Midpoint on a great circle

Average Error: 0.9 → 0.2

Time: 23.3s

Precision: binary64

Cost: 97344

Math

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

↓

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

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
 (+
  lambda1
  (atan2
   (*
    (cos phi2)
    (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2))))
   (+
    (cos phi1)
    (*
     (cos phi2)
     (+
      (log (pow (exp (sin lambda1)) (sin lambda2)))
      (* (cos lambda2) (cos lambda1))))))))

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) {
	return lambda1 + atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (cos(phi1) + (cos(phi2) * (log(pow(exp(sin(lambda1)), sin(lambda2))) + (cos(lambda2) * cos(lambda1))))));
}

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
    code = lambda1 + atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (cos(phi1) + (cos(phi2) * (log((exp(sin(lambda1)) ** sin(lambda2))) + (cos(lambda2) * cos(lambda1))))))
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) {
	return lambda1 + Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2)))), (Math.cos(phi1) + (Math.cos(phi2) * (Math.log(Math.pow(Math.exp(Math.sin(lambda1)), Math.sin(lambda2))) + (Math.cos(lambda2) * Math.cos(lambda1))))));
}

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):
	return lambda1 + math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2)))), (math.cos(phi1) + (math.cos(phi2) * (math.log(math.pow(math.exp(math.sin(lambda1)), math.sin(lambda2))) + (math.cos(lambda2) * math.cos(lambda1))))))

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)
	return Float64(lambda1 + atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2)))), Float64(cos(phi1) + Float64(cos(phi2) * Float64(log((exp(sin(lambda1)) ^ sin(lambda2))) + Float64(cos(lambda2) * cos(lambda1)))))))
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)
	tmp = lambda1 + atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (cos(phi1) + (cos(phi2) * (log((exp(sin(lambda1)) ^ sin(lambda2))) + (cos(lambda2) * cos(lambda1))))));
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_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Log[N[Power[N[Exp[N[Sin[lambda1], $MachinePrecision]], $MachinePrecision], N[Sin[lambda2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $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 0.8

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

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

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	78208

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

Alternative 2
Error	0.6
Cost	71808

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

Alternative 3
Error	0.8
Cost	58752

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

Alternative 4
Error	0.8
Cost	45568

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

Alternative 5
Error	1.6
Cost	39369

\[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.15 \cdot 10^{-8} \lor \neg \left(\lambda_2 \leq 48000000000\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \]

Alternative 6
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 7
Error	1.3
Cost	39168

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

Alternative 8
Error	7.1
Cost	33412

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t_0\\ \mathbf{if}\;\phi_1 \leq -30000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 + \left(-0.5 \cdot \left(\phi_2 \cdot \phi_2\right) + 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 9 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \end{array} \]

Alternative 9
Error	12.2
Cost	33284

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.4:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{\left(t_0 + 1\right) + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 + t_0}\\ \end{array} \]

Alternative 10
Error	12.6
Cost	33220

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq -0.01:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\left(t_0 + 1\right) + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + t_0}\\ \end{array} \]

Alternative 11
Error	7.6
Cost	33160

\[\begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{-8}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_1 + t_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 \cdot t_0 + 1}\\ \end{array} \]

Alternative 12
Error	7.7
Cost	33032

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t_0\\ \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{-8}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_1}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \]

Alternative 13
Error	7.7
Cost	33032

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{+20}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{-8}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \]

Alternative 14
Error	7.6
Cost	32905

\[\begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.7 \cdot 10^{+20} \lor \neg \left(\phi_2 \leq 2.6 \cdot 10^{-8}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 15
Error	17.7
Cost	32644

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

Alternative 16
Error	14.5
Cost	26240

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

Alternative 17
Error	14.7
Cost	26112

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

Alternative 18
Error	20.9
Cost	19840

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

Alternative 19
Error	28.3
Cost	19712

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

Alternative 20
Error	24.3
Cost	19712

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

Alternative 21
Error	21.1
Cost	19712

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

Reproduce

herbie shell --seed 2023027 
(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)))))))