Equirectangular approximation to distance on a great circle

Average Error: 38.9 → 0.1

Time: 18.5s

Precision: binary64

Cost: 39872

Math

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

↓

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

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

Derivation

1. Initial program 38.9

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

2. Simplified 3.7

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

3. Applied egg-rr 3.7

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

4. Applied egg-rr 0.2

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

5. Simplified 0.2

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}}\right), \phi_1 - \phi_2\right) \]
   Proof
   (fma.f64 (cos.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1)) (neg.f64 (*.f64 (sin.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (cos.f64 (*.f64 1/2 phi2)) (cos.f64 (*.f64 1/2 phi1))) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1))))): 7 points increase in error, 9 points decrease in error
   (-.f64 (*.f64 (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi2 1/2))) (cos.f64 (*.f64 1/2 phi1))) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 (*.f64 phi2 1/2)) (cos.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi1 1/2)))) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite=> *-commutative_binary64 (*.f64 (cos.f64 (*.f64 phi1 1/2)) (cos.f64 (*.f64 phi2 1/2)))) (*.f64 (sin.f64 (*.f64 1/2 phi2)) (sin.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 (*.f64 phi1 1/2)) (cos.f64 (*.f64 phi2 1/2))) (*.f64 (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi2 1/2))) (sin.f64 (*.f64 1/2 phi1)))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 (*.f64 phi1 1/2)) (cos.f64 (*.f64 phi2 1/2))) (*.f64 (sin.f64 (*.f64 phi2 1/2)) (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 phi1 1/2))))): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (cos.f64 (*.f64 phi1 1/2)) (cos.f64 (*.f64 phi2 1/2))) (Rewrite=> *-commutative_binary64 (*.f64 (sin.f64 (*.f64 phi1 1/2)) (sin.f64 (*.f64 phi2 1/2))))): 0 points increase in error, 0 points decrease in error

6. Applied egg-rr 0.1

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

7. Final simplification 0.1

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

Alternatives

Alternative 1
Error	7.4
Cost	33672

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-148}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.7 \cdot 10^{+190}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - t_0 \cdot \cos \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot t_0\right)\\ \end{array} \]

Alternative 2
Error	0.1
Cost	33536

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

Alternative 3
Error	14.4
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.2 \cdot 10^{-74}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 4
Error	6.3
Cost	13700

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 5
Error	3.7
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 6
Error	19.1
Cost	13572

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 7
Error	17.0
Cost	13572

\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00365:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 8
Error	46.7
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(\lambda_2 - \lambda_1\right)\\ t_1 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -2.4 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 9.8 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 3.2 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 9
Error	45.7
Cost	848

\[\begin{array}{l} t_0 := R \cdot \left(\lambda_2 - \lambda_1\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.7 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.42 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-11}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	48.1
Cost	652

\[\begin{array}{l} t_0 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 9.5 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-148}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 3.3 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 11
Error	50.1
Cost	388

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12
Error	50.2
Cost	324

\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.8 \cdot 10^{-64}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 13
Error	57.6
Cost	192

\[R \cdot \lambda_2 \]

Reproduce

herbie shell --seed 2022332 
(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))))))