Bearing on a great circle

Percentage Accurate: 79.9% → 99.7%

Time: 29.5s

Alternatives: 24

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}

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 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (sin (- lambda1 lambda2)) (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
}

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 = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (fma (sin lambda1) (sin lambda2) (* (cos lambda2) (cos lambda1)))
    (* (cos phi2) (sin phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (fma(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(lambda1))) * (cos(phi2) * sin(phi1)))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sin-diff 86.5%

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

   2. sub-neg 86.5%

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

6. Applied egg-rr 86.5%

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

   1. sub-neg 86.5%

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

8. Simplified 86.5%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* 86.5%

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

   2. cos-diff 99.8%

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

   3. distribute-lft-in 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. distribute-lft-out 99.8%

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

    2. *-commutative 99.8%

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

    3. +-commutative 99.8%

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

    4. fma-define 99.8%

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

    5. *-commutative 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (cos phi2)
    (*
     (sin phi1)
     (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sin-diff 86.5%

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

   2. sub-neg 86.5%

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

6. Applied egg-rr 86.5%

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

   1. sub-neg 86.5%

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

8. Simplified 86.5%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
9. Step-by-step derivation

   1. cos-diff 99.8%

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

   2. *-commutative 99.8%

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

   3. fma-define 99.8%

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

10. Applied egg-rr 99.8%

    \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\right)} \]
11. Add Preprocessing

Alternative 3: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -6.1 \lor \neg \left(\phi_2 \leq 2.8 \cdot 10^{-24}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{t\_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= phi2 -6.1) (not (<= phi2 2.8e-24)))
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
       (cos phi2))
      (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (atan2
      (*
       (cos phi2)
       (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
      (-
       t_0
       (*
        (sin phi1)
        (+
         (* (cos lambda2) (cos lambda1))
         (* (sin lambda1) (sin lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((phi2 <= -6.1) || !(phi2 <= 2.8e-24)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), (t_0 - (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((phi2 <= -6.1) || !(phi2 <= 2.8e-24))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), Float64(t_0 - Float64(sin(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -6.1], N[Not[LessEqual[phi2, 2.8e-24]], $MachinePrecision]], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -6.0999999999999996 or 2.8000000000000002e-24 < phi2

   1. Initial program 74.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 74.5%

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

      2. associate-*l* 74.5%

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

   3. Simplified 74.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 88.3%

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

      2. sub-neg 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. sub-neg 88.3%

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

   8. Simplified 88.3%

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

   if -6.0999999999999996 < phi2 < 2.8000000000000002e-24

   1. Initial program 77.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 77.7%

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

      2. associate-*l* 77.7%

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

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 84.6%

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

      2. cancel-sign-sub-inv 84.6%

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

      3. fma-define 84.6%

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

   6. Applied egg-rr 84.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
   7. Step-by-step derivation

      1. cos-diff 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in phi2 around 0 99.9%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\cos \lambda_1\right) \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.1 \lor \neg \left(\phi_2 \leq 2.8 \cdot 10^{-24}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 2.8 \cdot 10^{-24}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_1 \cdot \cos \phi_2}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))))
   (if (or (<= phi2 -1.25e-5) (not (<= phi2 2.8e-24)))
     (atan2
      (* t_1 (cos phi2))
      (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (atan2
      t_1
      (-
       t_0
       (*
        (fma (sin lambda1) (sin lambda2) (* (cos lambda2) (cos lambda1)))
        (* (cos phi2) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = (sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2));
	double tmp;
	if ((phi2 <= -1.25e-5) || !(phi2 <= 2.8e-24)) {
		tmp = atan2((t_1 * cos(phi2)), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2(t_1, (t_0 - (fma(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(lambda1))) * (cos(phi2) * sin(phi1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2)))
	tmp = 0.0
	if ((phi2 <= -1.25e-5) || !(phi2 <= 2.8e-24))
		tmp = atan(Float64(t_1 * cos(phi2)), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(lambda1))) * Float64(cos(phi2) * sin(phi1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -1.25e-5], N[Not[LessEqual[phi2, 2.8e-24]], $MachinePrecision]], N[ArcTan[N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.25000000000000006e-5 or 2.8000000000000002e-24 < phi2

   1. Initial program 73.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 73.9%

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

      2. associate-*l* 73.9%

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

   3. Simplified 73.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 88.4%

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

      2. sub-neg 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. sub-neg 88.4%

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

   8. Simplified 88.4%

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

   if -1.25000000000000006e-5 < phi2 < 2.8000000000000002e-24

   1. Initial program 78.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 78.3%

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

      2. associate-*l* 78.3%

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

   3. Simplified 78.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 84.5%

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

      2. sub-neg 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. sub-neg 84.5%

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

   8. Simplified 84.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 84.5%

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

      2. cos-diff 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. distribute-lft-out 99.9%

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

       2. *-commutative 99.9%

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

       3. +-commutative 99.9%

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

       4. fma-define 99.9%

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

       5. *-commutative 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in phi2 around 0 99.9%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\cos \lambda_2 \cdot \sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2}}{\cos \phi_1 \cdot \sin \phi_2 - \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 2.8 \cdot 10^{-24}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\cos \phi_1 \cdot \sin \phi_2 - \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (cos phi2)
    (*
     (sin phi1)
     (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda1) (sin lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))));
}

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sin-diff 86.5%

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

   2. sub-neg 86.5%

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

6. Applied egg-rr 86.5%

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

   1. sub-neg 86.5%

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

8. Simplified 86.5%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
9. Step-by-step derivation

   1. cos-diff 99.8%

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

   2. +-commutative 99.8%

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

   3. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 6: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -6.1 \lor \neg \left(\phi_2 \leq 10^{-24}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (*
          (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
          (cos phi2))))
   (if (or (<= phi2 -6.1) (not (<= phi2 1e-24)))
     (atan2
      t_1
      (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (atan2
      t_1
      (-
       t_0
       (*
        (sin phi1)
        (+
         (* (cos lambda2) (cos lambda1))
         (* (sin lambda1) (sin lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2);
	double tmp;
	if ((phi2 <= -6.1) || !(phi2 <= 1e-24)) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2(t_1, (t_0 - (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)
    if ((phi2 <= (-6.1d0)) .or. (.not. (phi2 <= 1d-24))) then
        tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
    else
        tmp = atan2(t_1, (t_0 - (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2);
	double tmp;
	if ((phi2 <= -6.1) || !(phi2 <= 1e-24)) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.cos((lambda1 - lambda2))))));
	} else {
		tmp = Math.atan2(t_1, (t_0 - (Math.sin(phi1) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = ((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)
	tmp = 0
	if (phi2 <= -6.1) or not (phi2 <= 1e-24):
		tmp = math.atan2(t_1, (t_0 - (math.cos(phi2) * (math.sin(phi1) * math.cos((lambda1 - lambda2))))))
	else:
		tmp = math.atan2(t_1, (t_0 - (math.sin(phi1) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2))
	tmp = 0.0
	if ((phi2 <= -6.1) || !(phi2 <= 1e-24))
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(sin(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2);
	tmp = 0.0;
	if ((phi2 <= -6.1) || ~((phi2 <= 1e-24)))
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	else
		tmp = atan2(t_1, (t_0 - (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -6.1], N[Not[LessEqual[phi2, 1e-24]], $MachinePrecision]], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -6.0999999999999996 or 9.99999999999999924e-25 < phi2

