Equirectangular approximation to distance on a great circle

Average Error: 38.8 → 0.2

Time: 24.8s

Precision: binary64

Cost: 46016

Math

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

↓

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

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
   (*
    (- lambda1 lambda2)
    (sqrt
     (+
      0.5
      (* 0.5 (fma (cos phi2) (cos phi1) (* (sin phi1) (- (sin phi2))))))))
   (- 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) {
	return R * hypot(((lambda1 - lambda2) * sqrt((0.5 + (0.5 * fma(cos(phi2), cos(phi1), (sin(phi1) * -sin(phi2))))))), (phi1 - phi2));
}

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

Derivation

1. Initial program 38.8

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

2. Simplified 3.8

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   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)))))): 155 points increase in error, 0 points decrease in error

3. Applied egg-rr 3.9

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

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	9.6
Cost	39684

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.318335671645753 \cdot 10^{-141}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \sqrt{0.5 + 0.5 \cdot \left(\cos \phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \sin \phi_2\right)}, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \sqrt{0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)}, \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 2
Error	0.2
Cost	39680

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

Alternative 3
Error	16.4
Cost	13704

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2012677809497.8635:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -1.9934329730175826 \cdot 10^{-97}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 4
Error	14.2
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.0835651482494671 \cdot 10^{-42}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 5
Error	6.5
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 350411.45250589313:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 6
Error	3.8
Cost	13696

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

Alternative 7
Error	13.1
Cost	13572

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2012677809497.8635:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 8
Error	32.2
Cost	7184

\[\begin{array}{l} t_0 := R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1\right)\\ t_1 := R \cdot \mathsf{hypot}\left(\lambda_2, \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -2.320436380497157 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 8.490674918195443 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 8.825344012140884 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 2.194504030752542 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.2084968598311036 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 9
Error	36.0
Cost	7052

\[\begin{array}{l} t_0 := R \cdot \left(\lambda_2 - \lambda_1\right)\\ t_1 := R \cdot \mathsf{hypot}\left(\lambda_2, \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -6.321633025278543 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq -1.212357382097531 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.7534628626627283 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.2084968598311036 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 10
Error	23.9
Cost	6916

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -134.72726975818338:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right)\\ \end{array} \]

Alternative 11
Error	23.6
Cost	6916

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.2084968598311036 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 12
Error	13.7
Cost	6912

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

Alternative 13
Error	50.0
Cost	1048

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -2.320436380497157 \cdot 10^{-97}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq -1.9351892136409803 \cdot 10^{-221}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.5245274969554803 \cdot 10^{-301}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 8.490674918195443 \cdot 10^{-193}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.7534628626627283 \cdot 10^{-104}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 2.970016784988563 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 14
Error	48.5
Cost	1048

\[\begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -6.321633025278543 \cdot 10^{-51}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq -1.212357382097531 \cdot 10^{-215}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5.258270202557508 \cdot 10^{-182}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 8.825344012140884 \cdot 10^{-112}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.7534628626627283 \cdot 10^{-104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 2.970016784988563 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 15
Error	46.6
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -2.149919429904484 \cdot 10^{+99}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.8448444049625803 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq -22159387361.577335:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq 2.2301410115946615 \cdot 10^{-202}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 16
Error	44.9
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(\lambda_2 - \lambda_1\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_1 \leq -134.72726975818338:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -3.556328502110379 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -1.9934329730175826 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq 2.2301410115946615 \cdot 10^{-202}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	49.7
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.2084968598311036 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 18
Error	57.0
Cost	192

\[R \cdot \lambda_2 \]

Reproduce

herbie shell --seed 2022295 
(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))))))