Bearing on a great circle

Percentage Accurate: 79.5% → 99.7%

Time: 38.0s

Alternatives: 32

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: 79.5% 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

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\sin \lambda_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \lambda_1\right)\right) + \cos \lambda_2 \cdot \cos \lambda_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (cos phi2)
   (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2))))
  (-
   (* (cos phi1) (sin phi2))
   (*
    (* (cos phi2) (sin phi1))
    (+
     (* (sin lambda2) (expm1 (log1p (sin lambda1))))
     (* (cos lambda2) (cos lambda1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * ((sin(lambda2) * expm1(log1p(sin(lambda1)))) + (cos(lambda2) * cos(lambda1))))));
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2)))), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos(phi2) * Math.sin(phi1)) * ((Math.sin(lambda2) * Math.expm1(Math.log1p(Math.sin(lambda1)))) + (Math.cos(lambda2) * Math.cos(lambda1))))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2)))), ((math.cos(phi1) * math.sin(phi2)) - ((math.cos(phi2) * math.sin(phi1)) * ((math.sin(lambda2) * math.expm1(math.log1p(math.sin(lambda1)))) + (math.cos(lambda2) * math.cos(lambda1))))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2)))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * Float64(Float64(sin(lambda2) * expm1(log1p(sin(lambda1)))) + Float64(cos(lambda2) * cos(lambda1))))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[(Exp[N[Log[1 + N[Sin[lambda1], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.7%

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

   1. sin-diff 89.7%

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

   2. flip-- 87.5%

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

4. Applied egg-rr 87.5%

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

   1. difference-of-squares 87.5%

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

   2. sub-neg 87.5%

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

   3. associate-/l* 89.7%

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

   4. cos-neg 89.7%

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

   5. *-commutative 89.7%

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

   6. fma-define 89.7%

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

   7. cos-neg 89.7%

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

6. Simplified 89.7%

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

   1. cos-diff 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in lambda2 around inf 99.7%

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

    1. *-commutative 99.7%

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

11. Simplified 99.7%

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

    1. expm1-log1p-u 99.7%

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

    2. expm1-undefine 99.7%

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

13. Applied egg-rr 99.7%

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

    1. expm1-define 99.7%

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

15. Simplified 99.7%

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

16. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (cos phi2)
   (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2))))
  (-
   (* (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((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), ((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((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), ((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.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2)))), ((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.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2)))), ((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(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2)))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), ((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[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $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. Initial program 79.7%

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

   1. sin-diff 89.7%

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

   2. flip-- 87.5%

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

4. Applied egg-rr 87.5%

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

   1. difference-of-squares 87.5%

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

   2. sub-neg 87.5%

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

   3. associate-/l* 89.7%

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

   4. cos-neg 89.7%

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

   5. *-commutative 89.7%

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

   6. fma-define 89.7%

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

   7. cos-neg 89.7%

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

6. Simplified 89.7%

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

   1. cos-diff 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

8. Applied egg-rr 99.7%

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

9. Taylor expanded in lambda2 around inf 99.7%

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

    1. *-commutative 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 3: 94.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\\ t_3 := \cos \phi_2 \cdot t\_2\\ t_4 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -6 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{t\_0 - t\_1 \cdot t\_4}\\ \mathbf{elif}\;\phi_2 \leq 2.75 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - t\_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_3}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot {\left(\sqrt[3]{t\_4}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin phi1)))
        (t_2
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2))))
        (t_3 (* (cos phi2) t_2))
        (t_4 (cos (- lambda1 lambda2))))
   (if (<= phi2 -6e-6)
     (atan2 t_3 (- t_0 (* t_1 t_4)))
     (if (<= phi2 2.75e-9)
       (atan2
        t_2
        (-
         t_0
         (*
          t_1
          (+
           (* (cos lambda2) (cos lambda1))
           (* (sin lambda1) (sin lambda2))))))
       (atan2
        t_3
        (- t_0 (* (cos phi2) (* (sin phi1) (pow (cbrt t_4) 3.0)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin(phi1);
	double t_2 = (sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2));
	double t_3 = cos(phi2) * t_2;
	double t_4 = cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -6e-6) {
		tmp = atan2(t_3, (t_0 - (t_1 * t_4)));
	} else if (phi2 <= 2.75e-9) {
		tmp = atan2(t_2, (t_0 - (t_1 * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2(t_3, (t_0 - (cos(phi2) * (sin(phi1) * pow(cbrt(t_4), 3.0)))));
	}
	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.cos(phi2) * Math.sin(phi1);
	double t_2 = (Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2));
	double t_3 = Math.cos(phi2) * t_2;
	double t_4 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -6e-6) {
		tmp = Math.atan2(t_3, (t_0 - (t_1 * t_4)));
	} else if (phi2 <= 2.75e-9) {
		tmp = Math.atan2(t_2, (t_0 - (t_1 * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2))))));
	} else {
		tmp = Math.atan2(t_3, (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.pow(Math.cbrt(t_4), 3.0)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(phi1))
	t_2 = Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2)))
	t_3 = Float64(cos(phi2) * t_2)
	t_4 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= -6e-6)
		tmp = atan(t_3, Float64(t_0 - Float64(t_1 * t_4)));
	elseif (phi2 <= 2.75e-9)
		tmp = atan(t_2, Float64(t_0 - Float64(t_1 * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(t_3, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * (cbrt(t_4) ^ 3.0)))));
	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[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -6e-6], N[ArcTan[t$95$3 / N[(t$95$0 - N[(t$95$1 * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 2.75e-9], N[ArcTan[t$95$2 / N[(t$95$0 - N[(t$95$1 * 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], N[ArcTan[t$95$3 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Power[N[Power[t$95$4, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.0000000000000002e-6

   1. Initial program 77.4%

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

      1. sin-diff 91.3%

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

   4. Applied egg-rr 91.3%

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

   if -6.0000000000000002e-6 < phi2 < 2.7499999999999998e-9

   1. Initial program 80.9%

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

      1. sin-diff 88.3%

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

      2. flip-- 86.1%

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

   4. Applied egg-rr 86.1%

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

      1. difference-of-squares 86.1%

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

      2. sub-neg 86.1%

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

      3. associate-/l* 88.3%

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

      4. cos-neg 88.3%

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

      5. *-commutative 88.3%

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

      6. fma-define 88.3%

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

      7. cos-neg 88.3%

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

   6. Simplified 88.3%

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

      1. cos-diff 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in phi2 around 0 99.9%

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

       1. *-commutative 99.9%

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

   11. Simplified 99.9%

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

   if 2.7499999999999998e-9 < phi2

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-*l* 79.8%

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

   3. Simplified 79.8%

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

      1. sin-diff 90.8%

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

   6. Applied egg-rr 90.8%

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

      1. add-cube-cbrt 90.9%

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

      2. pow3 90.8%

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

   8. Applied egg-rr 90.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 2.75 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot {\left(\sqrt[3]{\cos \left(\lambda_1 - \lambda_2\right)}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ t_3 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -7200000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - t\_2 \cdot t\_3}\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - t\_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot {\left(\sqrt[3]{t\_3}\right)}^{3}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (*
          (cos phi2)
          (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))))
        (t_2 (* (cos phi2) (sin phi1)))
        (t_3 (cos (- lambda1 lambda2))))
   (if (<= phi2 -7200000000.0)
     (atan2 t_1 (- t_0 (* t_2 t_3)))
     (if (<= phi2 2.2e-12)
       (atan2
        t_1
        (-
         (sin phi2)
         (*
          t_2
          (+
           (* (cos lambda2) (cos lambda1))
           (* (sin lambda1) (sin lambda2))))))
       (atan2
        t_1
        (- t_0 (* (cos phi2) (* (sin phi1) (pow (cbrt t_3) 3.0)))))))))

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) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)));
	double t_2 = cos(phi2) * sin(phi1);
	double t_3 = cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -7200000000.0) {
		tmp = atan2(t_1, (t_0 - (t_2 * t_3)));
	} else if (phi2 <= 2.2e-12) {
		tmp = atan2(t_1, (sin(phi2) - (t_2 * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * pow(cbrt(t_3), 3.0)))));
	}
	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.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2)));
	double t_2 = Math.cos(phi2) * Math.sin(phi1);
	double t_3 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -7200000000.0) {
		tmp = Math.atan2(t_1, (t_0 - (t_2 * t_3)));
	} else if (phi2 <= 2.2e-12) {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - (t_2 * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2))))));
	} else {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.pow(Math.cbrt(t_3), 3.0)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))))
	t_2 = Float64(cos(phi2) * sin(phi1))
	t_3 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= -7200000000.0)
		tmp = atan(t_1, Float64(t_0 - Float64(t_2 * t_3)));
	elseif (phi2 <= 2.2e-12)
		tmp = atan(t_1, Float64(sin(phi2) - Float64(t_2 * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * (cbrt(t_3) ^ 3.0)))));
	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[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -7200000000.0], N[ArcTan[t$95$1 / N[(t$95$0 - N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 2.2e-12], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(t$95$2 * 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], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Power[N[Power[t$95$3, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -7.2e9

   1. Initial program 78.2%

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

      1. sin-diff 91.1%

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

   4. Applied egg-rr 91.1%

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

   if -7.2e9 < phi2 < 2.19999999999999992e-12

   1. Initial program 80.4%

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

      1. sin-diff 88.5%

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

      2. flip-- 86.3%

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

   4. Applied egg-rr 86.3%

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

      1. difference-of-squares 86.3%

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

      2. sub-neg 86.3%

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

      3. associate-/l* 88.5%

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

      4. cos-neg 88.5%

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

      5. *-commutative 88.5%

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

      6. fma-define 88.5%

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

      7. cos-neg 88.5%

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

   6. Simplified 88.5%

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

      1. cos-diff 99.9%

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

      2. +-commutative 99.9%

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

      3. *-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in lambda2 around inf 99.9%