   1. Initial program 74.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 74.5%

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

      2. associate-*l* 74.5%

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

   3. Simplified 74.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 88.3%

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

      2. sub-neg 88.3%

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

   6. Applied egg-rr 88.3%

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

      1. sub-neg 88.3%

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

   8. Simplified 88.3%

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

   if -6.0999999999999996 < phi2 < 9.99999999999999924e-25

   1. Initial program 77.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 77.7%

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

      2. associate-*l* 77.7%

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

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 84.6%

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

      2. sub-neg 84.6%

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

   6. Applied egg-rr 84.6%

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

      1. sub-neg 84.6%

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

   8. Simplified 84.6%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 84.6%

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

      2. cos-diff 99.9%

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

      3. distribute-lft-in 99.9%

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

      4. *-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. distribute-lft-out 99.9%

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

       2. *-commutative 99.9%

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

       3. +-commutative 99.9%

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

       4. fma-define 99.9%

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

       5. *-commutative 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in phi2 around 0 99.9%

       \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.1 \lor \neg \left(\phi_2 \leq 10^{-24}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (cos phi2)
   (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
  (-
   (* (cos phi1) (sin phi2))
   (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sin-diff 86.5%

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

   2. cancel-sign-sub-inv 86.5%

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

   3. fma-define 86.5%

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

6. Applied egg-rr 86.5%

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

7. Final simplification 86.5%

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

Alternative 8: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -4.4 \cdot 10^{-8} \lor \neg \left(\lambda_1 \leq 0.000125\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda1 -4.4e-8) (not (<= lambda1 0.000125)))
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
       (cos phi2))
      (- t_0 (* (sin phi1) (* (cos lambda1) (cos phi2)))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (-
       t_0
       (*
        (cos phi2)
        (*
         (sin phi1)
         (+
          (* (cos lambda2) (cos lambda1))
          (* (sin lambda1) (sin lambda2))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda1 <= -4.4e-8) || !(lambda1 <= 0.000125)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * (cos(lambda1) * cos(phi2)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda1 <= (-4.4d-8)) .or. (.not. (lambda1 <= 0.000125d0))) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * (cos(lambda1) * cos(phi2)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda1 <= -4.4e-8) || !(lambda1 <= 0.000125)) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (t_0 - (Math.sin(phi1) * (Math.cos(lambda1) * Math.cos(phi2)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2)))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda1 <= -4.4e-8) or not (lambda1 <= 0.000125):
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (t_0 - (math.sin(phi1) * (math.cos(lambda1) * math.cos(phi2)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (math.cos(phi2) * (math.sin(phi1) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2)))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda1 <= -4.4e-8) || !(lambda1 <= 0.000125))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * Float64(cos(lambda1) * cos(phi2)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda1 <= -4.4e-8) || ~((lambda1 <= 0.000125)))
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * (cos(lambda1) * cos(phi2)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -4.4e-8], N[Not[LessEqual[lambda1, 0.000125]], $MachinePrecision]], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.3999999999999997e-8 or 1.25e-4 < lambda1

   1. Initial program 56.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 56.0%

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

      2. associate-*l* 56.0%

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

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 75.6%

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

      2. sub-neg 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. sub-neg 75.6%

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

   8. Simplified 75.6%

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

   9. Taylor expanded in lambda2 around 0 75.2%

      \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 75.2%

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

   11. Simplified 75.2%

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

   if -4.3999999999999997e-8 < lambda1 < 1.25e-4

   1. Initial program 98.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 98.8%

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

      2. associate-*l* 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

   6. Applied egg-rr 98.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.4 \cdot 10^{-8} \lor \neg \left(\lambda_1 \leq 0.000125\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.8 \cdot 10^{-6} \lor \neg \left(\lambda_1 \leq 0.0009\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(1 + \left(\cos \left(\lambda_1 + \lambda_2\right) + -1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda1 -3.8e-6) (not (<= lambda1 0.0009)))
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
       (cos phi2))
      (- t_0 (* (sin phi1) (* (cos lambda1) (cos phi2)))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (-
       t_0
       (*
        (cos phi2)
        (* (sin phi1) (+ 1.0 (+ (cos (+ lambda1 lambda2)) -1.0)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda1 <= -3.8e-6) || !(lambda1 <= 0.0009)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * (cos(lambda1) * cos(phi2)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * (1.0 + (cos((lambda1 + lambda2)) + -1.0))))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda1 <= (-3.8d-6)) .or. (.not. (lambda1 <= 0.0009d0))) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * (cos(lambda1) * cos(phi2)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * (1.0d0 + (cos((lambda1 + lambda2)) + (-1.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda1 <= -3.8e-6) || !(lambda1 <= 0.0009)) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (t_0 - (Math.sin(phi1) * (Math.cos(lambda1) * Math.cos(phi2)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * (1.0 + (Math.cos((lambda1 + lambda2)) + -1.0))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda1 <= -3.8e-6) or not (lambda1 <= 0.0009):
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (t_0 - (math.sin(phi1) * (math.cos(lambda1) * math.cos(phi2)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (math.cos(phi2) * (math.sin(phi1) * (1.0 + (math.cos((lambda1 + lambda2)) + -1.0))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda1 <= -3.8e-6) || !(lambda1 <= 0.0009))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * Float64(cos(lambda1) * cos(phi2)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * Float64(1.0 + Float64(cos(Float64(lambda1 + lambda2)) + -1.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda1 <= -3.8e-6) || ~((lambda1 <= 0.0009)))
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * (cos(lambda1) * cos(phi2)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * (1.0 + (cos((lambda1 + lambda2)) + -1.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -3.8e-6], N[Not[LessEqual[lambda1, 0.0009]], $MachinePrecision]], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(1.0 + N[(N[Cos[N[(lambda1 + lambda2), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.8e-6 or 8.9999999999999998e-4 < lambda1

