Average Error: 39.2 → 0.1
Time: 26.0s
Precision: binary64
Cost: 33792
\[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
↓
\[R \cdot \mathsf{hypot}\left(\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right) \cdot \left(\lambda_2 - \lambda_1\right) + \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right) \]
(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
   (+
    (* (* (sin (* 0.5 phi1)) (sin (* phi2 0.5))) (- lambda2 lambda1))
    (* (- lambda1 lambda2) (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))))
   (- 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((((sin((0.5 * phi1)) * sin((phi2 * 0.5))) * (lambda2 - lambda1)) + ((lambda1 - lambda2) * (cos((0.5 * phi1)) * cos((phi2 * 0.5))))), (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((((Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))) * (lambda2 - lambda1)) + ((lambda1 - lambda2) * (Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))))), (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((((math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))) * (lambda2 - lambda1)) + ((lambda1 - lambda2) * (math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))))), (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(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))) * Float64(lambda2 - lambda1)) + Float64(Float64(lambda1 - lambda2) * Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))))), 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((((sin((0.5 * phi1)) * sin((phi2 * 0.5))) * (lambda2 - lambda1)) + ((lambda1 - lambda2) * (cos((0.5 * phi1)) * cos((phi2 * 0.5))))), (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[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision] + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $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(\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) + \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), \phi_1 - \phi_2\right)

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.2

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

  2. Simplified3.7

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

  3. Applied egg-rr3.7

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

  4. Applied egg-rr0.2

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

  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

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

Alternatives

Alternative 1
Error9.0
Cost33800
\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -1 \cdot 10^{+224}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(t_1 - t_0\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq -6.457228250730709 \cdot 10^{-113}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot t_0 - \lambda_2 \cdot t_1, \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 2
Error9.0
Cost33800
\[\begin{array}{l} t_0 := \cos \left(\phi_2 \cdot 0.5\right)\\ t_1 := \sin \left(\phi_2 \cdot 0.5\right)\\ t_2 := \sin \left(0.5 \cdot \phi_1\right)\\ t_3 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\lambda_1 \leq -1 \cdot 10^{+224}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(t_3 \cdot \left(\lambda_1 \cdot t_0\right) - t_1 \cdot \left(\lambda_1 \cdot t_2\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq -6.457228250730709 \cdot 10^{-113}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(t_2 \cdot t_1\right) - \lambda_2 \cdot \left(t_3 \cdot t_0\right), \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 3
Error0.1
Cost33792
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right) + \sin \left(\phi_2 \cdot 0.5\right) \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_2 - \lambda_1\right)\right), \phi_1 - \phi_2\right) \]
Alternative 4
Error9.0
Cost33672
\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\lambda_1 \leq -1 \cdot 10^{+224}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(t_1 - t_0\right), \phi_1 - \phi_2\right)\\ \mathbf{elif}\;\lambda_1 \leq -6.457228250730709 \cdot 10^{-113}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(t_0 - t_1\right), \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 5
Error9.3
Cost33540
\[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.457228250730709 \cdot 10^{-113}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]
Alternative 6
Error24.0
Cost13708
\[\begin{array}{l} t_0 := R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 3.316191442604775 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.38848732481254 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 3719955018.244:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 7
Error3.7
Cost13696
\[R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 - \phi_1\right)\right), \phi_1 - \phi_2\right) \]
Alternative 8
Error3.7
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 9
Error18.4
Cost13572
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 3719955018.244:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 10
Error16.7
Cost13572
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.910710473738562 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
Alternative 11
Error24.4
Cost7180
\[\begin{array}{l} t_0 := R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 3.316191442604775 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_2 \leq 3.38848732481254 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 153310.93330732212:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]
Alternative 12
Error24.0
Cost6916
\[\begin{array}{l} \mathbf{if}\;\phi_2 \leq 153310.93330732212:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]
Alternative 13
Error48.6
Cost388
\[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.8183840418480905 \cdot 10^{+50}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
Alternative 14
Error46.9
Cost320
\[R \cdot \left(\phi_2 - \phi_1\right) \]
Alternative 15
Error54.2
Cost192
\[R \cdot \phi_2 \]

Error

Reproduce

herbie shell --seed 2022225 
(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))))))