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

       1. *-commutative 99.9%

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

   11. Simplified 99.9%

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

   12. Taylor expanded in phi1 around 0 99.8%

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

   if 2.19999999999999992e-12 < phi2

   1. Initial program 79.8%

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

      1. *-commutative 79.8%

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

      2. associate-*l* 79.8%

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

   3. Simplified 79.8%

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

      1. sin-diff 90.8%

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

   6. Applied egg-rr 90.8%

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

      1. add-cube-cbrt 90.9%

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

      2. pow3 90.8%

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

   8. Applied egg-rr 90.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -7200000000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-12}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot {\left(\sqrt[3]{\cos \left(\lambda_1 - \lambda_2\right)}\right)}^{3}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{+16} \lor \neg \left(\lambda_1 \leq 1850000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (*
          (cos phi2)
          (-
           (* (sin lambda1) (cos lambda2))
           (* (cos lambda1) (sin lambda2))))))
   (if (or (<= lambda1 -2.2e+16) (not (<= lambda1 1850000000.0)))
     (atan2 t_1 (- t_0 (* (cos lambda1) (* (cos phi2) (sin phi1)))))
     (atan2 t_1 (- 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 t_1 = cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)));
	double tmp;
	if ((lambda1 <= -2.2e+16) || !(lambda1 <= 1850000000.0)) {
		tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	} else {
		tmp = atan2(t_1, (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) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))
    if ((lambda1 <= (-2.2d+16)) .or. (.not. (lambda1 <= 1850000000.0d0))) then
        tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))))
    else
        tmp = atan2(t_1, (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 t_1 = Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2)));
	double tmp;
	if ((lambda1 <= -2.2e+16) || !(lambda1 <= 1850000000.0)) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(lambda1) * (Math.cos(phi2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2(t_1, (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)
	t_1 = math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2)))
	tmp = 0
	if (lambda1 <= -2.2e+16) or not (lambda1 <= 1850000000.0):
		tmp = math.atan2(t_1, (t_0 - (math.cos(lambda1) * (math.cos(phi2) * math.sin(phi1)))))
	else:
		tmp = math.atan2(t_1, (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))
	t_1 = Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))))
	tmp = 0.0
	if ((lambda1 <= -2.2e+16) || !(lambda1 <= 1850000000.0))
		tmp = atan(t_1, Float64(t_0 - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	else
		tmp = atan(t_1, 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);
	t_1 = cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)));
	tmp = 0.0;
	if ((lambda1 <= -2.2e+16) || ~((lambda1 <= 1850000000.0)))
		tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	else
		tmp = atan2(t_1, (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]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -2.2e+16], N[Not[LessEqual[lambda1, 1850000000.0]], $MachinePrecision]], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / 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 < -2.2e16 or 1.85e9 < lambda1

   1. Initial program 59.2%

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

      1. *-commutative 59.2%

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

      2. associate-*l* 59.2%

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

   3. Simplified 59.2%

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

      1. sin-diff 80.3%

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

   6. Applied egg-rr 80.3%

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

   7. Taylor expanded in lambda2 around 0 80.4%

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

      1. *-commutative 80.4%

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

   9. Simplified 80.4%

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

   if -2.2e16 < lambda1 < 1.85e9

   1. Initial program 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*l* 98.0%

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

   3. Simplified 98.0%

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

      1. sin-diff 98.2%

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

   6. Applied egg-rr 98.2%

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

   7. Taylor expanded in lambda1 around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. cos-neg 98.2%

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

   9. Simplified 98.2%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{+16} \lor \neg \left(\lambda_1 \leq 1850000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{+16} \lor \neg \left(\lambda_1 \leq 3.4 \cdot 10^{-65}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{t\_0 - \cos \lambda_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - t\_1 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \left(\sin \lambda_2 + -0.5 \cdot \left(\lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (* (cos phi2) (sin phi1))))
   (if (or (<= lambda1 -2.2e+16) (not (<= lambda1 3.4e-65)))
     (atan2
      (*
       (cos phi2)
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2))))
      (- t_0 (* (cos lambda1) t_1)))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (-
       t_0
       (*
        t_1
        (+
         (cos lambda2)
         (*
          lambda1
          (+ (sin lambda2) (* -0.5 (* lambda1 (cos 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(phi1);
	double tmp;
	if ((lambda1 <= -2.2e+16) || !(lambda1 <= 3.4e-65)) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (t_0 - (cos(lambda1) * t_1)));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (t_1 * (cos(lambda2) + (lambda1 * (sin(lambda2) + (-0.5 * (lambda1 * cos(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(phi1)
    if ((lambda1 <= (-2.2d+16)) .or. (.not. (lambda1 <= 3.4d-65))) then
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (t_0 - (cos(lambda1) * t_1)))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (t_1 * (cos(lambda2) + (lambda1 * (sin(lambda2) + ((-0.5d0) * (lambda1 * cos(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(phi1);
	double tmp;
	if ((lambda1 <= -2.2e+16) || !(lambda1 <= 3.4e-65)) {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2)))), (t_0 - (Math.cos(lambda1) * t_1)));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (t_1 * (Math.cos(lambda2) + (lambda1 * (Math.sin(lambda2) + (-0.5 * (lambda1 * Math.cos(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(phi1)
	tmp = 0
	if (lambda1 <= -2.2e+16) or not (lambda1 <= 3.4e-65):
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2)))), (t_0 - (math.cos(lambda1) * t_1)))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (t_1 * (math.cos(lambda2) + (lambda1 * (math.sin(lambda2) + (-0.5 * (lambda1 * math.cos(lambda2)))))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(phi1))
	tmp = 0.0
	if ((lambda1 <= -2.2e+16) || !(lambda1 <= 3.4e-65))
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2)))), Float64(t_0 - Float64(cos(lambda1) * t_1)));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(t_1 * Float64(cos(lambda2) + Float64(lambda1 * Float64(sin(lambda2) + Float64(-0.5 * Float64(lambda1 * cos(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(phi1);
	tmp = 0.0;
	if ((lambda1 <= -2.2e+16) || ~((lambda1 <= 3.4e-65)))
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (t_0 - (cos(lambda1) * t_1)));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (t_1 * (cos(lambda2) + (lambda1 * (sin(lambda2) + (-0.5 * (lambda1 * cos(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[phi1], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -2.2e+16], N[Not[LessEqual[lambda1, 3.4e-65]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[(N[Cos[lambda2], $MachinePrecision] + N[(lambda1 * N[(N[Sin[lambda2], $MachinePrecision] + N[(-0.5 * N[(lambda1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.2e16 or 3.39999999999999987e-65 < lambda1

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l* 62.9%

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

   3. Simplified 62.9%

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

      1. sin-diff 81.6%

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

   6. Applied egg-rr 81.6%

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

   7. Taylor expanded in lambda2 around 0 81.6%

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

      1. *-commutative 81.6%

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

   9. Simplified 81.6%

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

   if -2.2e16 < lambda1 < 3.39999999999999987e-65

   1. Initial program 99.2%

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

   3. Taylor expanded in lambda1 around 0 99.3%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \left(-\lambda_2\right) + \lambda_1 \cdot \left(-0.5 \cdot \left(\lambda_1 \cdot \cos \left(-\lambda_2\right)\right) - \sin \left(-\lambda_2\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. cos-neg 99.3%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \lambda_2} + \lambda_1 \cdot \left(-0.5 \cdot \left(\lambda_1 \cdot \cos \left(-\lambda_2\right)\right) - \sin \left(-\lambda_2\right)\right)\right)} \]

      2. *-commutative 99.3%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \left(-0.5 \cdot \color{blue}{\left(\cos \left(-\lambda_2\right) \cdot \lambda_1\right)} - \sin \left(-\lambda_2\right)\right)\right)} \]

      3. cos-neg 99.3%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \left(-0.5 \cdot \left(\color{blue}{\cos \lambda_2} \cdot \lambda_1\right) - \sin \left(-\lambda_2\right)\right)\right)} \]

      4. sin-neg 99.3%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \left(-0.5 \cdot \left(\cos \lambda_2 \cdot \lambda_1\right) - \color{blue}{\left(-\sin \lambda_2\right)}\right)\right)} \]

   5. Simplified 99.3%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \lambda_2 + \lambda_1 \cdot \left(-0.5 \cdot \left(\cos \lambda_2 \cdot \lambda_1\right) - \left(-\sin \lambda_2\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{+16} \lor \neg \left(\lambda_1 \leq 3.4 \cdot 10^{-65}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \left(\sin \lambda_2 + -0.5 \cdot \left(\lambda_1 \cdot \cos \lambda_2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.4% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (cos phi2)
   (- (* (sin lambda1) (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) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(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) * cos(lambda2)) - (cos(lambda1) * sin(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) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(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) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(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) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(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) * cos(lambda2)) - (cos(lambda1) * sin(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[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $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 79.7%

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

   1. *-commutative 79.7%

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

   2. associate-*l* 79.7%

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

3. Simplified 79.7%

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

   1. sin-diff 89.7%

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

6. Applied egg-rr 89.7%

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

7. Final simplification 89.7%

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

Alternative 8: 87.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.5e-8)
     (atan2
      t_0
      (-
       (* (cos phi1) (sin phi2))
       (*
        (* (cos phi2) (sin phi1))
        (expm1 (log1p (cos (- lambda2 lambda1)))))))
     (if (<= phi1 2e-27)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2))))
        (- (sin phi2) (* (cos phi2) (* (cos lambda2) (sin phi1)))))
       (atan2
        t_0
        (fma
         (sin phi2)
         (cos phi1)
         (* (sin phi1) (* (cos phi2) (- (cos (- lambda1 lambda2)))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.5e-8) {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * expm1(log1p(cos((lambda2 - lambda1)))))));
	} else if (phi1 <= 2e-27) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (sin(phi2) - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	} else {
		tmp = atan2(t_0, fma(sin(phi2), cos(phi1), (sin(phi1) * (cos(phi2) * -cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.5e-8)
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * expm1(log1p(cos(Float64(lambda2 - lambda1)))))));
	elseif (phi1 <= 2e-27)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2)))), Float64(sin(phi2) - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	else
		tmp = atan(t_0, fma(sin(phi2), cos(phi1), Float64(sin(phi1) * Float64(cos(phi2) * Float64(-cos(Float64(lambda1 - lambda2)))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.5e-8], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[(Exp[N[Log[1 + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 2e-27], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * (-N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.4999999999999999e-8