   1. Initial program 56.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 56.0%

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

      2. associate-*l* 56.0%

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

   3. Simplified 56.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 75.6%

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

      2. sub-neg 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. sub-neg 75.6%

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

   8. Simplified 75.6%

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

   9. Taylor expanded in lambda2 around 0 75.2%

      \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 75.2%

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

   11. Simplified 75.2%

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

   if -3.8e-6 < lambda1 < 8.9999999999999998e-4

   1. Initial program 98.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 98.8%

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

      2. associate-*l* 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

   6. Applied egg-rr 98.9%

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

      1. +-commutative 98.9%

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

      2. *-commutative 98.9%

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

      3. cos-diff 98.8%

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

      4. expm1-log1p-u 98.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

      5. expm1-undefine 98.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}\right)} \]

   8. Applied egg-rr 98.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}\right)} \]
   9. Step-by-step derivation

      1. expm1-define 98.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

      2. cos-neg 98.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right)\right)\right)} \]

      3. sub-neg 98.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)\right)\right)} \]

      4. +-commutative 98.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right)\right)\right)} \]

      5. distribute-neg-in 98.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right)\right)\right)} \]

      6. remove-double-neg 98.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right)\right)\right)} \]

      7. sub-neg 98.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right)\right)} \]

   10. Simplified 98.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)} \]
   11. Step-by-step derivation

       1. expm1-undefine 98.8%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)} - 1\right)}\right)} \]

   12. Applied egg-rr 98.8%

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

       1. associate--l+ 98.8%

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

       2. +-commutative 98.8%

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

   14. Simplified 98.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(1 + \left(\cos \left(\lambda_1 + \lambda_2\right) - 1\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.8 \cdot 10^{-6} \lor \neg \left(\lambda_1 \leq 0.0009\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(1 + \left(\cos \left(\lambda_1 + \lambda_2\right) + -1\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
   (cos phi2))
  (-
   (* (cos phi1) (sin phi2))
   (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
}

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 = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sin-diff 86.5%

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

   2. sub-neg 86.5%

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

6. Applied egg-rr 86.5%

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

   1. sub-neg 86.5%

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

8. Simplified 86.5%

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

Alternative 11: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(1 + \left(\cos \left(\lambda_1 + \lambda_2\right) + -1\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 10^{-51}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t\_0 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -1.85e-5)
     (atan2
      t_1
      (-
       t_0
       (*
        (cos phi2)
        (* (sin phi1) (+ 1.0 (+ (cos (+ lambda1 lambda2)) -1.0))))))
     (if (<= phi1 1e-51)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (- t_0 (* (cos (- lambda2 lambda1)) (* (cos phi2) phi1))))
       (atan2
        t_1
        (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.85e-5) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * (1.0 + (cos((lambda1 + lambda2)) + -1.0))))));
	} else if (phi1 <= 1e-51) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos((lambda2 - lambda1)) * (cos(phi2) * phi1))));
	} else {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= (-1.85d-5)) then
        tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * (1.0d0 + (cos((lambda1 + lambda2)) + (-1.0d0)))))))
    else if (phi1 <= 1d-51) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos((lambda2 - lambda1)) * (cos(phi2) * phi1))))
    else
        tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.85e-5) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * (1.0 + (Math.cos((lambda1 + lambda2)) + -1.0))))));
	} else if (phi1 <= 1e-51) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (t_0 - (Math.cos((lambda2 - lambda1)) * (Math.cos(phi2) * phi1))));
	} else {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -1.85e-5:
		tmp = math.atan2(t_1, (t_0 - (math.cos(phi2) * (math.sin(phi1) * (1.0 + (math.cos((lambda1 + lambda2)) + -1.0))))))
	elif phi1 <= 1e-51:
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (t_0 - (math.cos((lambda2 - lambda1)) * (math.cos(phi2) * phi1))))
	else:
		tmp = math.atan2(t_1, (t_0 - (math.cos(phi2) * (math.sin(phi1) * math.cos((lambda1 - lambda2))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -1.85e-5)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * Float64(1.0 + Float64(cos(Float64(lambda1 + lambda2)) + -1.0))))));
	elseif (phi1 <= 1e-51)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(cos(Float64(lambda2 - lambda1)) * Float64(cos(phi2) * phi1))));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -1.85e-5)
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * (1.0 + (cos((lambda1 + lambda2)) + -1.0))))));
	elseif (phi1 <= 1e-51)
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos((lambda2 - lambda1)) * (cos(phi2) * phi1))));
	else
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.85e-5], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(1.0 + N[(N[Cos[N[(lambda1 + lambda2), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1e-51], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.84999999999999991e-5

   1. Initial program 70.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 70.8%

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

      2. associate-*l* 70.8%

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

   3. Simplified 70.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

   6. Applied egg-rr 71.4%

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

      1. +-commutative 71.4%

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

      2. *-commutative 71.4%

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

      3. cos-diff 70.8%

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

      4. expm1-log1p-u 70.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

      5. expm1-undefine 70.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}\right)} \]

   8. Applied egg-rr 70.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}\right)} \]
   9. Step-by-step derivation

      1. expm1-define 70.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

      2. cos-neg 70.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right)\right)\right)} \]

      3. sub-neg 70.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)\right)\right)} \]

      4. +-commutative 70.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right)\right)\right)} \]

      5. distribute-neg-in 70.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right)\right)\right)} \]

      6. remove-double-neg 70.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right)\right)\right)} \]

      7. sub-neg 70.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right)\right)} \]

   10. Simplified 70.9%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)} \]
   11. Step-by-step derivation

       1. expm1-undefine 70.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)} - 1\right)}\right)} \]

   12. Applied egg-rr 70.8%

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

       1. associate--l+ 70.9%

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

       2. +-commutative 70.9%

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

   14. Simplified 70.9%

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

   if -1.84999999999999991e-5 < phi1 < 1e-51

   1. Initial program 79.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 79.6%

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

      2. associate-*l* 79.6%

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

   3. Simplified 79.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 99.7%

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

      2. sub-neg 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. sub-neg 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in phi1 around 0 99.7%

      \[\leadsto \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   10. Step-by-step derivation

