?

Average Accuracy: 26.2% → 68.5%
Time: 18.3s
Precision: binary64
Cost: 33792

?

\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
↓
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) + \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \]
(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) (* (cos (* phi2 0.5)) (cos (* 0.5 phi1))))
    (* (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))) (- lambda2 lambda1)))
   (- phi1 phi2))))
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) * (cos((phi2 * 0.5)) * cos((0.5 * phi1)))) + ((sin((0.5 * phi1)) * sin((phi2 * 0.5))) * (lambda2 - lambda1))), (phi1 - phi2));
}

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

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):
	return R * math.hypot((((lambda1 - lambda2) * (math.cos((phi2 * 0.5)) * math.cos((0.5 * phi1)))) + ((math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))) * (lambda2 - lambda1))), (phi1 - phi2))

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(Float64(lambda1 - lambda2) * Float64(cos(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1)))) + Float64(Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))) * Float64(lambda2 - lambda1))), Float64(phi1 - phi2)))
end

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 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((((lambda1 - lambda2) * (cos((phi2 * 0.5)) * cos((0.5 * phi1)))) + ((sin((0.5 * phi1)) * sin((phi2 * 0.5))) * (lambda2 - lambda1))), (phi1 - phi2));
end

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[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

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 \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) + \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right)

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 27.2%

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

  2. Simplified67.6%

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

  3. Applied egg-rr67.5%

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

  4. Applied egg-rr70.9%

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

  5. Applied egg-rr71.0%

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

  6. Final simplification71.0%

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

Alternatives

Alternative 1
Accuracy51.5%
Cost13700
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -4250000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 2
Accuracy64.6%
Cost13696
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \]
Alternative 3
Accuracy48.5%
Cost13572
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
Alternative 4
Accuracy50.7%
Cost13572
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -95000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 5
Accuracy43.2%
Cost6916
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 9 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
Alternative 6
Accuracy17.9%
Cost388
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1700000:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 7
Accuracy19.5%
Cost320
\[R \cdot \left(\phi_2 - \phi_1\right) \]
Alternative 8
Accuracy10.9%
Cost192
\[R \cdot \phi_1 \]
Alternative 9
Accuracy11.4%
Cost192
\[R \cdot \phi_2 \]

Error

Reproduce?

herbie shell --seed 2023157 -o generate:proofs
(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))))))