   1. Initial program 81.9%

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

      1. expm1-log1p-u 81.9%

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

      2. expm1-undefine 81.9%

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

   4. Applied egg-rr 81.9%

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

      1. expm1-define 81.9%

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

      2. sub-neg 81.9%

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

      3. neg-mul-1 81.9%

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

      4. remove-double-neg 81.9%

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

      5. mul-1-neg 81.9%

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

      6. neg-mul-1 81.9%

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

      7. distribute-neg-in 81.9%

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

      8. +-commutative 81.9%

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

      9. cos-neg 81.9%

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

      10. mul-1-neg 81.9%

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

      11. unsub-neg 81.9%

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

   6. Simplified 81.9%

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

   if -2.4999999999999999e-8 < phi1 < 2.0000000000000001e-27

   1. Initial program 81.3%

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

      1. *-commutative 81.3%

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

      2. associate-*l* 81.3%

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

   3. Simplified 81.3%

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

      1. sin-diff 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in lambda1 around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. cos-neg 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in phi1 around 0 99.8%

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

   if 2.0000000000000001e-27 < phi1

   1. Initial program 74.3%

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

      1. add-exp-log 22.2%

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

   4. Applied egg-rr 22.2%

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

      1. *-commutative 22.2%

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

      2. rem-exp-log 74.3%

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

      3. fma-neg 74.3%

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

      4. *-commutative 74.3%

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

      5. *-commutative 74.3%

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

      6. associate-*l* 74.3%

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

   6. Applied egg-rr 74.3%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 2 \cdot 10^{-27}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (cos phi2) (sin (- lambda1 lambda2)))
  (-
   (* (cos phi1) (sin phi2))
   (* (* (cos phi2) (sin phi1)) (expm1 (log1p (cos (- lambda2 lambda1))))))))

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)) * expm1(log1p(cos((lambda2 - lambda1)))))));
}

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.expm1(Math.log1p(Math.cos((lambda2 - lambda1)))))));
}

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.expm1(math.log1p(math.cos((lambda2 - lambda1)))))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * expm1(log1p(cos(Float64(lambda2 - lambda1)))))))
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[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[(Exp[N[Log[1 + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.7%

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

   1. expm1-log1p-u 79.7%

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

   2. expm1-undefine 79.7%

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

4. Applied egg-rr 79.7%

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

   1. expm1-define 79.7%

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

   2. sub-neg 79.7%

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

   3. neg-mul-1 79.7%

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

   4. remove-double-neg 79.7%

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

   5. mul-1-neg 79.7%

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

   6. neg-mul-1 79.7%

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

   7. distribute-neg-in 79.7%

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

   8. +-commutative 79.7%

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

   9. cos-neg 79.7%

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

   10. mul-1-neg 79.7%

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

   11. unsub-neg 79.7%

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

6. Simplified 79.7%

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

7. Final simplification 79.7%

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

Alternative 10: 79.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (* (cos phi2) (sin (- lambda1 lambda2)))
  (-
   (* (cos phi1) (sin phi2))
   (* (cos phi2) (expm1 (log1p (* (sin phi1) (cos (- lambda2 lambda1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * expm1(log1p((sin(phi1) * cos((lambda2 - lambda1))))))));
}

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.expm1(Math.log1p((Math.sin(phi1) * Math.cos((lambda2 - lambda1))))))));
}

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.expm1(math.log1p((math.sin(phi1) * math.cos((lambda2 - lambda1))))))))

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) * expm1(log1p(Float64(sin(phi1) * cos(Float64(lambda2 - lambda1))))))))
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[(Exp[N[Log[1 + N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.7%

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

   1. *-commutative 79.7%

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

   2. associate-*l* 79.7%

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

3. Simplified 79.7%

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

   1. expm1-log1p-u 79.7%

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

   2. expm1-undefine 78.2%

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

   3. *-commutative 78.2%

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

6. Applied egg-rr 78.2%

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

   1. expm1-define 79.7%

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

   2. sub-neg 79.7%

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

   3. neg-mul-1 79.7%

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

   4. remove-double-neg 79.7%

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

   5. mul-1-neg 79.7%

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

   6. neg-mul-1 79.7%

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

   7. distribute-neg-in 79.7%

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

   8. +-commutative 79.7%

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

   9. cos-neg 79.7%

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

   10. mul-1-neg 79.7%

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

   11. unsub-neg 79.7%

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

8. Simplified 79.7%

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

9. Final simplification 79.7%

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

Alternative 11: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_2 \leq -1600000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \lambda_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t\_0 - t\_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (* (cos phi2) (sin phi1))))
   (if (<= lambda2 -1600000000.0)
     (atan2 t_1 (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1)))))
     (if (<= lambda2 9.2e-10)
       (atan2 t_1 (- t_0 (* (cos lambda1) t_2)))
       (atan2
        (* (cos phi2) (sin (- lambda2)))
        (- t_0 (* t_2 (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 t_2 = cos(phi2) * sin(phi1);
	double tmp;
	if (lambda2 <= -1600000000.0) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	} else if (lambda2 <= 9.2e-10) {
		tmp = atan2(t_1, (t_0 - (cos(lambda1) * t_2)));
	} else {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (t_2 * 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) :: t_2
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    t_2 = cos(phi2) * sin(phi1)
    if (lambda2 <= (-1600000000.0d0)) then
        tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))))
    else if (lambda2 <= 9.2d-10) then
        tmp = atan2(t_1, (t_0 - (cos(lambda1) * t_2)))
    else
        tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (t_2 * 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 t_2 = Math.cos(phi2) * Math.sin(phi1);
	double tmp;
	if (lambda2 <= -1600000000.0) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)))));
	} else if (lambda2 <= 9.2e-10) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(lambda1) * t_2)));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (t_0 - (t_2 * 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))
	t_2 = math.cos(phi2) * math.sin(phi1)
	tmp = 0
	if lambda2 <= -1600000000.0:
		tmp = math.atan2(t_1, (t_0 - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))))
	elif lambda2 <= 9.2e-10:
		tmp = math.atan2(t_1, (t_0 - (math.cos(lambda1) * t_2)))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), (t_0 - (t_2 * 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)))
	t_2 = Float64(cos(phi2) * sin(phi1))
	tmp = 0.0
	if (lambda2 <= -1600000000.0)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	elseif (lambda2 <= 9.2e-10)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(lambda1) * t_2)));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_0 - Float64(t_2 * 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));
	t_2 = cos(phi2) * sin(phi1);
	tmp = 0.0;
	if (lambda2 <= -1600000000.0)
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	elseif (lambda2 <= 9.2e-10)
		tmp = atan2(t_1, (t_0 - (cos(lambda1) * t_2)));
	else
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (t_2 * 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]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -1600000000.0], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 9.2e-10], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$2 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.6e9

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in lambda1 around 0 49.2%

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

      1. *-commutative 75.9%

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

      2. cos-neg 75.9%

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

   7. Simplified 49.2%

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

   if -1.6e9 < lambda2 < 9.20000000000000028e-10

   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. Add Preprocessing

   3. Taylor expanded in lambda2 around 0 99.8%

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

   if 9.20000000000000028e-10 < lambda2

   1. Initial program 65.9%

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

   3. Taylor expanded in lambda1 around 0 67.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1600000000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -1600000000 \lor \neg \left(\lambda_2 \leq 9.2 \cdot 10^{-10}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda2 -1600000000.0) (not (<= lambda2 9.2e-10)))
     (atan2
      (* (cos phi2) (sin (- lambda2)))
      (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1)))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* (cos phi2) (* (cos lambda1) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda2 <= -1600000000.0) || !(lambda2 <= 9.2e-10)) {
		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(phi2) * (cos(lambda1) * 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 <= (-1600000000.0d0)) .or. (.not. (lambda2 <= 9.2d-10))) 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(phi2) * (cos(lambda1) * 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 <= -1600000000.0) || !(lambda2 <= 9.2e-10)) {
		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(phi2) * (Math.cos(lambda1) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda2 <= -1600000000.0) or not (lambda2 <= 9.2e-10):
		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(phi2) * (math.cos(lambda1) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda2 <= -1600000000.0) || !(lambda2 <= 9.2e-10))
		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(phi2) * Float64(cos(lambda1) * 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 <= -1600000000.0) || ~((lambda2 <= 9.2e-10)))
		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(phi2) * (cos(lambda1) * 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, -1600000000.0], N[Not[LessEqual[lambda2, 9.2e-10]], $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[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.6e9 or 9.20000000000000028e-10 < lambda2

   1. Initial program 57.9%

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

      1. *-commutative 57.9%

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

      2. associate-*l* 57.9%

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

   3. Simplified 57.9%

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

      1. sin-diff 78.9%

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

   6. Applied egg-rr 78.9%

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

   7. Taylor expanded in lambda1 around 0 78.8%

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

      1. *-commutative 78.8%

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

      2. cos-neg 78.8%

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

   9. Simplified 78.8%

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

   10. Taylor expanded in lambda1 around 0 57.8%

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

       1. neg-mul-1 57.8%

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

       2. sin-neg 57.8%

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

   12. Simplified 57.8%

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

   if -1.6e9 < lambda2 < 9.20000000000000028e-10

   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* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in lambda2 around 0 99.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1600000000 \lor \neg \left(\lambda_2 \leq 9.2 \cdot 10^{-10}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq -1600000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_2}\\ \mathbf{elif}\;\lambda_2 \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1))))))
   (if (<= lambda2 -1600000000.0)
     (atan2 t_1 t_2)
     (if (<= lambda2 9.2e-10)
       (atan2 t_1 (- t_0 (* (cos lambda1) (* (cos phi2) (sin phi1)))))
       (atan2 (* (cos phi2) (sin (- lambda2))) t_2)))))