       1. associate-*r* 99.7%

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

       2. *-commutative 99.7%

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

       3. sub-neg 99.7%

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

       4. remove-double-neg 99.7%

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

       5. mul-1-neg 99.7%

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

       6. distribute-neg-in 99.7%

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

       7. +-commutative 99.7%

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

       8. cos-neg 99.7%

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

       9. mul-1-neg 99.7%

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

       10. unsub-neg 99.7%

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

   11. Simplified 99.7%

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

   if 1e-51 < phi1

   1. Initial program 75.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 75.2%

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

      2. associate-*l* 75.2%

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

   3. Simplified 75.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(1 + \left(\cos \left(\lambda_1 + \lambda_2\right) + -1\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 10^{-51}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -6.1:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \phi_2 \cdot \left(e^{\mathsf{log1p}\left(t\_1\right)} + -1\right)}\\ \mathbf{elif}\;\phi_2 \leq 2.8 \cdot 10^{-24}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{t\_0 - \cos \phi_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos (- lambda1 lambda2))))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi2 -6.1)
     (atan2 t_2 (- t_0 (* (cos phi2) (+ (exp (log1p t_1)) -1.0))))
     (if (<= phi2 2.8e-24)
       (atan2
        (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
        (- t_0 (* (cos phi2) t_1)))
       (atan2
        t_2
        (-
         t_0
         (*
          (cos phi2)
          (* (sin phi1) (expm1 (log1p (cos (- lambda2 lambda1))))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(phi1) * cos((lambda1 - lambda2));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -6.1) {
		tmp = atan2(t_2, (t_0 - (cos(phi2) * (exp(log1p(t_1)) + -1.0))));
	} else if (phi2 <= 2.8e-24) {
		tmp = atan2(((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))), (t_0 - (cos(phi2) * t_1)));
	} else {
		tmp = atan2(t_2, (t_0 - (cos(phi2) * (sin(phi1) * expm1(log1p(cos((lambda2 - lambda1))))))));
	}
	return tmp;
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin(phi1) * Math.cos((lambda1 - lambda2));
	double t_2 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -6.1) {
		tmp = Math.atan2(t_2, (t_0 - (Math.cos(phi2) * (Math.exp(Math.log1p(t_1)) + -1.0))));
	} else if (phi2 <= 2.8e-24) {
		tmp = Math.atan2(((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))), (t_0 - (Math.cos(phi2) * t_1)));
	} else {
		tmp = Math.atan2(t_2, (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.expm1(Math.log1p(Math.cos((lambda2 - lambda1))))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.sin(phi1) * math.cos((lambda1 - lambda2))
	t_2 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= -6.1:
		tmp = math.atan2(t_2, (t_0 - (math.cos(phi2) * (math.exp(math.log1p(t_1)) + -1.0))))
	elif phi2 <= 2.8e-24:
		tmp = math.atan2(((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))), (t_0 - (math.cos(phi2) * t_1)))
	else:
		tmp = math.atan2(t_2, (t_0 - (math.cos(phi2) * (math.sin(phi1) * math.expm1(math.log1p(math.cos((lambda2 - lambda1))))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -6.1)
		tmp = atan(t_2, Float64(t_0 - Float64(cos(phi2) * Float64(exp(log1p(t_1)) + -1.0))));
	elseif (phi2 <= 2.8e-24)
		tmp = atan(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))), Float64(t_0 - Float64(cos(phi2) * t_1)));
	else
		tmp = atan(t_2, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * expm1(log1p(cos(Float64(lambda2 - lambda1))))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -6.1], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Exp[N[Log[1 + t$95$1], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 2.8e-24], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(Exp[N[Log[1 + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.0999999999999996

   1. Initial program 63.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 63.7%

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

      2. associate-*l* 63.7%

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

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 63.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      2. expm1-undefine 63.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}} \]

      3. *-commutative 63.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\right)} - 1\right)} \]

   6. Applied egg-rr 63.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)} - 1\right)}} \]

   if -6.0999999999999996 < phi2 < 2.8000000000000002e-24

   1. Initial program 77.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 77.7%

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

      2. associate-*l* 77.7%

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

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sin-diff 84.6%

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

      2. cancel-sign-sub-inv 84.6%

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

      3. fma-define 84.6%

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

   6. Applied egg-rr 84.6%

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

   7. Taylor expanded in phi2 around 0 84.1%

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

      1. mul-1-neg 84.1%

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

      2. +-commutative 84.1%

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

      3. sub-neg 84.1%

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

      4. *-commutative 84.1%

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

   9. Simplified 84.1%

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

   if 2.8000000000000002e-24 < phi2

   1. Initial program 81.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 81.5%

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

      2. associate-*l* 81.5%

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

   3. Simplified 81.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

   6. Applied egg-rr 81.6%

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

      1. +-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. cos-diff 81.5%

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

      4. expm1-log1p-u 81.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

      5. expm1-undefine 81.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}\right)} \]

   8. Applied egg-rr 81.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}\right)} \]
   9. Step-by-step derivation

      1. expm1-define 81.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

      2. cos-neg 81.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right)\right)\right)} \]

      3. sub-neg 81.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)\right)\right)} \]

      4. +-commutative 81.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right)\right)\right)} \]

      5. distribute-neg-in 81.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right)\right)\right)} \]

      6. remove-double-neg 81.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right)\right)\right)} \]

      7. sub-neg 81.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right)\right)} \]

