Average Error: 38.7 → 0.1
Time: 27.3s
Precision: binary64
Cost: 33792
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
↓
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) + \left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \left(\lambda_2 - \lambda_1\right), \phi_1 - \phi_2\right) \]
(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 (* 0.5 phi1)) (cos (* phi2 0.5))))
    (* (* (sin (* phi2 0.5)) (sin (* 0.5 phi1))) (- 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((0.5 * phi1)) * cos((phi2 * 0.5)))) + ((sin((phi2 * 0.5)) * sin((0.5 * phi1))) * (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((0.5 * phi1)) * Math.cos((phi2 * 0.5)))) + ((Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1))) * (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((0.5 * phi1)) * math.cos((phi2 * 0.5)))) + ((math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1))) * (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(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))) + Float64(Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))) * 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((0.5 * phi1)) * cos((phi2 * 0.5)))) + ((sin((phi2 * 0.5)) * sin((0.5 * phi1))) * (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[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $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(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) + \left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\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 38.7

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

  2. Simplified3.8

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

  3. Applied egg-rr3.8

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)\right)\right)}, \phi_1 - \phi_2\right) \]

  4. Applied egg-rr0.2

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}\right)\right), \phi_1 - \phi_2\right) \]

  5. Applied egg-rr0.1

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

  6. Final simplification0.1

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

Alternatives

Alternative 1
Error2.1
Cost33664
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) + \sin \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_2 \cdot \sin \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right) \]
Alternative 2
Error14.0
Cost13708
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.0592555540658346 \cdot 10^{+129}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 10^{+245}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 10^{+258}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_2 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 3
Error19.0
Cost13704
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.11279800189672 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -3.2401508871842196 \cdot 10^{-153}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 4
Error15.0
Cost13700
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.950327075352609 \cdot 10^{-146}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 5
Error3.8
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 6
Error17.2
Cost13572
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.04590703279181849:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 7
Error13.5
Cost13444
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{+258}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_2 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 8
Error24.0
Cost7048
\[\begin{array}{l} t_0 := R \cdot \mathsf{hypot}\left(\lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq 2.881157387165448 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 1.1965760360404852 \cdot 10^{-127}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error19.9
Cost6916
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.5437724869068735 \cdot 10^{-71}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 10
Error13.8
Cost6912
\[R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right) \]
Alternative 11
Error49.5
Cost1248
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.04590703279181849:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.1878083319295299 \cdot 10^{-111}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_1 \leq -7.848520367225946 \cdot 10^{-178}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq -1.1411441771515235 \cdot 10^{-232}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_1 \leq -1.7313333895618456 \cdot 10^{-267}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 2.4394855180385192 \cdot 10^{-296}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_1 \leq 8.371537876318213 \cdot 10^{-265}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 3.972179648773149 \cdot 10^{-224}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 12
Error52.3
Cost852
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.605451215693404 \cdot 10^{-202}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\lambda_2 \leq -2.3951425286732574 \cdot 10^{-294}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 4.53832332539785 \cdot 10^{-55}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\lambda_2 \leq 5.130366966328912 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\lambda_2 \leq 1.1143187230952869 \cdot 10^{+54}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
Alternative 13
Error45.2
Cost848
\[\begin{array}{l} t_0 := R \cdot \left(\lambda_2 - \lambda_1\right)\\ t_1 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -2.3153818749767235 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 1.667950775216395 \cdot 10^{-267}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.7645127488947535 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 0.026801255497214203:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 14
Error45.6
Cost716
\[\begin{array}{l} t_0 := R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq 4.53832332539785 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 5.130366966328912 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\lambda_2 \leq 4.323359406883525 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
Alternative 15
Error52.0
Cost588
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.53832332539785 \cdot 10^{-55}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\lambda_2 \leq 5.130366966328912 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\lambda_2 \leq 1.1143187230952869 \cdot 10^{+54}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
Alternative 16
Error57.2
Cost192
\[R \cdot \lambda_2 \]

Error

Reproduce

herbie shell --seed 2022308 
(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))))))