

Time: 17.7s

Precision: binary64

Cost: 13700

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-37}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

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 (<= phi2 3e-37)
   (* 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 (phi2 <= 3e-37) {
		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 (phi2 <= 3e-37) {
		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 phi2 <= 3e-37:
		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 (phi2 <= 3e-37)
		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 (phi2 <= 3e-37)
		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[phi2, 3e-37], 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 phi2 < 3e-37

   1. Initial program 52.76

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

   2. Simplified 3.34

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

      [Start] 52.76	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      hypot-def [=>] 3.34	\[ R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi2 around 0 3.35

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

   if 3e-37 < phi2

   1. Initial program 69.91

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

   2. Simplified 9.7

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

      [Start] 69.91	\[ R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
      hypot-def [=>] 9.7	\[ R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Taylor expanded in phi1 around 0 10.72

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

4. Final simplification 6.71

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-37}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	50.3%
Cost	14168

\[\begin{array}{l} t_0 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \left(R \cdot \lambda_2\right) - R \cdot \lambda_1\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -4.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_1 \leq -1.6 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.18 \cdot 10^{-153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -5 \cdot 10^{-211}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.45 \cdot 10^{-236}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -7 \cdot 10^{-284}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \phi_2}\\ \mathbf{elif}\;\phi_1 \leq 4.1 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	50.74%
Cost	14032

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -7.4 \cdot 10^{+19}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 3.6 \cdot 10^{+71}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\lambda_2 \leq 1.85 \cdot 10^{+93}:\\ \;\;\;\;\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(R \cdot \lambda_2\right) - R \cdot \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 1.2 \cdot 10^{+154}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\phi_1 + \phi_2\right)}\\ \end{array} \]

Alternative 3
Error	6.23%
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
Error	14.07%
Cost	13568

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

Alternative 5
Error	47.47%
Cost	7496

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.2 \cdot 10^{-251}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 3.7 \cdot 10^{-57}:\\ \;\;\;\;\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \left(R \cdot \lambda_2\right) - R \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 6
Error	49.9%
Cost	7436

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.5 \cdot 10^{-242}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 6 \cdot 10^{-140}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.6 \cdot 10^{-119}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(-\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{-100}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 7
Error	47.98%
Cost	7368

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.22 \cdot 10^{-250}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-57}:\\ \;\;\;\;\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(R \cdot \lambda_2\right) - R \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 8
Error	49.89%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.4 \cdot 10^{-244}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{-100}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 9
Error	50.34%
Cost	585

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2 \cdot 10^{-241} \lor \neg \left(\phi_2 \leq 2.9 \cdot 10^{-96}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]

Alternative 10
Error	50.27%
Cost	584

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.9 \cdot 10^{-241}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{-100}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 11
Error	56.67%
Cost	456

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.85 \cdot 10^{-245}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.45 \cdot 10^{-40}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12
Error	68.12%
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.05 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 13
Error	89.17%
Cost	192

\[R \cdot \lambda_2 \]

Reproduce

herbie shell --seed 2023115 
(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))))))