   10. Simplified 81.5%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.1:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(e^{\mathsf{log1p}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} + -1\right)}\\ \mathbf{elif}\;\phi_2 \leq 2.8 \cdot 10^{-24}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_1 - \cos \phi_2 \cdot t\_0}\\ t_3 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_1 - t\_0}\\ \mathbf{if}\;\lambda_1 \leq -0.118:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_1 \leq -9.5 \cdot 10^{-161}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\lambda_1 \leq 1.12 \cdot 10^{-284}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t\_1 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_1 \leq 3.4 \cdot 10^{-26}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (cos (- lambda1 lambda2))))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (atan2 (* (sin lambda1) (cos phi2)) (- t_1 (* (cos phi2) t_0))))
        (t_3 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (- t_1 t_0))))
   (if (<= lambda1 -0.118)
     t_2
     (if (<= lambda1 -9.5e-161)
       t_3
       (if (<= lambda1 1.12e-284)
         (atan2
          (* (cos phi2) (sin (- lambda2)))
          (- t_1 (* (cos phi2) (* (cos lambda2) (sin phi1)))))
         (if (<= lambda1 3.4e-26) t_3 t_2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = atan2((sin(lambda1) * cos(phi2)), (t_1 - (cos(phi2) * t_0)));
	double t_3 = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_1 - t_0));
	double tmp;
	if (lambda1 <= -0.118) {
		tmp = t_2;
	} else if (lambda1 <= -9.5e-161) {
		tmp = t_3;
	} else if (lambda1 <= 1.12e-284) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_1 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	} else if (lambda1 <= 3.4e-26) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin(phi1) * cos((lambda1 - lambda2))
    t_1 = cos(phi1) * sin(phi2)
    t_2 = atan2((sin(lambda1) * cos(phi2)), (t_1 - (cos(phi2) * t_0)))
    t_3 = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_1 - t_0))
    if (lambda1 <= (-0.118d0)) then
        tmp = t_2
    else if (lambda1 <= (-9.5d-161)) then
        tmp = t_3
    else if (lambda1 <= 1.12d-284) then
        tmp = atan2((cos(phi2) * sin(-lambda2)), (t_1 - (cos(phi2) * (cos(lambda2) * sin(phi1)))))
    else if (lambda1 <= 3.4d-26) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double t_2 = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_1 - (Math.cos(phi2) * t_0)));
	double t_3 = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_1 - t_0));
	double tmp;
	if (lambda1 <= -0.118) {
		tmp = t_2;
	} else if (lambda1 <= -9.5e-161) {
		tmp = t_3;
	} else if (lambda1 <= 1.12e-284) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (t_1 - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)))));
	} else if (lambda1 <= 3.4e-26) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi1) * math.sin(phi2)
	t_2 = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_1 - (math.cos(phi2) * t_0)))
	t_3 = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_1 - t_0))
	tmp = 0
	if lambda1 <= -0.118:
		tmp = t_2
	elif lambda1 <= -9.5e-161:
		tmp = t_3
	elif lambda1 <= 1.12e-284:
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), (t_1 - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))))
	elif lambda1 <= 3.4e-26:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_1 - Float64(cos(phi2) * t_0)))
	t_3 = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_1 - t_0))
	tmp = 0.0
	if (lambda1 <= -0.118)
		tmp = t_2;
	elseif (lambda1 <= -9.5e-161)
		tmp = t_3;
	elseif (lambda1 <= 1.12e-284)
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_1 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	elseif (lambda1 <= 3.4e-26)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * cos((lambda1 - lambda2));
	t_1 = cos(phi1) * sin(phi2);
	t_2 = atan2((sin(lambda1) * cos(phi2)), (t_1 - (cos(phi2) * t_0)));
	t_3 = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_1 - t_0));
	tmp = 0.0;
	if (lambda1 <= -0.118)
		tmp = t_2;
	elseif (lambda1 <= -9.5e-161)
		tmp = t_3;
	elseif (lambda1 <= 1.12e-284)
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_1 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	elseif (lambda1 <= 3.4e-26)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda1, -0.118], t$95$2, If[LessEqual[lambda1, -9.5e-161], t$95$3, If[LessEqual[lambda1, 1.12e-284], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 3.4e-26], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.11799999999999999 or 3.40000000000000013e-26 < lambda1

   1. Initial program 57.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 57.3%

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

      2. associate-*l* 57.3%

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

   3. Simplified 57.3%

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

   5. Taylor expanded in lambda2 around 0 60.1%

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

   if -0.11799999999999999 < lambda1 < -9.4999999999999996e-161 or 1.11999999999999996e-284 < lambda1 < 3.40000000000000013e-26

   1. Initial program 99.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. log1p-expm1-u 99.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      2. *-commutative 99.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\right)\right)} \]

   6. Applied egg-rr 99.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)\right)}} \]

   7. Taylor expanded in phi2 around 0 87.9%

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

   if -9.4999999999999996e-161 < lambda1 < 1.11999999999999996e-284

   1. Initial program 99.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in lambda1 around 0 99.8%

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

   6. Taylor expanded in lambda1 around 0 99.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 99.8%

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

      2. cos-neg 99.8%

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

      3. *-commutative 99.8%

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

   8. Simplified 99.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.118:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\lambda_1 \leq -9.5 \cdot 10^{-161}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_1 \leq 1.12 \cdot 10^{-284}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_1 \leq 3.4 \cdot 10^{-26}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\lambda_1 \leq -0.0255:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_1}\\ \mathbf{elif}\;\lambda_1 \leq 0.04:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 + \lambda_2\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2)))))))
   (if (<= lambda1 -0.0255)
     (atan2 (* (sin lambda1) (cos phi2)) t_1)
     (if (<= lambda1 0.04)
       (atan2
        (* (cos phi2) (sin (- lambda1 lambda2)))
        (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1)))))
       (atan2 (* (cos phi2) (sin (+ lambda1 lambda2))) t_1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))));
	double tmp;
	if (lambda1 <= -0.0255) {
		tmp = atan2((sin(lambda1) * cos(phi2)), t_1);
	} else if (lambda1 <= 0.04) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 + lambda2))), t_1);
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))
    if (lambda1 <= (-0.0255d0)) then
        tmp = atan2((sin(lambda1) * cos(phi2)), t_1)
    else if (lambda1 <= 0.04d0) then
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 + lambda2))), t_1)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.cos((lambda1 - lambda2))));
	double tmp;
	if (lambda1 <= -0.0255) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), t_1);
	} else if (lambda1 <= 0.04) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 + lambda2))), t_1);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = t_0 - (math.cos(phi2) * (math.sin(phi1) * math.cos((lambda1 - lambda2))))
	tmp = 0
	if lambda1 <= -0.0255:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), t_1)
	elif lambda1 <= 0.04:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 + lambda2))), t_1)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))))
	tmp = 0.0
	if (lambda1 <= -0.0255)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), t_1);
	elseif (lambda1 <= 0.04)
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 + lambda2))), t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))));
	tmp = 0.0;
	if (lambda1 <= -0.0255)
		tmp = atan2((sin(lambda1) * cos(phi2)), t_1);
	elseif (lambda1 <= 0.04)
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 + lambda2))), t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.0255], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision], If[LessEqual[lambda1, 0.04], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 + lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.0254999999999999984

   1. Initial program 65.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 65.7%

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

      2. associate-*l* 65.7%

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

   3. Simplified 65.7%

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

   5. Taylor expanded in lambda2 around 0 70.3%

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

   if -0.0254999999999999984 < lambda1 < 0.0400000000000000008

   1. Initial program 98.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 98.8%

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

      2. associate-*l* 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in lambda1 around 0 98.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 98.7%

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

      2. cos-neg 98.7%

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

      3. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   if 0.0400000000000000008 < lambda1

