# ?

Average Error: 6.9 → 0.0
Time: 26.8s
Precision: binary64
Cost: 39168

# ?

$\left(\left(-1.6 \leq lat1 \land lat1 \leq 1.6\right) \land \left(-1.6 \leq lat2 \land lat2 \leq 1.6\right)\right) \land \left(-6.4 \leq dlon \land dlon \leq 6.4\right)$
$2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(\frac{lat1 - lat2}{2}\right)}^{2} + \left(\cos lat1 \cdot \cos lat2\right) \cdot {\sin \left(\frac{dlon}{2}\right)}^{2}}\right)$
$2 \cdot \sin^{-1} \left(\mathsf{hypot}\left(\sqrt{\cos lat2} \cdot \sin \left(dlon \cdot 0.5\right), \sin \left(0.5 \cdot \left(lat1 - lat2\right)\right)\right)\right)$
(FPCore (lat1 lat2 dlon)
:precision binary64
(*
2.0
(asin
(sqrt
(+
(pow (sin (/ (- lat1 lat2) 2.0)) 2.0)
(* (* (cos lat1) (cos lat2)) (pow (sin (/ dlon 2.0)) 2.0)))))))
(FPCore (lat1 lat2 dlon)
:precision binary64
(*
2.0
(asin
(hypot
(* (sqrt (cos lat2)) (sin (* dlon 0.5)))
(sin (* 0.5 (- lat1 lat2)))))))
double code(double lat1, double lat2, double dlon) {
return 2.0 * asin(sqrt((pow(sin(((lat1 - lat2) / 2.0)), 2.0) + ((cos(lat1) * cos(lat2)) * pow(sin((dlon / 2.0)), 2.0)))));
}

double code(double lat1, double lat2, double dlon) {
return 2.0 * asin(hypot((sqrt(cos(lat2)) * sin((dlon * 0.5))), sin((0.5 * (lat1 - lat2)))));
}

public static double code(double lat1, double lat2, double dlon) {
return 2.0 * Math.asin(Math.sqrt((Math.pow(Math.sin(((lat1 - lat2) / 2.0)), 2.0) + ((Math.cos(lat1) * Math.cos(lat2)) * Math.pow(Math.sin((dlon / 2.0)), 2.0)))));
}

public static double code(double lat1, double lat2, double dlon) {
return 2.0 * Math.asin(Math.hypot((Math.sqrt(Math.cos(lat2)) * Math.sin((dlon * 0.5))), Math.sin((0.5 * (lat1 - lat2)))));
}

def code(lat1, lat2, dlon):
return 2.0 * math.asin(math.sqrt((math.pow(math.sin(((lat1 - lat2) / 2.0)), 2.0) + ((math.cos(lat1) * math.cos(lat2)) * math.pow(math.sin((dlon / 2.0)), 2.0)))))

def code(lat1, lat2, dlon):
return 2.0 * math.asin(math.hypot((math.sqrt(math.cos(lat2)) * math.sin((dlon * 0.5))), math.sin((0.5 * (lat1 - lat2)))))

function code(lat1, lat2, dlon)
return Float64(2.0 * asin(sqrt(Float64((sin(Float64(Float64(lat1 - lat2) / 2.0)) ^ 2.0) + Float64(Float64(cos(lat1) * cos(lat2)) * (sin(Float64(dlon / 2.0)) ^ 2.0))))))
end

function code(lat1, lat2, dlon)
return Float64(2.0 * asin(hypot(Float64(sqrt(cos(lat2)) * sin(Float64(dlon * 0.5))), sin(Float64(0.5 * Float64(lat1 - lat2))))))
end

function tmp = code(lat1, lat2, dlon)
tmp = 2.0 * asin(sqrt(((sin(((lat1 - lat2) / 2.0)) ^ 2.0) + ((cos(lat1) * cos(lat2)) * (sin((dlon / 2.0)) ^ 2.0)))));
end

function tmp = code(lat1, lat2, dlon)
tmp = 2.0 * asin(hypot((sqrt(cos(lat2)) * sin((dlon * 0.5))), sin((0.5 * (lat1 - lat2)))));
end

code[lat1_, lat2_, dlon_] := N[(2.0 * N[ArcSin[N[Sqrt[N[(N[Power[N[Sin[N[(N[(lat1 - lat2), $MachinePrecision] / 2.0),$MachinePrecision]], $MachinePrecision], 2.0],$MachinePrecision] + N[(N[(N[Cos[lat1], $MachinePrecision] * N[Cos[lat2],$MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[N[(dlon / 2.0),$MachinePrecision]], $MachinePrecision], 2.0],$MachinePrecision]), $MachinePrecision]),$MachinePrecision]], $MachinePrecision]],$MachinePrecision]), $MachinePrecision]  code[lat1_, lat2_, dlon_] := N[(2.0 * N[ArcSin[N[Sqrt[N[(N[Sqrt[N[Cos[lat2],$MachinePrecision]], $MachinePrecision] * N[Sin[N[(dlon * 0.5),$MachinePrecision]], $MachinePrecision]),$MachinePrecision] ^ 2 + N[Sin[N[(0.5 * N[(lat1 - lat2), $MachinePrecision]),$MachinePrecision]], $MachinePrecision] ^ 2],$MachinePrecision]], $MachinePrecision]),$MachinePrecision]

2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(\frac{lat1 - lat2}{2}\right)}^{2} + \left(\cos lat1 \cdot \cos lat2\right) \cdot {\sin \left(\frac{dlon}{2}\right)}^{2}}\right)