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 t_2 = t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)));
	double tmp;
	if (lambda2 <= -1600000000.0) {
		tmp = atan2(t_1, t_2);
	} else if (lambda2 <= 9.2e-10) {
		tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * sin(-lambda2)), 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) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    t_2 = t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))
    if (lambda2 <= (-1600000000.0d0)) then
        tmp = atan2(t_1, t_2)
    else if (lambda2 <= 9.2d-10) then
        tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))))
    else
        tmp = atan2((cos(phi2) * sin(-lambda2)), 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.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double t_2 = t_0 - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)));
	double tmp;
	if (lambda2 <= -1600000000.0) {
		tmp = Math.atan2(t_1, t_2);
	} else if (lambda2 <= 9.2e-10) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(lambda1) * (Math.cos(phi2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), t_2);
	}
	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))
	t_2 = t_0 - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))
	tmp = 0
	if lambda2 <= -1600000000.0:
		tmp = math.atan2(t_1, t_2)
	elif lambda2 <= 9.2e-10:
		tmp = math.atan2(t_1, (t_0 - (math.cos(lambda1) * (math.cos(phi2) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), t_2)
	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)))
	t_2 = Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1))))
	tmp = 0.0
	if (lambda2 <= -1600000000.0)
		tmp = atan(t_1, t_2);
	elseif (lambda2 <= 9.2e-10)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), t_2);
	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));
	t_2 = t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)));
	tmp = 0.0;
	if (lambda2 <= -1600000000.0)
		tmp = atan2(t_1, t_2);
	elseif (lambda2 <= 9.2e-10)
		tmp = atan2(t_1, (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	else
		tmp = atan2((cos(phi2) * sin(-lambda2)), t_2);
	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]}, Block[{t$95$2 = N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -1600000000.0], N[ArcTan[t$95$1 / t$95$2], $MachinePrecision], If[LessEqual[lambda2, 9.2e-10], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / t$95$2], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.6e9

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in lambda1 around 0 49.2%

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

      1. *-commutative 75.9%

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

      2. cos-neg 75.9%

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

   7. Simplified 49.2%

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

   if -1.6e9 < lambda2 < 9.20000000000000028e-10

   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. Add Preprocessing

   3. Taylor expanded in lambda2 around 0 99.8%

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

   if 9.20000000000000028e-10 < lambda2

   1. Initial program 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-*l* 65.9%

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

   3. Simplified 65.9%

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

      1. sin-diff 81.4%

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

   6. Applied egg-rr 81.4%

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

   7. Taylor expanded in lambda1 around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. cos-neg 81.6%

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

   9. Simplified 81.6%

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

   10. Taylor expanded in lambda1 around 0 67.0%

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

       1. neg-mul-1 67.0%

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

       2. sin-neg 67.0%

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

   12. Simplified 67.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1600000000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)\\ \mathbf{if}\;\lambda_2 \leq -1600000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_2}\\ \mathbf{elif}\;\lambda_2 \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1))))))
   (if (<= lambda2 -1600000000.0)
     (atan2 t_1 t_2)
     (if (<= lambda2 9.2e-10)
       (atan2 t_1 (- t_0 (* (cos phi2) (* (cos lambda1) (sin phi1)))))
       (atan2 (* (cos phi2) (sin (- lambda2))) t_2)))))

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 t_2 = t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)));
	double tmp;
	if (lambda2 <= -1600000000.0) {
		tmp = atan2(t_1, t_2);
	} else if (lambda2 <= 9.2e-10) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * sin(-lambda2)), 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) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    t_2 = t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))
    if (lambda2 <= (-1600000000.0d0)) then
        tmp = atan2(t_1, t_2)
    else if (lambda2 <= 9.2d-10) then
        tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))))
    else
        tmp = atan2((cos(phi2) * sin(-lambda2)), 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.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double t_2 = t_0 - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)));
	double tmp;
	if (lambda2 <= -1600000000.0) {
		tmp = Math.atan2(t_1, t_2);
	} else if (lambda2 <= 9.2e-10) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * (Math.cos(lambda1) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), t_2);
	}
	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))
	t_2 = t_0 - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))
	tmp = 0
	if lambda2 <= -1600000000.0:
		tmp = math.atan2(t_1, t_2)
	elif lambda2 <= 9.2e-10:
		tmp = math.atan2(t_1, (t_0 - (math.cos(phi2) * (math.cos(lambda1) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), t_2)
	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)))
	t_2 = Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1))))
	tmp = 0.0
	if (lambda2 <= -1600000000.0)
		tmp = atan(t_1, t_2);
	elseif (lambda2 <= 9.2e-10)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda1) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), t_2);
	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));
	t_2 = t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)));
	tmp = 0.0;
	if (lambda2 <= -1600000000.0)
		tmp = atan2(t_1, t_2);
	elseif (lambda2 <= 9.2e-10)
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	else
		tmp = atan2((cos(phi2) * sin(-lambda2)), t_2);
	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]}, Block[{t$95$2 = N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -1600000000.0], N[ArcTan[t$95$1 / t$95$2], $MachinePrecision], If[LessEqual[lambda2, 9.2e-10], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / t$95$2], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.6e9

   1. Initial program 49.5%

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

      1. *-commutative 49.5%

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

      2. associate-*l* 49.5%

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

   3. Simplified 49.5%

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

   5. Taylor expanded in lambda1 around 0 49.2%

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

      1. *-commutative 75.9%

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

      2. cos-neg 75.9%

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

   7. Simplified 49.2%

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

   if -1.6e9 < lambda2 < 9.20000000000000028e-10

   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* 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in lambda2 around 0 99.7%

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

   if 9.20000000000000028e-10 < lambda2

   1. Initial program 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-*l* 65.9%

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

   3. Simplified 65.9%

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

      1. sin-diff 81.4%

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

   6. Applied egg-rr 81.4%

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

   7. Taylor expanded in lambda1 around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. cos-neg 81.6%

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

   9. Simplified 81.6%

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

   10. Taylor expanded in lambda1 around 0 67.0%

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

       1. neg-mul-1 67.0%

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

       2. sin-neg 67.0%

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

   12. Simplified 67.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1600000000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 9.2 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1750000000 \lor \neg \left(\lambda_2 \leq 9.2 \cdot 10^{-10}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -1750000000.0) (not (<= lambda2 9.2e-10)))
   (atan2
    (* (cos phi2) (sin (- lambda2)))
    (- (* (cos phi1) (sin phi2)) (* (cos phi2) (* (cos lambda2) (sin phi1)))))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (fma (sin phi2) (cos phi1) (* (cos phi2) (- (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -1750000000.0) || !(lambda2 <= 9.2e-10)) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), fma(sin(phi2), cos(phi1), (cos(phi2) * -sin(phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -1750000000.0) || !(lambda2 <= 9.2e-10))
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), fma(sin(phi2), cos(phi1), Float64(cos(phi2) * Float64(-sin(phi1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -1750000000.0], N[Not[LessEqual[lambda2, 9.2e-10]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - 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[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.75e9 or 9.20000000000000028e-10 < lambda2

   1. Initial program 57.9%

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

      1. *-commutative 57.9%

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

      2. associate-*l* 57.9%

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

   3. Simplified 57.9%

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

      1. sin-diff 78.9%

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

   6. Applied egg-rr 78.9%

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

   7. Taylor expanded in lambda1 around 0 78.8%

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

      1. *-commutative 78.8%

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

      2. cos-neg 78.8%

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

   9. Simplified 78.8%

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

   10. Taylor expanded in lambda1 around 0 57.8%

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

       1. neg-mul-1 57.8%

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

       2. sin-neg 57.8%

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

   12. Simplified 57.8%

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

   if -1.75e9 < lambda2 < 9.20000000000000028e-10

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 45.5%

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

   4. Applied egg-rr 45.5%

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

   5. Taylor expanded in lambda1 around 0 85.1%

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

      1. *-commutative 85.1%

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

      2. fma-neg 85.1%

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

      3. *-commutative 85.1%

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

      4. distribute-rgt-neg-in 85.1%

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

      5. cos-neg 85.1%

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

      6. *-commutative 85.1%

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

   7. Simplified 85.1%

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

   8. Taylor expanded in lambda2 around 0 85.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1750000000 \lor \neg \left(\lambda_2 \leq 9.2 \cdot 10^{-10}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.028 \lor \neg \left(\phi_2 \leq 4.9 \cdot 10^{-15}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi2 -0.028) (not (<= phi2 4.9e-15)))
     (atan2 t_0 (fma (sin phi2) (cos phi1) (* (cos phi2) (- (sin phi1)))))
     (atan2
      t_0
      (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -0.028) || !(phi2 <= 4.9e-15)) {
		tmp = atan2(t_0, fma(sin(phi2), cos(phi1), (cos(phi2) * -sin(phi1))));
	} else {
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if ((phi2 <= -0.028) || !(phi2 <= 4.9e-15))
		tmp = atan(t_0, fma(sin(phi2), cos(phi1), Float64(cos(phi2) * Float64(-sin(phi1)))));
	else
		tmp = atan(t_0, Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -0.028], N[Not[LessEqual[phi2, 4.9e-15]], $MachinePrecision]], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.0280000000000000006 or 4.8999999999999999e-15 < phi2