   1. Initial program 45.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 45.5%

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

      2. associate-*l* 45.5%

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

   3. Simplified 45.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 45.5%

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt[3]{\left(\sin \left(\lambda_1 - \lambda_2\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\lambda_1 - \lambda_2\right)}} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. pow3 45.5%

         \[\leadsto \tan^{-1}_* \frac{\sqrt[3]{\color{blue}{{\sin \left(\lambda_1 - \lambda_2\right)}^{3}}} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   6. Applied egg-rr 45.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt[3]{{\sin \left(\lambda_1 - \lambda_2\right)}^{3}}} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
   7. Step-by-step derivation

      1. rem-cbrt-cube 45.5%

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

      2. sin-diff 71.2%

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

      3. cancel-sign-sub-inv 71.2%

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

      4. add-sqr-sqrt 35.2%

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

      5. sqrt-unprod 63.0%

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

      6. sqr-neg 63.0%

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

      7. sqrt-unprod 27.8%

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

      8. add-sqr-sqrt 47.2%

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

      9. sin-sum 48.8%

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

   8. Applied egg-rr 48.8%

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

      1. +-commutative 48.8%

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

   10. Simplified 48.8%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_2 + \lambda_1\right)} \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.0255:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\lambda_1 \leq 0.04:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 + \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.006 \lor \neg \left(\lambda_1 \leq 0.0125\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda1 -0.006) (not (<= lambda1 0.0125)))
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda1 <= -0.006) || !(lambda1 <= 0.0125)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda1 <= (-0.006d0)) .or. (.not. (lambda1 <= 0.0125d0))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda1 <= -0.006) || !(lambda1 <= 0.0125)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.cos((lambda1 - lambda2))))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda1 <= -0.006) or not (lambda1 <= 0.0125):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - (math.cos(phi2) * (math.sin(phi1) * math.cos((lambda1 - lambda2))))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda1 <= -0.006) || !(lambda1 <= 0.0125))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda1 <= -0.006) || ~((lambda1 <= 0.0125)))
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -0.006], N[Not[LessEqual[lambda1, 0.0125]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -0.0060000000000000001 or 0.012500000000000001 < lambda1

   1. Initial program 55.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 55.6%

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

      2. associate-*l* 55.6%

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

   3. Simplified 55.6%

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

   5. Taylor expanded in lambda2 around 0 59.2%

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

   if -0.0060000000000000001 < lambda1 < 0.012500000000000001

   1. Initial program 98.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 98.8%

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

      2. associate-*l* 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in lambda1 around 0 98.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 98.7%

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

      2. cos-neg 98.7%

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

      3. *-commutative 98.7%

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

   7. Simplified 98.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\sin \phi_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.006 \lor \neg \left(\lambda_1 \leq 0.0125\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -0.0285 \lor \neg \left(\lambda_2 \leq 4.4 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda2 -0.0285) (not (<= lambda2 4.4e-5)))
     (atan2
      (* (cos phi2) (sin (- lambda2)))
      (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1)))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* (cos lambda1) (* (cos phi2) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda2 <= -0.0285) || !(lambda2 <= 4.4e-5)) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda2 <= (-0.0285d0)) .or. (.not. (lambda2 <= 4.4d-5))) then
        tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda2 <= -0.0285) || !(lambda2 <= 4.4e-5)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (t_0 - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (Math.cos(lambda1) * (Math.cos(phi2) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda2 <= -0.0285) or not (lambda2 <= 4.4e-5):
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), (t_0 - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (math.cos(lambda1) * (math.cos(phi2) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda2 <= -0.0285) || !(lambda2 <= 4.4e-5))
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda2 <= -0.0285) || ~((lambda2 <= 4.4e-5)))
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda2, -0.0285], N[Not[LessEqual[lambda2, 4.4e-5]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -0.028500000000000001 or 4.3999999999999999e-5 < lambda2

   1. Initial program 54.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 54.6%

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

      2. associate-*l* 54.6%

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

   3. Simplified 54.6%

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

   5. Taylor expanded in lambda1 around 0 55.0%

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

   6. Taylor expanded in lambda1 around 0 55.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 54.4%

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

      2. cos-neg 54.4%

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

      3. *-commutative 54.4%

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

   8. Simplified 55.0%

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

   if -0.028500000000000001 < lambda2 < 4.3999999999999999e-5

   1. Initial program 98.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 98.2%

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

      2. associate-*l* 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in lambda2 around 0 98.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \lambda_1 \cdot \left(\sin \phi_1 \cdot \cos \phi_2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.0285 \lor \neg \left(\lambda_2 \leq 4.4 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -5.5 \cdot 10^{-5} \lor \neg \left(\lambda_2 \leq 4.4 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda2 -5.5e-5) (not (<= lambda2 4.4e-5)))
     (atan2
      (* (cos phi2) (sin (- lambda2)))
      (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1)))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* (sin phi1) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda2 <= -5.5e-5) || !(lambda2 <= 4.4e-5)) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (sin(phi1) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda2 <= (-5.5d-5)) .or. (.not. (lambda2 <= 4.4d-5))) then
        tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (sin(phi1) * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda2 <= -5.5e-5) || !(lambda2 <= 4.4e-5)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (t_0 - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda2 <= -5.5e-5) or not (lambda2 <= 4.4e-5):
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), (t_0 - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda2 <= -5.5e-5) || !(lambda2 <= 4.4e-5))
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda2 <= -5.5e-5) || ~((lambda2 <= 4.4e-5)))
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (sin(phi1) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda2, -5.5e-5], N[Not[LessEqual[lambda2, 4.4e-5]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -5.5000000000000002e-5 or 4.3999999999999999e-5 < lambda2

   1. Initial program 54.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 54.5%

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

      2. associate-*l* 54.5%

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

   3. Simplified 54.5%

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

   5. Taylor expanded in lambda1 around 0 54.8%

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

   6. Taylor expanded in lambda1 around 0 54.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(-\lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\cos \left(-\lambda_2\right) \cdot \sin \phi_1\right)}} \]
   7. Step-by-step derivation

      1. *-commutative 54.3%

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

      2. cos-neg 54.3%

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

      3. *-commutative 54.3%

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

   8. Simplified 54.8%

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

   if -5.5000000000000002e-5 < lambda2 < 4.3999999999999999e-5

   1. Initial program 98.7%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 98.7%

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

      2. associate-*l* 98.7%

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

   3. Simplified 98.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. log1p-expm1-u 98.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

      2. *-commutative 98.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\right)\right)} \]

