Average Error: 39.3 → 0.1
Time: 36.9s
Precision: binary64
Cost: 33856
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
↓
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) + \left(-\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\right), \phi_1 - \phi_2\right) \]
(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 phi2)) (cos (* 0.5 phi1))))
    (- (* (* (sin (* 0.5 phi1)) (sin (* 0.5 phi2))) (- lambda1 lambda2))))
   (- 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 * phi2)) * cos((0.5 * phi1)))) + -((sin((0.5 * phi1)) * sin((0.5 * phi2))) * (lambda1 - lambda2))), (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 * phi2)) * Math.cos((0.5 * phi1)))) + -((Math.sin((0.5 * phi1)) * Math.sin((0.5 * phi2))) * (lambda1 - lambda2))), (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 * phi2)) * math.cos((0.5 * phi1)))) + -((math.sin((0.5 * phi1)) * math.sin((0.5 * phi2))) * (lambda1 - lambda2))), (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 * phi2)) * cos(Float64(0.5 * phi1)))) + Float64(-Float64(Float64(sin(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2))) * Float64(lambda1 - lambda2)))), 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 * phi2)) * cos((0.5 * phi1)))) + -((sin((0.5 * phi1)) * sin((0.5 * phi2))) * (lambda1 - lambda2))), (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 * phi2), $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[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(lambda1 - lambda2), $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_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) + \left(-\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) \cdot \left(\lambda_1 - \lambda_2\right)\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 39.3

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

  2. Simplified3.7

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

  3. Applied egg-rr0.1

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

  4. Simplified0.1

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

  5. Applied egg-rr0.1

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

Alternatives

Alternative 1
Error3.6
Cost33540
\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2 \cdot 10^{+144}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_1, \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 2
Error0.1
Cost33536
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right) \]
Alternative 3
Error3.8
Cost13832
\[\begin{array}{l} t_0 := R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \mathbf{if}\;\phi_2 \leq -9.2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error13.3
Cost13768
\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq -1.7 \cdot 10^{-84}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 10^{+17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(t_0 \cdot \lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(-\lambda_2\right) \cdot t_0, \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 5
Error13.6
Cost13768
\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1 \cdot 10^{-86}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.15 \cdot 10^{+89}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(-\lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 6
Error13.6
Cost13704
\[\begin{array}{l} t_0 := R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{if}\;\lambda_2 \leq -6 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 3.2 \cdot 10^{+16}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 7
Error12.9
Cost13704
\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -7 \cdot 10^{+192}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq 6.8 \cdot 10^{+214}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 8
Error8.1
Cost13700
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -19500000000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(-\lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 9
Error3.7
Cost13696
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right) \]
Alternative 10
Error14.0
Cost6912
\[R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right) \]
Alternative 11
Error39.3
Cost1032
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{-118}:\\ \;\;\;\;\left(-R \cdot \phi_2\right) + \phi_1 \cdot R\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-64}:\\ \;\;\;\;\left(-\lambda_2\right) \cdot \left(R \cdot \left(1 + \left(\phi_1 \cdot \phi_1\right) \cdot -0.125\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.2 \cdot 10^{-35}:\\ \;\;\;\;\phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
Alternative 12
Error42.7
Cost720
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.2 \cdot 10^{+45}:\\ \;\;\;\;-R \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq 4.4 \cdot 10^{-118}:\\ \;\;\;\;\phi_1 \cdot R\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;\left(-\lambda_2\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 3.1 \cdot 10^{-32}:\\ \;\;\;\;\phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 13
Error38.5
Cost644
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{-204}:\\ \;\;\;\;\left(-R \cdot \phi_2\right) + \phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
Alternative 14
Error40.6
Cost588
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-118}:\\ \;\;\;\;R \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 7 \cdot 10^{-64}:\\ \;\;\;\;\left(-\lambda_2\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{-26}:\\ \;\;\;\;\phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 15
Error42.6
Cost456
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.22 \cdot 10^{+45}:\\ \;\;\;\;-R \cdot \phi_2\\ \mathbf{elif}\;\phi_2 \leq 10^{-30}:\\ \;\;\;\;\phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 16
Error38.5
Cost452
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.1 \cdot 10^{-204}:\\ \;\;\;\;R \cdot \left(\phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
Alternative 17
Error48.2
Cost324
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.8 \cdot 10^{-27}:\\ \;\;\;\;\phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 18
Error54.1
Cost192
\[R \cdot \phi_2 \]

Error

Reproduce

herbie shell --seed 2023010 
(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))))))