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 33.0%

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

   4. Applied egg-rr 33.0%

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

   5. Taylor expanded in lambda1 around 0 69.6%

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

      1. *-commutative 69.6%

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

      2. fma-neg 69.6%

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

      3. *-commutative 69.6%

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

      4. distribute-rgt-neg-in 69.6%

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

      5. cos-neg 69.6%

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

      6. *-commutative 69.6%

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

   7. Simplified 69.6%

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

   8. Taylor expanded in lambda2 around 0 59.5%

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

   if -0.0280000000000000006 < phi2 < 4.8999999999999999e-15

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 28.2%

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

   4. Applied egg-rr 28.2%

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

   5. Taylor expanded in phi2 around 0 81.5%

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

      1. sub-neg 81.5%

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

      2. neg-mul-1 81.5%

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

      3. remove-double-neg 81.5%

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

      4. neg-mul-1 81.5%

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

      5. distribute-neg-in 81.5%

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

      6. +-commutative 81.5%

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

      7. neg-mul-1 81.5%

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

      8. cos-neg 81.5%

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

      9. neg-mul-1 81.5%

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

      10. sub-neg 81.5%

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

   7. Simplified 81.5%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.028 \lor \neg \left(\phi_2 \leq 4.9 \cdot 10^{-15}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \phi_2 \cdot \left(-\sin \phi_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.00182:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{elif}\;\phi_2 \leq 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \sin \phi_1 \cdot \left(-\cos \lambda_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi2 -0.00182)
     (atan2
      t_0
      (+
       (* (cos phi1) (sin phi2))
       (/ (* (cos phi2) (* (sin lambda1) 0.0)) 2.0)))
     (if (<= phi2 1e-9)
       (atan2
        t_0
        (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda2 lambda1)))))
       (atan2
        t_0
        (fma (sin phi2) (cos phi1) (* (sin phi1) (- (cos lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -0.00182) {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0)) / 2.0)));
	} else if (phi2 <= 1e-9) {
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2(t_0, fma(sin(phi2), cos(phi1), (sin(phi1) * -cos(lambda2))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -0.00182)
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) + Float64(Float64(cos(phi2) * Float64(sin(lambda1) * 0.0)) / 2.0)));
	elseif (phi2 <= 1e-9)
		tmp = atan(t_0, Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(t_0, fma(sin(phi2), cos(phi1), Float64(sin(phi1) * Float64(-cos(lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.00182], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 1e-9], N[ArcTan[t$95$0 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.00182

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

      1. *-commutative 77.4%

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

      2. sin-cos-mult 61.0%

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

      3. associate-*l/ 61.0%

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

      4. associate--r- 61.0%

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

      5. +-commutative 61.0%

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in phi1 around 0 53.0%

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

      1. +-commutative 53.0%

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

   9. Simplified 53.0%

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

   10. Taylor expanded in lambda2 around 0 53.0%

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

       1. *-lft-identity 53.0%

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

       2. sin-neg 53.0%

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

       3. neg-mul-1 53.0%

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

       4. distribute-rgt-out 53.0%

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

       5. metadata-eval 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \color{blue}{0}\right)}{2}} \]

   12. Simplified 53.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\color{blue}{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}}{2}} \]

   if -0.00182 < phi2 < 1.00000000000000006e-9

   1. Initial program 80.9%

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

      1. add-exp-log 28.0%

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

   4. Applied egg-rr 28.0%

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

   5. Taylor expanded in phi2 around 0 80.9%

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

      1. sub-neg 80.9%

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

      2. neg-mul-1 80.9%

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

      3. remove-double-neg 80.9%

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

      4. neg-mul-1 80.9%

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

      5. distribute-neg-in 80.9%

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

      6. +-commutative 80.9%

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

      7. neg-mul-1 80.9%

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

      8. cos-neg 80.9%

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

      9. neg-mul-1 80.9%

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

      10. sub-neg 80.9%

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

   7. Simplified 80.9%

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

   if 1.00000000000000006e-9 < phi2

   1. Initial program 79.8%

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

      1. add-exp-log 32.2%

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

   4. Applied egg-rr 32.2%

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

   5. Taylor expanded in lambda1 around 0 68.7%

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

      1. *-commutative 68.7%

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

      2. fma-neg 68.7%

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

      3. *-commutative 68.7%

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

      4. distribute-rgt-neg-in 68.7%

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

      5. cos-neg 68.7%

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

      6. *-commutative 68.7%

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

   7. Simplified 68.7%

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

   8. Taylor expanded in phi2 around 0 55.4%

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

      1. mul-1-neg 55.4%

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

      2. distribute-rgt-neg-in 55.4%

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

   10. Simplified 55.4%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00182:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{elif}\;\phi_2 \leq 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \sin \phi_1 \cdot \left(-\cos \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 79.5% 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 79.7%

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

   1. *-commutative 79.7%

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

   2. associate-*l* 79.7%

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

3. Simplified 79.7%

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

5. Final simplification 79.7%

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

Alternative 19: 66.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.27:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{elif}\;\phi_2 \leq 4.9 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \frac{\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi2 -0.27)
     (atan2 t_1 (+ t_0 (/ (* (cos phi2) (* (sin lambda1) 0.0)) 2.0)))
     (if (<= phi2 4.9e-15)
       (atan2
        t_1
        (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda2 lambda1)))))
       (atan2
        t_1
        (-
         t_0
         (/
          (+
           (sin (+ lambda2 (- phi1 lambda1)))
           (sin (+ phi1 (- lambda1 lambda2))))
          2.0)))))))

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 (phi2 <= -0.27) {
		tmp = atan2(t_1, (t_0 + ((cos(phi2) * (sin(lambda1) * 0.0)) / 2.0)));
	} else if (phi2 <= 4.9e-15) {
		tmp = atan2(t_1, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2(t_1, (t_0 - ((sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2)))) / 2.0)));
	}
	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 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi2 <= (-0.27d0)) then
        tmp = atan2(t_1, (t_0 + ((cos(phi2) * (sin(lambda1) * 0.0d0)) / 2.0d0)))
    else if (phi2 <= 4.9d-15) then
        tmp = atan2(t_1, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))))
    else
        tmp = atan2(t_1, (t_0 - ((sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2)))) / 2.0d0)))
    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 t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -0.27) {
		tmp = Math.atan2(t_1, (t_0 + ((Math.cos(phi2) * (Math.sin(lambda1) * 0.0)) / 2.0)));
	} else if (phi2 <= 4.9e-15) {
		tmp = Math.atan2(t_1, ((phi2 * Math.cos(phi1)) - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = Math.atan2(t_1, (t_0 - ((Math.sin((lambda2 + (phi1 - lambda1))) + Math.sin((phi1 + (lambda1 - lambda2)))) / 2.0)));
	}
	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 phi2 <= -0.27:
		tmp = math.atan2(t_1, (t_0 + ((math.cos(phi2) * (math.sin(lambda1) * 0.0)) / 2.0)))
	elif phi2 <= 4.9e-15:
		tmp = math.atan2(t_1, ((phi2 * math.cos(phi1)) - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	else:
		tmp = math.atan2(t_1, (t_0 - ((math.sin((lambda2 + (phi1 - lambda1))) + math.sin((phi1 + (lambda1 - lambda2)))) / 2.0)))
	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 (phi2 <= -0.27)
		tmp = atan(t_1, Float64(t_0 + Float64(Float64(cos(phi2) * Float64(sin(lambda1) * 0.0)) / 2.0)));
	elseif (phi2 <= 4.9e-15)
		tmp = atan(t_1, Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(Float64(sin(Float64(lambda2 + Float64(phi1 - lambda1))) + sin(Float64(phi1 + Float64(lambda1 - lambda2)))) / 2.0)));
	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 (phi2 <= -0.27)
		tmp = atan2(t_1, (t_0 + ((cos(phi2) * (sin(lambda1) * 0.0)) / 2.0)));
	elseif (phi2 <= 4.9e-15)
		tmp = atan2(t_1, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	else
		tmp = atan2(t_1, (t_0 - ((sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2)))) / 2.0)));
	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]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.27], N[ArcTan[t$95$1 / N[(t$95$0 + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 4.9e-15], N[ArcTan[t$95$1 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[(N[Sin[N[(lambda2 + N[(phi1 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(phi1 + N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.27000000000000002

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

      1. *-commutative 77.4%

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

      2. sin-cos-mult 61.0%

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

      3. associate-*l/ 61.0%

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

      4. associate--r- 61.0%

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

      5. +-commutative 61.0%

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in phi1 around 0 53.0%

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

      1. +-commutative 53.0%

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

   9. Simplified 53.0%

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

   10. Taylor expanded in lambda2 around 0 53.0%

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

       1. *-lft-identity 53.0%

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

       2. sin-neg 53.0%

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

       3. neg-mul-1 53.0%

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

       4. distribute-rgt-out 53.0%

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

       5. metadata-eval 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \color{blue}{0}\right)}{2}} \]

