Equirectangular approximation to distance on a great circle

Average Error: 38.8 → 4.0

Time: 20.9s

Precision: binary64

Cost: 13700

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -7 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

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 -7e-8)
   (* 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 <= -7e-8) {
		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 <= -7e-8) {
		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 <= -7e-8:
		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 <= -7e-8)
		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 <= -7e-8)
		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, -7e-8], 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 < -7.00000000000000048e-8

   1. Initial program 46.9

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

   2. Simplified 7.2

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
      Proof
      (*.f64 R (hypot.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (-.f64 phi1 phi2))): 0 points increase in error, 0 points decrease in error
      (*.f64 R (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in phi2 around 0 7.2

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

   if -7.00000000000000048e-8 < phi1

   1. Initial program 32.7

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

   2. Simplified 1.6

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
      Proof
      (*.f64 R (hypot.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (-.f64 phi1 phi2))): 0 points increase in error, 0 points decrease in error
      (*.f64 R (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2))) (*.f64 (-.f64 lambda1 lambda2) (cos.f64 (/.f64 (+.f64 phi1 phi2) 2)))) (*.f64 (-.f64 phi1 phi2) (-.f64 phi1 phi2)))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in phi1 around 0 1.6

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	28.5
Cost	13968

\[\begin{array}{l} t_0 := R \cdot \sqrt{\phi_2 \cdot \phi_2 + {\left(\lambda_1 - \lambda_2\right)}^{2}}\\ \mathbf{if}\;\phi_1 \leq -4.3 \cdot 10^{-63}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_1 \leq 9.6 \cdot 10^{-194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 3.8 \cdot 10^{-151}:\\ \;\;\;\;\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq 2.35 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 2
Error	19.2
Cost	13837

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.6 \cdot 10^{+100} \lor \neg \left(\lambda_2 \leq 7.2 \cdot 10^{+125}\right) \land \lambda_2 \leq 7.8 \cdot 10^{+192}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3
Error	19.4
Cost	13837

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.75 \cdot 10^{+100}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.1 \cdot 10^{+126} \lor \neg \left(\lambda_2 \leq 1.32 \cdot 10^{+193}\right):\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 4
Error	7.2
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0205:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 5
Error	4.0
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 6
Error	31.6
Cost	7637

\[\begin{array}{l} t_0 := \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{+224}:\\ \;\;\;\;t_0 \cdot \left(R \cdot \left(-\lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -1.5 \cdot 10^{-77}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\lambda_1 \leq -6.4 \cdot 10^{-103}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -4.4 \cdot 10^{-147} \lor \neg \left(\lambda_1 \leq -3.1 \cdot 10^{-199}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot t_0\right)\\ \end{array} \]

Alternative 7
Error	31.6
Cost	7637

\[\begin{array}{l} t_0 := \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -3.2 \cdot 10^{+223}:\\ \;\;\;\;\left(\lambda_1 \cdot t_0\right) \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_1 \leq -1.5 \cdot 10^{-77}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\lambda_1 \leq -6.4 \cdot 10^{-103}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -4.4 \cdot 10^{-147} \lor \neg \left(\lambda_1 \leq -3.1 \cdot 10^{-199}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot t_0\right)\\ \end{array} \]

Alternative 8
Error	33.9
Cost	7508

\[\begin{array}{l} t_0 := \cos \left(\phi_1 \cdot 0.5\right)\\ t_1 := t_0 \cdot \left(R \cdot \left(-\lambda_1\right)\right)\\ t_2 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -2.7 \cdot 10^{-191}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\phi_2 \leq 6 \cdot 10^{-273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 5.5 \cdot 10^{-187}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot t_0\right)\\ \mathbf{elif}\;\phi_2 \leq 2.9 \cdot 10^{-72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.85 \cdot 10^{-53}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	29.7
Cost	7373

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.7 \cdot 10^{+100} \lor \neg \left(\lambda_2 \leq 9.2 \cdot 10^{+125}\right) \land \lambda_2 \leq 1.7 \cdot 10^{+195}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 10
Error	29.6
Cost	7373

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.45 \cdot 10^{+100} \lor \neg \left(\lambda_2 \leq 1.1 \cdot 10^{+126}\right) \land \lambda_2 \leq 2.45 \cdot 10^{+188}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 11
Error	29.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.3 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 12
Error	29.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.5 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\phi_1 \cdot 0.5\right) \cdot \left(R \cdot \lambda_2\right)\\ \end{array} \]

Alternative 13
Error	29.7
Cost	6980

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.02 \cdot 10^{+193}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot \left(R \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 14
Error	35.5
Cost	388

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-119}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 15
Error	30.6
Cost	320

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

Alternative 16
Error	46.3
Cost	192

\[R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2022340 
(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))))))