   6. Applied egg-rr 98.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)\right)}} \]

   7. Taylor expanded in phi2 around 0 84.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -5.5 \cdot 10^{-5} \lor \neg \left(\lambda_2 \leq 4.4 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (cos phi2) (sin (- lambda1 lambda2)))
  (-
   (* (cos phi1) (sin phi2))
   (* (cos phi2) (* (sin phi1) (+ 1.0 (+ (cos (+ lambda1 lambda2)) -1.0)))))))

C

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

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 = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * (1.0d0 + (cos((lambda1 + lambda2)) + (-1.0d0)))))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), ((Math.cos(phi1) * Math.sin(phi2)) - (Math.cos(phi2) * (Math.sin(phi1) * (1.0 + (Math.cos((lambda1 + lambda2)) + -1.0))))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), ((math.cos(phi1) * math.sin(phi2)) - (math.cos(phi2) * (math.sin(phi1) * (1.0 + (math.cos((lambda1 + lambda2)) + -1.0))))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(sin(phi1) * Float64(1.0 + Float64(cos(Float64(lambda1 + lambda2)) + -1.0))))))
end

MATLAB

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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(1.0 + N[(N[Cos[N[(lambda1 + lambda2), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. cos-diff 99.8%

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

   2. +-commutative 99.8%

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

   3. *-commutative 99.8%

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

6. Applied egg-rr 76.4%

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

   1. +-commutative 76.4%

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

   2. *-commutative 76.4%

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

   3. cos-diff 76.1%

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

   4. expm1-log1p-u 76.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

   5. expm1-undefine 76.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}\right)} \]

8. Applied egg-rr 76.0%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}\right)} \]
9. Step-by-step derivation

   1. expm1-define 76.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

   2. cos-neg 76.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right)\right)\right)} \]

   3. sub-neg 76.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)\right)\right)} \]

   4. +-commutative 76.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right)\right)\right)} \]

   5. distribute-neg-in 76.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right)\right)\right)} \]

   6. remove-double-neg 76.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right)\right)\right)} \]

   7. sub-neg 76.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right)\right)} \]

10. Simplified 76.1%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)} \]
11. Step-by-step derivation

    1. expm1-undefine 76.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)} - 1\right)}\right)} \]

12. Applied egg-rr 76.0%

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

    1. associate--l+ 76.1%

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

    2. +-commutative 76.1%

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

14. Simplified 76.1%

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

15. Final simplification 76.1%

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

Alternative 19: 79.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (cos phi2) (sin (- lambda1 lambda2)))
  (-
   (* (cos phi1) (sin phi2))
   (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
}

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 = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

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

5. Final simplification 76.1%

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

Alternative 20: 54.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -24000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (<= phi1 -24000.0)
     (atan2
      (sin (- lambda2))
      (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* phi1 (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -24000.0) {
		tmp = atan2(sin(-lambda2), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (phi1 * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if (phi1 <= (-24000.0d0)) then
        tmp = atan2(sin(-lambda2), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (phi1 * cos((lambda2 - lambda1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if (phi1 <= -24000.0) {
		tmp = Math.atan2(Math.sin(-lambda2), (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.cos((lambda1 - lambda2))))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (phi1 * Math.cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if phi1 <= -24000.0:
		tmp = math.atan2(math.sin(-lambda2), (t_0 - (math.cos(phi2) * (math.sin(phi1) * math.cos((lambda1 - lambda2))))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (phi1 * math.cos((lambda2 - lambda1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -24000.0)
		tmp = atan(sin(Float64(-lambda2)), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(phi1 * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi1 <= -24000.0)
		tmp = atan2(sin(-lambda2), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (phi1 * cos((lambda2 - lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -24000.0], N[ArcTan[N[Sin[(-lambda2)], $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -24000

   1. Initial program 69.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 69.8%

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

      2. associate-*l* 69.8%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in lambda1 around 0 37.4%

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

   6. Taylor expanded in phi2 around 0 26.8%

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

   if -24000 < phi1

   1. Initial program 77.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 77.9%

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

      2. associate-*l* 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in phi1 around 0 59.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   6. Step-by-step derivation

      1. sub-neg 59.7%

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

      2. neg-mul-1 59.7%

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

      3. neg-mul-1 59.7%

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

      4. remove-double-neg 59.7%

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

      5. mul-1-neg 59.7%

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

      6. distribute-neg-in 59.7%

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

      7. +-commutative 59.7%

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

      8. *-commutative 59.7%

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

      9. cos-neg 59.7%

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

      10. mul-1-neg 59.7%

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

      11. unsub-neg 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in phi2 around 0 60.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -24000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 67.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (cos phi2) (sin (- lambda1 lambda2)))
  (- (* (cos phi1) (sin phi2)) (* (sin phi1) (cos (- lambda1 lambda2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
}

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 = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. log1p-expm1-u 76.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}} \]

   2. *-commutative 76.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}\right)\right)} \]

6. Applied egg-rr 76.0%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)\right)}} \]

7. Taylor expanded in phi2 around 0 65.2%

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

8. Final simplification 65.2%

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

Alternative 22: 52.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -21:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - \left|\cos \phi_2 \cdot t\_0\right|}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_1}{\cos \phi_1 \cdot \sin \phi_2 - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* phi1 (cos (- lambda2 lambda1))))
        (t_1 (sin (- lambda1 lambda2))))
   (if (<= phi1 -21.0)
     (atan2 t_1 (- (sin phi2) (fabs (* (cos phi2) t_0))))
     (atan2 (* (cos phi2) t_1) (- (* (cos phi1) (sin phi2)) t_0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * cos((lambda2 - lambda1));
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -21.0) {
		tmp = atan2(t_1, (sin(phi2) - fabs((cos(phi2) * t_0))));
	} else {
		tmp = atan2((cos(phi2) * t_1), ((cos(phi1) * sin(phi2)) - t_0));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = phi1 * cos((lambda2 - lambda1))
    t_1 = sin((lambda1 - lambda2))
    if (phi1 <= (-21.0d0)) then
        tmp = atan2(t_1, (sin(phi2) - abs((cos(phi2) * t_0))))
    else
        tmp = atan2((cos(phi2) * t_1), ((cos(phi1) * sin(phi2)) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * Math.cos((lambda2 - lambda1));
	double t_1 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -21.0) {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - Math.abs((Math.cos(phi2) * t_0))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * t_1), ((Math.cos(phi1) * Math.sin(phi2)) - t_0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = phi1 * math.cos((lambda2 - lambda1))
	t_1 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -21.0:
		tmp = math.atan2(t_1, (math.sin(phi2) - math.fabs((math.cos(phi2) * t_0))))
	else:
		tmp = math.atan2((math.cos(phi2) * t_1), ((math.cos(phi1) * math.sin(phi2)) - t_0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(phi1 * cos(Float64(lambda2 - lambda1)))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -21.0)
		tmp = atan(t_1, Float64(sin(phi2) - abs(Float64(cos(phi2) * t_0))));
	else
		tmp = atan(Float64(cos(phi2) * t_1), Float64(Float64(cos(phi1) * sin(phi2)) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = phi1 * cos((lambda2 - lambda1));
	t_1 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -21.0)
		tmp = atan2(t_1, (sin(phi2) - abs((cos(phi2) * t_0))));
	else
		tmp = atan2((cos(phi2) * t_1), ((cos(phi1) * sin(phi2)) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -21.0], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[Abs[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -21