   12. Simplified 53.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\color{blue}{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}}{2}} \]

   if -0.27000000000000002 < phi2 < 4.8999999999999999e-15

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 28.2%

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

   4. Applied egg-rr 28.2%

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

   5. Taylor expanded in phi2 around 0 81.5%

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

      1. sub-neg 81.5%

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

      2. neg-mul-1 81.5%

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

      3. remove-double-neg 81.5%

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

      4. neg-mul-1 81.5%

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

      5. distribute-neg-in 81.5%

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

      6. +-commutative 81.5%

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

      7. neg-mul-1 81.5%

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

      8. cos-neg 81.5%

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

      9. neg-mul-1 81.5%

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

      10. sub-neg 81.5%

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

   7. Simplified 81.5%

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

   if 4.8999999999999999e-15 < phi2

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-*l* 78.4%

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

   3. Simplified 78.4%

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

      1. *-commutative 78.4%

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

      2. sin-cos-mult 57.7%

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

      3. associate-*l/ 57.7%

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

      4. associate--r- 57.7%

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

      5. +-commutative 57.7%

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

   6. Applied egg-rr 57.7%

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

   7. Taylor expanded in phi2 around 0 54.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.27:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{elif}\;\phi_2 \leq 4.9 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 66.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0096:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{elif}\;\phi_2 \leq 0.00017:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi2 -0.0096)
     (atan2
      t_0
      (+
       (* (cos phi1) (sin phi2))
       (/ (* (cos phi2) (* (sin lambda1) 0.0)) 2.0)))
     (if (<= phi2 0.00017)
       (atan2
        t_0
        (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda2 lambda1)))))
       (atan2
        t_0
        (-
         (sin phi2)
         (/
          (*
           (cos phi2)
           (+
            (sin (+ lambda2 (- phi1 lambda1)))
            (sin (+ phi1 (- lambda1 lambda2)))))
          2.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -0.0096) {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0)) / 2.0)));
	} else if (phi2 <= 0.00017) {
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2(t_0, (sin(phi2) - ((cos(phi2) * (sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2))))) / 2.0)));
	}
	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(phi2) * sin((lambda1 - lambda2))
    if (phi2 <= (-0.0096d0)) then
        tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0d0)) / 2.0d0)))
    else if (phi2 <= 0.00017d0) then
        tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))))
    else
        tmp = atan2(t_0, (sin(phi2) - ((cos(phi2) * (sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2))))) / 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -0.0096) {
		tmp = Math.atan2(t_0, ((Math.cos(phi1) * Math.sin(phi2)) + ((Math.cos(phi2) * (Math.sin(lambda1) * 0.0)) / 2.0)));
	} else if (phi2 <= 0.00017) {
		tmp = Math.atan2(t_0, ((phi2 * Math.cos(phi1)) - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - ((Math.cos(phi2) * (Math.sin((lambda2 + (phi1 - lambda1))) + Math.sin((phi1 + (lambda1 - lambda2))))) / 2.0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= -0.0096:
		tmp = math.atan2(t_0, ((math.cos(phi1) * math.sin(phi2)) + ((math.cos(phi2) * (math.sin(lambda1) * 0.0)) / 2.0)))
	elif phi2 <= 0.00017:
		tmp = math.atan2(t_0, ((phi2 * math.cos(phi1)) - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	else:
		tmp = math.atan2(t_0, (math.sin(phi2) - ((math.cos(phi2) * (math.sin((lambda2 + (phi1 - lambda1))) + math.sin((phi1 + (lambda1 - lambda2))))) / 2.0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -0.0096)
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) + Float64(Float64(cos(phi2) * Float64(sin(lambda1) * 0.0)) / 2.0)));
	elseif (phi2 <= 0.00017)
		tmp = atan(t_0, Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(t_0, Float64(sin(phi2) - Float64(Float64(cos(phi2) * Float64(sin(Float64(lambda2 + Float64(phi1 - lambda1))) + sin(Float64(phi1 + Float64(lambda1 - lambda2))))) / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= -0.0096)
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0)) / 2.0)));
	elseif (phi2 <= 0.00017)
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	else
		tmp = atan2(t_0, (sin(phi2) - ((cos(phi2) * (sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2))))) / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.0096], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 0.00017], N[ArcTan[t$95$0 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[N[(lambda2 + N[(phi1 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(phi1 + N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.00959999999999999916

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

      1. *-commutative 77.4%

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

      2. sin-cos-mult 61.0%

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

      3. associate-*l/ 61.0%

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

      4. associate--r- 61.0%

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

      5. +-commutative 61.0%

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in phi1 around 0 53.0%

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

      1. +-commutative 53.0%

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

   9. Simplified 53.0%

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

   10. Taylor expanded in lambda2 around 0 53.0%

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

       1. *-lft-identity 53.0%

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

       2. sin-neg 53.0%

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

       3. neg-mul-1 53.0%

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

       4. distribute-rgt-out 53.0%

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

       5. metadata-eval 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \color{blue}{0}\right)}{2}} \]

   12. Simplified 53.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\color{blue}{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}}{2}} \]

   if -0.00959999999999999916 < phi2 < 1.7e-4

   1. Initial program 81.0%

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

      1. add-exp-log 27.7%

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

   4. Applied egg-rr 27.7%

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

   5. Taylor expanded in phi2 around 0 81.0%

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

      1. sub-neg 81.0%

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

      2. neg-mul-1 81.0%

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

      3. remove-double-neg 81.0%

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

      4. neg-mul-1 81.0%

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

      5. distribute-neg-in 81.0%

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

      6. +-commutative 81.0%

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

      7. neg-mul-1 81.0%

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

      8. cos-neg 81.0%

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

      9. neg-mul-1 81.0%

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

      10. sub-neg 81.0%

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

   7. Simplified 81.0%

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

   if 1.7e-4 < phi2

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*l* 79.4%

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

   3. Simplified 79.4%

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

      1. *-commutative 79.4%

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

      2. sin-cos-mult 57.9%

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

      3. associate-*l/ 57.9%

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

      4. associate--r- 57.9%

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

      5. +-commutative 57.9%

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

   6. Applied egg-rr 57.9%

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

   7. Taylor expanded in phi1 around 0 50.6%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.0096:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{elif}\;\phi_2 \leq 0.00017:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)\right)}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 65.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t\_0\\ \mathbf{if}\;\phi_2 \leq -0.039:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{elif}\;\phi_2 \leq 0.064:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - \frac{\cos \phi_2 \cdot \left(t\_0 + \sin \left(\lambda_2 - \lambda_1\right)\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (* (cos phi2) t_0)))
   (if (<= phi2 -0.039)
     (atan2
      t_1
      (+
       (* (cos phi1) (sin phi2))
       (/ (* (cos phi2) (* (sin lambda1) 0.0)) 2.0)))
     (if (<= phi2 0.064)
       (atan2
        t_1
        (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda2 lambda1)))))
       (atan2
        t_1
        (-
         (sin phi2)
         (/ (* (cos phi2) (+ t_0 (sin (- lambda2 lambda1)))) 2.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos(phi2) * t_0;
	double tmp;
	if (phi2 <= -0.039) {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0)) / 2.0)));
	} else if (phi2 <= 0.064) {
		tmp = atan2(t_1, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2(t_1, (sin(phi2) - ((cos(phi2) * (t_0 + sin((lambda2 - lambda1)))) / 2.0)));
	}
	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 = sin((lambda1 - lambda2))
    t_1 = cos(phi2) * t_0
    if (phi2 <= (-0.039d0)) then
        tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0d0)) / 2.0d0)))
    else if (phi2 <= 0.064d0) then
        tmp = atan2(t_1, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))))
    else
        tmp = atan2(t_1, (sin(phi2) - ((cos(phi2) * (t_0 + sin((lambda2 - lambda1)))) / 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * t_0;
	double tmp;
	if (phi2 <= -0.039) {
		tmp = Math.atan2(t_1, ((Math.cos(phi1) * Math.sin(phi2)) + ((Math.cos(phi2) * (Math.sin(lambda1) * 0.0)) / 2.0)));
	} else if (phi2 <= 0.064) {
		tmp = Math.atan2(t_1, ((phi2 * Math.cos(phi1)) - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - ((Math.cos(phi2) * (t_0 + Math.sin((lambda2 - lambda1)))) / 2.0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.cos(phi2) * t_0
	tmp = 0
	if phi2 <= -0.039:
		tmp = math.atan2(t_1, ((math.cos(phi1) * math.sin(phi2)) + ((math.cos(phi2) * (math.sin(lambda1) * 0.0)) / 2.0)))
	elif phi2 <= 0.064:
		tmp = math.atan2(t_1, ((phi2 * math.cos(phi1)) - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	else:
		tmp = math.atan2(t_1, (math.sin(phi2) - ((math.cos(phi2) * (t_0 + math.sin((lambda2 - lambda1)))) / 2.0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * t_0)
	tmp = 0.0
	if (phi2 <= -0.039)
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) + Float64(Float64(cos(phi2) * Float64(sin(lambda1) * 0.0)) / 2.0)));
	elseif (phi2 <= 0.064)
		tmp = atan(t_1, Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(t_1, Float64(sin(phi2) - Float64(Float64(cos(phi2) * Float64(t_0 + sin(Float64(lambda2 - lambda1)))) / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = cos(phi2) * t_0;
	tmp = 0.0;
	if (phi2 <= -0.039)
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0)) / 2.0)));
	elseif (phi2 <= 0.064)
		tmp = atan2(t_1, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	else
		tmp = atan2(t_1, (sin(phi2) - ((cos(phi2) * (t_0 + sin((lambda2 - lambda1)))) / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[phi2, -0.039], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 0.064], N[ArcTan[t$95$1 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$0 + N[Sin[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0389999999999999999

   1. Initial program 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

      1. *-commutative 77.4%

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

      2. sin-cos-mult 61.0%

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

      3. associate-*l/ 61.0%

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

      4. associate--r- 61.0%

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

      5. +-commutative 61.0%

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in phi1 around 0 53.0%

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

      1. +-commutative 53.0%

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

   9. Simplified 53.0%

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

   10. Taylor expanded in lambda2 around 0 53.0%

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

       1. *-lft-identity 53.0%

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

       2. sin-neg 53.0%

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

       3. neg-mul-1 53.0%

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

       4. distribute-rgt-out 53.0%

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

       5. metadata-eval 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \color{blue}{0}\right)}{2}} \]

