?

Average Error: 0.9 → 0.9
Time: 26.8s
Precision: binary64
Cost: 39296

?

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
↓
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2) (sin (- lambda1 lambda2)))
   (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
↓
(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi2) (sin (- lambda1 lambda2)))
   (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
}
↓
double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
}

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function
↓
real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
end function

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
}
↓
public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
}

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))
↓
def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end
↓
function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))
end

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
end
↓
function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
end

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}
↓
\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 0.9

    \[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

  2. Final simplification0.9

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

Alternatives

Alternative 1
Error7.6
Cost39432
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.056:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{1 + t_0 \cdot \cos \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 1.16 \cdot 10^{+24}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{t_0 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 \cdot \cos \lambda_1 + \cos \phi_1}\\ \end{array} \]
Alternative 2
Error1.4
Cost39232
\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(-\lambda_2\right) \cdot \cos \phi_2} \]
Alternative 3
Error16.0
Cost33036
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right) + 1\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -0.0155:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0}\\ \mathbf{elif}\;\lambda_2 \leq -1.3 \cdot 10^{-194}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\lambda_2 \leq 6.2 \cdot 10^{-11}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 \cdot \cos \lambda_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0}\\ \end{array} \]
Alternative 4
Error7.5
Cost33032
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -0.00045:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \left(-\lambda_2\right) + \cos \phi_1}\\ \mathbf{elif}\;\lambda_2 \leq 1.18 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \end{array} \]
Alternative 5
Error7.8
Cost33032
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.102:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{1 + t_0 \cdot \cos \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{t_0 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \]
Alternative 6
Error7.5
Cost32968
\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \left(-\lambda_2\right) + \cos \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error10.6
Cost32904
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right) + 1\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -0.025:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0}\\ \mathbf{elif}\;\lambda_2 \leq 2.65 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{t_0}\\ \end{array} \]
Alternative 8
Error14.2
Cost26896
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right) + 1\\ t_1 := \cos \phi_2 + \cos \phi_1\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\lambda_2\right)}{t_1}\\ \mathbf{if}\;\lambda_2 \leq -0.064:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0}\\ \mathbf{elif}\;\lambda_2 \leq -2.5 \cdot 10^{-195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 2.1 \cdot 10^{-139}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \lambda_1}{t_1}\\ \mathbf{elif}\;\lambda_2 \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0}\\ \end{array} \]
Alternative 9
Error14.2
Cost26832
\[\begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + 1}\\ t_1 := \cos \phi_2 + \cos \phi_1\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\lambda_2\right)}{t_1}\\ \mathbf{if}\;\lambda_2 \leq -0.0255:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq -3.4 \cdot 10^{-201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\lambda_2 \leq 5.2 \cdot 10^{-140}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \lambda_1}{t_1}\\ \mathbf{elif}\;\lambda_2 \leq 5.6 \cdot 10^{-5}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error19.1
Cost26440
\[\begin{array}{l} t_0 := \cos \phi_2 + \cos \phi_1\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \lambda_1}{t_0}\\ \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 5.3 \cdot 10^{-126}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(-\lambda_2\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 11
Error21.0
Cost26376
\[\begin{array}{l} t_0 := \cos \phi_2 + \cos \phi_1\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \lambda_1}{t_0}\\ \mathbf{if}\;\lambda_1 \leq -9.2 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_1 \leq 1.85 \cdot 10^{-128}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 12
Error22.9
Cost26312
\[\begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -1.35 \cdot 10^{-197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 4.3 \cdot 10^{-147}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 13
Error25.0
Cost20168
\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{\lambda_2 \cdot \left(-\cos \phi_2\right)}{t_0 + 1}\\ \mathbf{if}\;\lambda_2 \leq -2.25 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\lambda_2 \leq 1.7 \cdot 10^{-139}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1}{t_0 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 14
Error26.9
Cost19712
\[\lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1} \]

Error

Reproduce?

herbie shell --seed 2023090 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))