2 \cdot \sin^{-1} \left(\mathsf{hypot}\left(\sqrt{\cos lat2} \cdot \sin \left(dlon \cdot 0.5\right), \sin \left(0.5 \cdot \left(lat1 - lat2\right)\right)\right)\right)


# Try it out?

Results

 In Out
Enter valid numbers for all inputs

# Derivation?

1. Initial program 6.9

$2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(\frac{lat1 - lat2}{2}\right)}^{2} + \left(\cos lat1 \cdot \cos lat2\right) \cdot {\sin \left(\frac{dlon}{2}\right)}^{2}}\right)$
2. Applied egg-rr0.0

$\leadsto 2 \cdot \sin^{-1} \color{blue}{\left(\mathsf{hypot}\left(\sin \left(dlon \cdot 0.5\right) \cdot \sqrt{\cos lat1 \cdot \cos lat2}, \sin \left(\left(lat1 - lat2\right) \cdot 0.5\right)\right)\right)}$
3. Taylor expanded in lat1 around 0 0.0

$\leadsto 2 \cdot \sin^{-1} \left(\mathsf{hypot}\left(\sin \left(dlon \cdot 0.5\right) \cdot \color{blue}{\sqrt{\cos lat2}}, \sin \left(\left(lat1 - lat2\right) \cdot 0.5\right)\right)\right)$
4. Final simplification0.0

$\leadsto 2 \cdot \sin^{-1} \left(\mathsf{hypot}\left(\sqrt{\cos lat2} \cdot \sin \left(dlon \cdot 0.5\right), \sin \left(0.5 \cdot \left(lat1 - lat2\right)\right)\right)\right)$

# Alternatives

Alternative 1
Error15.5
Cost39300
$\begin{array}{l} \mathbf{if}\;lat1 \leq -7.4 \cdot 10^{-151}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\sqrt{\mathsf{fma}\left(0.25, \left(dlon \cdot dlon\right) \cdot \cos lat1, {\sin \left(0.5 \cdot lat1\right)}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\mathsf{hypot}\left(\sqrt{\cos lat2} \cdot \sin \left(dlon \cdot 0.5\right), \sin \left(lat2 \cdot -0.5\right)\right)\right)\\ \end{array}$
Alternative 2
Error15.2
Cost39172
$\begin{array}{l} \mathbf{if}\;lat1 \leq -9 \cdot 10^{-152}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(0.5 \cdot \left(lat1 - lat2\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\mathsf{hypot}\left(\sqrt{\cos lat2} \cdot \sin \left(dlon \cdot 0.5\right), \sin \left(lat2 \cdot -0.5\right)\right)\right)\\ \end{array}$
Alternative 3
Error14.4
Cost33426
$\begin{array}{l} \mathbf{if}\;dlon \leq -2.1 \cdot 10^{-101} \lor \neg \left(dlon \leq -4.4 \cdot 10^{-118}\right) \land \left(dlon \leq -1.4 \cdot 10^{-139} \lor \neg \left(dlon \leq 2.3 \cdot 10^{-132}\right)\right):\\ \;\;\;\;2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(lat2 \cdot -0.5\right)}^{2} + \cos lat2 \cdot \left(dlon \cdot \left(dlon \cdot 0.25\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(0.5 \cdot \left(lat1 - lat2\right)\right)}^{2}}\right)\\ \end{array}$
Alternative 4
Error17.9
Cost26376
$\begin{array}{l} t_0 := \sin \left(dlon \cdot 0.5\right)\\ \mathbf{if}\;dlon \leq -1.45 \cdot 10^{-101}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\sqrt{{t_0}^{2}}\right)\\ \mathbf{elif}\;dlon \leq 1.2 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(0.5 \cdot \left(lat1 - lat2\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sin^{-1} t_0\\ \end{array}$
Alternative 5
Error44.0
Cost26116
$\begin{array}{l} \mathbf{if}\;lat2 \leq 4.5 \cdot 10^{-139}:\\ \;\;\;\;2 \cdot \sin^{-1} \sin \left(dlon \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(lat2 \cdot -0.5\right)}^{2}}\right)\\ \end{array}$
Alternative 6
Error37.2
Cost26116
$\begin{array}{l} \mathbf{if}\;lat2 \leq 5.9 \cdot 10^{-124}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(dlon \cdot 0.5\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sin^{-1} \left(\sqrt{{\sin \left(lat2 \cdot -0.5\right)}^{2}}\right)\\ \end{array}$
Alternative 7
Error51.7
Cost13120
$2 \cdot \sin^{-1} \sin \left(lat2 \cdot -0.5\right)$
Alternative 8
Error51.4
Cost13120
$2 \cdot \sin^{-1} \sin \left(dlon \cdot 0.5\right)$

# Reproduce?

herbie shell --seed 1
(FPCore (lat1 lat2 dlon)
:name "2*asin(sqrt(pow(sin((lat1-lat2)/2),2) + cos(lat1)*cos(lat2)*pow(sin(dlon/2),2)))"
:precision binary64
:pre (and (and (and (<= -1.6 lat1) (<= lat1 1.6)) (and (<= -1.6 lat2) (<= lat2 1.6))) (and (<= -6.4 dlon) (<= dlon 6.4)))
(* 2.0 (asin (sqrt (+ (pow (sin (/ (- lat1 lat2) 2.0)) 2.0) (* (* (cos lat1) (cos lat2)) (pow (sin (/ dlon 2.0)) 2.0)))))))