   12. Simplified 53.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\color{blue}{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}}{2}} \]

   if -0.0389999999999999999 < phi2 < 0.064000000000000001

   1. Initial program 81.0%

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

      1. add-exp-log 27.7%

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

   4. Applied egg-rr 27.7%

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

   5. Taylor expanded in phi2 around 0 81.0%

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

      1. sub-neg 81.0%

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

      2. neg-mul-1 81.0%

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

      3. remove-double-neg 81.0%

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

      4. neg-mul-1 81.0%

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

      5. distribute-neg-in 81.0%

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

      6. +-commutative 81.0%

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

      7. neg-mul-1 81.0%

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

      8. cos-neg 81.0%

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

      9. neg-mul-1 81.0%

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

      10. sub-neg 81.0%

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

   7. Simplified 81.0%

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

   if 0.064000000000000001 < phi2

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*l* 79.4%

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

   3. Simplified 79.4%

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

      1. *-commutative 79.4%

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

      2. sin-cos-mult 57.9%

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

      3. associate-*l/ 57.9%

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

      4. associate--r- 57.9%

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

      5. +-commutative 57.9%

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

   6. Applied egg-rr 57.9%

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

   7. Taylor expanded in phi1 around 0 48.2%

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

      1. +-commutative 48.2%

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

   9. Simplified 48.2%

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

   10. Taylor expanded in phi1 around 0 48.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.039:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{elif}\;\phi_2 \leq 0.064:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \left(\lambda_1 - \lambda_2\right) + \sin \left(\lambda_2 - \lambda_1\right)\right)}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 66.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0235 \lor \neg \left(\phi_2 \leq 0.0039\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi2 -0.0235) (not (<= phi2 0.0039)))
     (atan2
      t_0
      (+
       (* (cos phi1) (sin phi2))
       (/ (* (cos phi2) (* (sin lambda1) 0.0)) 2.0)))
     (atan2
      t_0
      (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -0.0235) || !(phi2 <= 0.0039)) {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0)) / 2.0)));
	} else {
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(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(phi2) * sin((lambda1 - lambda2))
    if ((phi2 <= (-0.0235d0)) .or. (.not. (phi2 <= 0.0039d0))) then
        tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0d0)) / 2.0d0)))
    else
        tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(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(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -0.0235) || !(phi2 <= 0.0039)) {
		tmp = Math.atan2(t_0, ((Math.cos(phi1) * Math.sin(phi2)) + ((Math.cos(phi2) * (Math.sin(lambda1) * 0.0)) / 2.0)));
	} else {
		tmp = Math.atan2(t_0, ((phi2 * Math.cos(phi1)) - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -0.0235) or not (phi2 <= 0.0039):
		tmp = math.atan2(t_0, ((math.cos(phi1) * math.sin(phi2)) + ((math.cos(phi2) * (math.sin(lambda1) * 0.0)) / 2.0)))
	else:
		tmp = math.atan2(t_0, ((phi2 * math.cos(phi1)) - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if ((phi2 <= -0.0235) || !(phi2 <= 0.0039))
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) + Float64(Float64(cos(phi2) * Float64(sin(lambda1) * 0.0)) / 2.0)));
	else
		tmp = atan(t_0, Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -0.0235) || ~((phi2 <= 0.0039)))
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) + ((cos(phi2) * (sin(lambda1) * 0.0)) / 2.0)));
	else
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -0.0235], N[Not[LessEqual[phi2, 0.0039]], $MachinePrecision]], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.0235 or 0.0038999999999999998 < phi2

   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. *-commutative 78.3%

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

      2. sin-cos-mult 59.7%

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

      3. associate-*l/ 59.7%

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

      4. associate--r- 59.7%

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

      5. +-commutative 59.7%

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

   6. Applied egg-rr 59.7%

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

   7. Taylor expanded in phi1 around 0 51.0%

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

      1. +-commutative 51.0%

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

   9. Simplified 51.0%

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

   10. Taylor expanded in lambda2 around 0 51.0%

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

       1. *-lft-identity 51.0%

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

       2. sin-neg 51.0%

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

       3. neg-mul-1 51.0%

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

       4. distribute-rgt-out 51.0%

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

       5. metadata-eval 51.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \color{blue}{0}\right)}{2}} \]

   12. Simplified 51.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\color{blue}{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}}{2}} \]

   if -0.0235 < phi2 < 0.0038999999999999998

   1. Initial program 81.0%

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

      1. add-exp-log 27.7%

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

   4. Applied egg-rr 27.7%

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

   5. Taylor expanded in phi2 around 0 81.0%

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

      1. sub-neg 81.0%

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

      2. neg-mul-1 81.0%

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

      3. remove-double-neg 81.0%

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

      4. neg-mul-1 81.0%

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

      5. distribute-neg-in 81.0%

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

      6. +-commutative 81.0%

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

      7. neg-mul-1 81.0%

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

      8. cos-neg 81.0%

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

      9. neg-mul-1 81.0%

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

      10. sub-neg 81.0%

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

   7. Simplified 81.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.0235 \lor \neg \left(\phi_2 \leq 0.0039\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot 0\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 64.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.00024:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_1 \cdot \left(-t\_0\right)}\\ \mathbf{elif}\;\phi_1 \leq 0.005:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_1 \cdot \left(-\mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -0.00024)
     (atan2 t_1 (* (sin phi1) (- t_0)))
     (if (<= phi1 0.005)
       (atan2 t_1 (- (sin phi2) (* phi1 (* (cos phi2) t_0))))
       (atan2 t_1 (* (sin phi1) (- (expm1 (log1p t_0)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.00024) {
		tmp = atan2(t_1, (sin(phi1) * -t_0));
	} else if (phi1 <= 0.005) {
		tmp = atan2(t_1, (sin(phi2) - (phi1 * (cos(phi2) * t_0))));
	} else {
		tmp = atan2(t_1, (sin(phi1) * -expm1(log1p(t_0))));
	}
	return tmp;
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.00024) {
		tmp = Math.atan2(t_1, (Math.sin(phi1) * -t_0));
	} else if (phi1 <= 0.005) {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - (phi1 * (Math.cos(phi2) * t_0))));
	} else {
		tmp = Math.atan2(t_1, (Math.sin(phi1) * -Math.expm1(Math.log1p(t_0))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -0.00024:
		tmp = math.atan2(t_1, (math.sin(phi1) * -t_0))
	elif phi1 <= 0.005:
		tmp = math.atan2(t_1, (math.sin(phi2) - (phi1 * (math.cos(phi2) * t_0))))
	else:
		tmp = math.atan2(t_1, (math.sin(phi1) * -math.expm1(math.log1p(t_0))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -0.00024)
		tmp = atan(t_1, Float64(sin(phi1) * Float64(-t_0)));
	elseif (phi1 <= 0.005)
		tmp = atan(t_1, Float64(sin(phi2) - Float64(phi1 * Float64(cos(phi2) * t_0))));
	else
		tmp = atan(t_1, Float64(sin(phi1) * Float64(-expm1(log1p(t_0)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.00024], N[ArcTan[t$95$1 / N[(N[Sin[phi1], $MachinePrecision] * (-t$95$0)), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 0.005], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(phi1 * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi1], $MachinePrecision] * (-N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.40000000000000006e-4

   1. Initial program 81.6%

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

      1. add-exp-log 23.9%

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

   4. Applied egg-rr 23.9%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

      1. mul-1-neg 51.0%

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

      2. *-commutative 51.0%

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

      3. distribute-rgt-neg-in 51.0%

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

      4. cos-neg 51.0%

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

      5. sub-neg 51.0%

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

      6. distribute-neg-in 51.0%

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

      7. remove-double-neg 51.0%

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

      8. +-commutative 51.0%

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

      9. sub-neg 51.0%

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

   7. Simplified 51.0%

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

   if -2.40000000000000006e-4 < phi1 < 0.0050000000000000001

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. associate-*l* 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in phi1 around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. sub-neg 81.6%

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

      3. neg-mul-1 81.6%

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

      4. remove-double-neg 81.6%

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

      5. mul-1-neg 81.6%

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

      6. neg-mul-1 81.6%

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

      7. distribute-neg-in 81.6%

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

      8. +-commutative 81.6%

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

      9. cos-neg 81.6%

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

      10. mul-1-neg 81.6%

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

      11. unsub-neg 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in phi1 around 0 81.6%

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

   if 0.0050000000000000001 < phi1

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 22.5%

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

   4. Applied egg-rr 22.5%

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

   5. Taylor expanded in phi2 around 0 48.4%

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

      1. mul-1-neg 48.4%

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

      2. *-commutative 48.4%

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

      3. distribute-rgt-neg-in 48.4%

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

      4. cos-neg 48.4%

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

      5. sub-neg 48.4%

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

      6. distribute-neg-in 48.4%

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

      7. remove-double-neg 48.4%

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

      8. +-commutative 48.4%

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

      9. sub-neg 48.4%

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

   7. Simplified 48.4%

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

      1. expm1-log1p-u 48.4%

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

      2. expm1-undefine 48.4%

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

   9. Applied egg-rr 48.4%

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

       1. expm1-define 48.4%

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

   11. Simplified 48.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00024:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 0.005:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 64.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.019 \lor \neg \left(\phi_1 \leq 0.000215\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_1 \cdot \left(-t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi1 -0.019) (not (<= phi1 0.000215)))
     (atan2 t_1 (* (sin phi1) (- t_0)))
     (atan2 t_1 (- (sin phi2) (* phi1 (* (cos phi2) t_0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -0.019) || !(phi1 <= 0.000215)) {
		tmp = atan2(t_1, (sin(phi1) * -t_0));
	} else {
		tmp = atan2(t_1, (sin(phi2) - (phi1 * (cos(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 = cos((lambda2 - lambda1))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if ((phi1 <= (-0.019d0)) .or. (.not. (phi1 <= 0.000215d0))) then
        tmp = atan2(t_1, (sin(phi1) * -t_0))
    else
        tmp = atan2(t_1, (sin(phi2) - (phi1 * (cos(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 = Math.cos((lambda2 - lambda1));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -0.019) || !(phi1 <= 0.000215)) {
		tmp = Math.atan2(t_1, (Math.sin(phi1) * -t_0));
	} else {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - (phi1 * (Math.cos(phi2) * t_0))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi1 <= -0.019) or not (phi1 <= 0.000215):
		tmp = math.atan2(t_1, (math.sin(phi1) * -t_0))
	else:
		tmp = math.atan2(t_1, (math.sin(phi2) - (phi1 * (math.cos(phi2) * t_0))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if ((phi1 <= -0.019) || !(phi1 <= 0.000215))
		tmp = atan(t_1, Float64(sin(phi1) * Float64(-t_0)));
	else
		tmp = atan(t_1, Float64(sin(phi2) - Float64(phi1 * Float64(cos(phi2) * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi1 <= -0.019) || ~((phi1 <= 0.000215)))
		tmp = atan2(t_1, (sin(phi1) * -t_0));
	else
		tmp = atan2(t_1, (sin(phi2) - (phi1 * (cos(phi2) * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -0.019], N[Not[LessEqual[phi1, 0.000215]], $MachinePrecision]], N[ArcTan[t$95$1 / N[(N[Sin[phi1], $MachinePrecision] * (-t$95$0)), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(phi1 * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0189999999999999995 or 2.14999999999999995e-4 < 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. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 23.2%