   1. Initial program 70.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 70.3%

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

      2. associate-*l* 70.3%

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

   3. Simplified 70.3%

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

   5. Taylor expanded in phi1 around 0 13.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   6. Step-by-step derivation

      1. sub-neg 13.7%

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

      2. neg-mul-1 13.7%

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

      3. neg-mul-1 13.7%

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

      4. remove-double-neg 13.7%

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

      5. mul-1-neg 13.7%

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

      6. distribute-neg-in 13.7%

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

      7. +-commutative 13.7%

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

      8. *-commutative 13.7%

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

      9. cos-neg 13.7%

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

      10. mul-1-neg 13.7%

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

      11. unsub-neg 13.7%

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

   7. Simplified 13.7%

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

   8. Taylor expanded in phi2 around 0 8.7%

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

   9. Taylor expanded in phi1 around 0 8.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2} - \phi_1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 6.2%

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

       2. sqrt-unprod 21.5%

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

       3. pow2 21.5%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sqrt{\color{blue}{{\left(\phi_1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)\right)}^{2}}}} \]

       4. associate-*r* 21.5%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sqrt{{\color{blue}{\left(\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2\right)}}^{2}}} \]

   11. Applied egg-rr 21.5%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \color{blue}{\sqrt{{\left(\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2\right)}^{2}}}} \]
   12. Step-by-step derivation

       1. unpow2 21.5%

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

       2. rem-sqrt-square 21.5%

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

       3. *-commutative 21.5%

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

   13. Simplified 21.5%

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

   if -21 < phi1

   1. Initial program 77.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 77.8%

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

      2. associate-*l* 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in phi1 around 0 59.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   6. Step-by-step derivation

      1. sub-neg 59.9%

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

      2. neg-mul-1 59.9%

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

      3. neg-mul-1 59.9%

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

      4. remove-double-neg 59.9%

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

      5. mul-1-neg 59.9%

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

      6. distribute-neg-in 59.9%

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

      7. +-commutative 59.9%

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

      8. *-commutative 59.9%

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

      9. cos-neg 59.9%

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

      10. mul-1-neg 59.9%

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

      11. unsub-neg 59.9%

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

   7. Simplified 59.9%

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

   8. Taylor expanded in phi2 around 0 61.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -21:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \left|\cos \phi_2 \cdot \left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right|}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 35.4% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (sin (- lambda1 lambda2))
  (- (sin phi2) (fabs (* (cos phi2) (* phi1 (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin((lambda1 - lambda2)), (sin(phi2) - fabs((cos(phi2) * (phi1 * cos((lambda2 - 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 = atan2(sin((lambda1 - lambda2)), (sin(phi2) - abs((cos(phi2) * (phi1 * cos((lambda2 - lambda1)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

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

5. Taylor expanded in phi1 around 0 49.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
6. Step-by-step derivation

   1. sub-neg 49.3%

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

   2. neg-mul-1 49.3%

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

   3. neg-mul-1 49.3%

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

   4. remove-double-neg 49.3%

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

   5. mul-1-neg 49.3%

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

   6. distribute-neg-in 49.3%

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

   7. +-commutative 49.3%

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

   8. *-commutative 49.3%

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

   9. cos-neg 49.3%

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

   10. mul-1-neg 49.3%

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

   11. unsub-neg 49.3%

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

7. Simplified 49.3%

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

8. Taylor expanded in phi2 around 0 32.6%

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

9. Taylor expanded in phi1 around 0 32.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2} - \phi_1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 24.5%

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

    2. sqrt-unprod 35.9%

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

    3. pow2 35.9%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sqrt{\color{blue}{{\left(\phi_1 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)\right)}^{2}}}} \]

    4. associate-*r* 35.9%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sqrt{{\color{blue}{\left(\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2\right)}}^{2}}} \]

11. Applied egg-rr 35.9%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \color{blue}{\sqrt{{\left(\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2\right)}^{2}}}} \]
12. Step-by-step derivation

    1. unpow2 35.9%

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

    2. rem-sqrt-square 35.9%

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

    3. *-commutative 35.9%

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

13. Simplified 35.9%

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

Alternative 24: 32.7% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (sin (- lambda1 lambda2))
  (- (sin phi2) (* phi1 (cos (- lambda2 lambda1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin((lambda1 - lambda2)), (sin(phi2) - (phi1 * cos((lambda2 - 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 = atan2(sin((lambda1 - lambda2)), (sin(phi2) - (phi1 * cos((lambda2 - lambda1)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.1%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 76.1%

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

   2. associate-*l* 76.1%

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

3. Simplified 76.1%

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

5. Taylor expanded in phi1 around 0 49.3%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
6. Step-by-step derivation

   1. sub-neg 49.3%

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

   2. neg-mul-1 49.3%

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

   3. neg-mul-1 49.3%

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

   4. remove-double-neg 49.3%

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

   5. mul-1-neg 49.3%

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

   6. distribute-neg-in 49.3%

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

   7. +-commutative 49.3%

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

   8. *-commutative 49.3%

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

   9. cos-neg 49.3%

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

   10. mul-1-neg 49.3%

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

   11. unsub-neg 49.3%

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

7. Simplified 49.3%

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

8. Taylor expanded in phi2 around 0 32.6%

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

9. Taylor expanded in phi1 around 0 32.6%

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

10. Taylor expanded in phi2 around 0 32.7%

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

Reproduce

herbie shell --seed 2024094 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Bearing on a great circle"
  :precision binary64
  (atan2 (* (sin (- lambda1 lambda2)) (cos phi2)) (- (* (cos phi1) (sin phi2)) (* (* (sin phi1) (cos phi2)) (cos (- lambda1 lambda2))))))