Midpoint on a great circle

Average Error: 0.9 → 0.9

Time: 27.7s

Precision: binary64

Cost: 78592

Math

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

↓

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \lambda_1 + \cos \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 \cdot \left(\left(\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right) \cdot \frac{1}{t_0}\right)} \end{array} \]

FPCore

(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
 (let* ((t_0 (+ (* (cos phi2) (cos lambda1)) (cos phi1))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (*
      t_0
      (*
       (+ (cos phi1) (* (cos (- lambda1 lambda2)) (cos phi2)))
       (/ 1.0 t_0)))))))

C

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) {
	double t_0 = (cos(phi2) * cos(lambda1)) + cos(phi1);
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 * ((cos(phi1) + (cos((lambda1 - lambda2)) * cos(phi2))) * (1.0 / t_0))));
}

Fortran

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
    real(8) :: t_0
    t_0 = (cos(phi2) * cos(lambda1)) + cos(phi1)
    code = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 * ((cos(phi1) + (cos((lambda1 - lambda2)) * cos(phi2))) * (1.0d0 / t_0))))
end function

Java

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) {
	double t_0 = (Math.cos(phi2) * Math.cos(lambda1)) + Math.cos(phi1);
	return lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 * ((Math.cos(phi1) + (Math.cos((lambda1 - lambda2)) * Math.cos(phi2))) * (1.0 / t_0))));
}

Python

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):
	t_0 = (math.cos(phi2) * math.cos(lambda1)) + math.cos(phi1)
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 * ((math.cos(phi1) + (math.cos((lambda1 - lambda2)) * math.cos(phi2))) * (1.0 / t_0))))

Julia

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)
	t_0 = Float64(Float64(cos(phi2) * cos(lambda1)) + cos(phi1))
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 * Float64(Float64(cos(phi1) + Float64(cos(Float64(lambda1 - lambda2)) * cos(phi2))) * Float64(1.0 / t_0)))))
end

MATLAB

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)
	t_0 = (cos(phi2) * cos(lambda1)) + cos(phi1);
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 * ((cos(phi1) + (cos((lambda1 - lambda2)) * cos(phi2))) * (1.0 / t_0))));
end

Wolfram

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_] := Block[{t$95$0 = N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

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. Applied egg-rr 0.9

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

3. Taylor expanded in lambda2 around 0 13.2

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

4. Taylor expanded in lambda2 around 0 0.9

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

5. Final simplification 0.9

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

Alternatives

Alternative 1
Error	0.9
Cost	52736

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \left(\frac{1}{t_0} \cdot \left(t_0 \cdot t_0\right)\right)} \end{array} \]

Alternative 2
Error	7.2
Cost	45960

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.56:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \lambda_1 + \cos \phi_1}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{1 + t_1 \cdot \cos \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{t_1 + \cos \phi_1}\\ \end{array} \]

Alternative 3
Error	7.8
Cost	45896

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.56:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\cos \phi_2 \leq 0.999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{1 + t_1 \cdot \cos \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{t_1 + \cos \phi_1}\\ \end{array} \]

Alternative 4
Error	1.1
Cost	39432

\[\begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 + \cos \left(-\lambda_2\right) \cdot \cos \phi_2}\\ \mathbf{if}\;\lambda_2 \leq -3.9 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 10^{-49}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \lambda_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	0.9
Cost	39296

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

Alternative 6
Error	17.1
Cost	39172

\[\begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.9885:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(-\lambda_2\right) + \lambda_1\right)}{\cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]

Alternative 7
Error	7.5
Cost	33160

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3500000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 3.7 \cdot 10^{-9}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{t_0 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{1 + t_0 \cdot \cos \phi_2}\\ \end{array} \]

Alternative 8
Error	7.4
Cost	33032

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{if}\;\phi_2 \leq -3500000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_2 \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	8.3
Cost	32968

\[\begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{if}\;\lambda_2 \leq -38000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 0.115:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	7.6
Cost	32968

\[\begin{array}{l} t_0 := \lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{if}\;\lambda_2 \leq -38000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\lambda_2 \leq 0.115:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	12.3
Cost	32772

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.2:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(-\lambda_2\right) + \lambda_1\right)}{\cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]

Alternative 12
Error	20.6
Cost	26432

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

Alternative 13
Error	20.5
Cost	26368

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

Alternative 14
Error	22.4
Cost	26304

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

Alternative 15
Error	23.6
Cost	26240

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

Alternative 16
Error	23.7
Cost	19712

\[\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{2} \]

Reproduce

herbie shell --seed 2023092 
(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)))))))