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

   4. Applied egg-rr 23.2%

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

   5. Taylor expanded in phi2 around 0 49.8%

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

      1. mul-1-neg 49.8%

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

      2. *-commutative 49.8%

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

      3. distribute-rgt-neg-in 49.8%

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

      4. cos-neg 49.8%

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

      5. sub-neg 49.8%

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

      6. distribute-neg-in 49.8%

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

      7. remove-double-neg 49.8%

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

      8. +-commutative 49.8%

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

      9. sub-neg 49.8%

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

   7. Simplified 49.8%

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

   if -0.0189999999999999995 < phi1 < 2.14999999999999995e-4

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. associate-*l* 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in phi1 around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. sub-neg 81.6%

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

      3. neg-mul-1 81.6%

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

      4. remove-double-neg 81.6%

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

      5. mul-1-neg 81.6%

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

      6. neg-mul-1 81.6%

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

      7. distribute-neg-in 81.6%

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

      8. +-commutative 81.6%

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

      9. cos-neg 81.6%

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

      10. mul-1-neg 81.6%

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

      11. unsub-neg 81.6%

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

   7. Simplified 81.6%

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

   8. Taylor expanded in phi1 around 0 81.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.019 \lor \neg \left(\phi_1 \leq 0.000215\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 48.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{if}\;\lambda_2 \leq -8 \cdot 10^{+34}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (- (cos (- lambda2 lambda1))))))
   (if (<= lambda2 -8e+34)
     (atan2 (* (cos phi2) (- (* lambda1 (cos lambda2)) (sin lambda2))) t_0)
     (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) t_0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * -cos((lambda2 - lambda1));
	double tmp;
	if (lambda2 <= -8e+34) {
		tmp = atan2((cos(phi2) * ((lambda1 * cos(lambda2)) - sin(lambda2))), t_0);
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), 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) :: tmp
    t_0 = sin(phi1) * -cos((lambda2 - lambda1))
    if (lambda2 <= (-8d+34)) then
        tmp = atan2((cos(phi2) * ((lambda1 * cos(lambda2)) - sin(lambda2))), t_0)
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), t_0)
    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((lambda2 - lambda1));
	double tmp;
	if (lambda2 <= -8e+34) {
		tmp = Math.atan2((Math.cos(phi2) * ((lambda1 * Math.cos(lambda2)) - Math.sin(lambda2))), t_0);
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), t_0);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * -math.cos((lambda2 - lambda1))
	tmp = 0
	if lambda2 <= -8e+34:
		tmp = math.atan2((math.cos(phi2) * ((lambda1 * math.cos(lambda2)) - math.sin(lambda2))), t_0)
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), t_0)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1))))
	tmp = 0.0
	if (lambda2 <= -8e+34)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(lambda1 * cos(lambda2)) - sin(lambda2))), t_0);
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * -cos((lambda2 - lambda1));
	tmp = 0.0;
	if (lambda2 <= -8e+34)
		tmp = atan2((cos(phi2) * ((lambda1 * cos(lambda2)) - sin(lambda2))), t_0);
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]}, If[LessEqual[lambda2, -8e+34], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(lambda1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -7.99999999999999956e34

   1. Initial program 47.4%

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

      1. add-exp-log 24.2%

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

   4. Applied egg-rr 24.2%

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

   5. Taylor expanded in phi2 around 0 29.9%

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

      1. mul-1-neg 29.9%

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

      2. *-commutative 29.9%

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

      3. distribute-rgt-neg-in 29.9%

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

      4. cos-neg 29.9%

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

      5. sub-neg 29.9%

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

      6. distribute-neg-in 29.9%

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

      7. remove-double-neg 29.9%

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

      8. +-commutative 29.9%

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

      9. sub-neg 29.9%

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

   7. Simplified 29.9%

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

   8. Taylor expanded in lambda1 around 0 37.4%

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

      1. +-commutative 37.4%

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

      2. sin-neg 37.4%

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

      3. unsub-neg 37.4%

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

      4. *-commutative 37.4%

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

      5. cos-neg 37.4%

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

   10. Simplified 37.4%

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

   if -7.99999999999999956e34 < lambda2

   1. Initial program 88.7%

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

      1. add-exp-log 32.4%

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

   4. Applied egg-rr 32.4%

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

   5. Taylor expanded in phi2 around 0 54.5%

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

      1. mul-1-neg 54.5%

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

      2. *-commutative 54.5%

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

      3. distribute-rgt-neg-in 54.5%

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

      4. cos-neg 54.5%

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

      5. sub-neg 54.5%

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

      6. distribute-neg-in 54.5%

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

      7. remove-double-neg 54.5%

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

      8. +-commutative 54.5%

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

      9. sub-neg 54.5%

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

   7. Simplified 54.5%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8 \cdot 10^{+34}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 38.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -55000000000000 \lor \neg \left(\phi_1 \leq 8.5 \cdot 10^{-46}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -55000000000000.0) (not (<= phi1 8.5e-46)))
   (atan2
    (* (cos phi2) (sin lambda1))
    (* (sin phi1) (- (cos (- lambda2 lambda1)))))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (* (cos (- lambda1 lambda2)) (- phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -55000000000000.0) || !(phi1 <= 8.5e-46)) {
		tmp = atan2((cos(phi2) * sin(lambda1)), (sin(phi1) * -cos((lambda2 - lambda1))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos((lambda1 - lambda2)) * -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) :: tmp
    if ((phi1 <= (-55000000000000.0d0)) .or. (.not. (phi1 <= 8.5d-46))) then
        tmp = atan2((cos(phi2) * sin(lambda1)), (sin(phi1) * -cos((lambda2 - lambda1))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos((lambda1 - lambda2)) * -phi1))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -55000000000000.0) || !(phi1 <= 8.5e-46)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(lambda1)), (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos((lambda1 - lambda2)) * -phi1));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -55000000000000.0) or not (phi1 <= 8.5e-46):
		tmp = math.atan2((math.cos(phi2) * math.sin(lambda1)), (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos((lambda1 - lambda2)) * -phi1))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -55000000000000.0) || !(phi1 <= 8.5e-46))
		tmp = atan(Float64(cos(phi2) * sin(lambda1)), Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(Float64(lambda1 - lambda2)) * Float64(-phi1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -55000000000000.0) || ~((phi1 <= 8.5e-46)))
		tmp = atan2((cos(phi2) * sin(lambda1)), (sin(phi1) * -cos((lambda2 - lambda1))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos((lambda1 - lambda2)) * -phi1));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -55000000000000.0], N[Not[LessEqual[phi1, 8.5e-46]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * (-phi1)), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.5e13 or 8.5000000000000001e-46 < phi1

   1. Initial program 78.1%

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

      1. add-exp-log 23.7%

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

   4. Applied egg-rr 23.7%

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

   5. Taylor expanded in phi2 around 0 50.0%

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

      1. mul-1-neg 50.0%

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

      2. *-commutative 50.0%

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

      3. distribute-rgt-neg-in 50.0%

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

      4. cos-neg 50.0%

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

      5. sub-neg 50.0%

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

      6. distribute-neg-in 50.0%

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

      7. remove-double-neg 50.0%

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

      8. +-commutative 50.0%

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

      9. sub-neg 50.0%

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

   7. Simplified 50.0%

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

   8. Taylor expanded in lambda2 around 0 30.9%

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

   if -5.5e13 < phi1 < 8.5000000000000001e-46

   1. Initial program 81.3%

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

      1. add-exp-log 38.0%

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

   4. Applied egg-rr 38.0%

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

   5. Taylor expanded in phi2 around 0 48.2%

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

      1. mul-1-neg 48.2%

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

      2. *-commutative 48.2%

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

      3. distribute-rgt-neg-in 48.2%

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

      4. cos-neg 48.2%

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

      5. sub-neg 48.2%

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

      6. distribute-neg-in 48.2%

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

      7. remove-double-neg 48.2%

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

      8. +-commutative 48.2%

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

      9. sub-neg 48.2%

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

   7. Simplified 48.2%

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

   8. Taylor expanded in phi1 around 0 47.8%

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

      1. mul-1-neg 47.8%

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

      2. distribute-rgt-neg-in 47.8%

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

      3. sub-neg 47.8%

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

      4. remove-double-neg 47.8%

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

      5. distribute-neg-in 47.8%

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

      6. +-commutative 47.8%

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

      7. neg-mul-1 47.8%

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

      8. cos-neg 47.8%

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

      9. neg-mul-1 47.8%

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

      10. sub-neg 47.8%

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

   10. Simplified 47.8%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -55000000000000 \lor \neg \left(\phi_1 \leq 8.5 \cdot 10^{-46}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 45.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \lambda_1 \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_1 \cdot \left(-\cos \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= lambda1 -2.2e+16)
     (atan2 t_0 (* (cos lambda1) (- (sin phi1))))
     (atan2 t_0 (* (sin phi1) (- (cos lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (lambda1 <= -2.2e+16) {
		tmp = atan2(t_0, (cos(lambda1) * -sin(phi1)));
	} else {
		tmp = atan2(t_0, (sin(phi1) * -cos(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(phi2) * sin((lambda1 - lambda2))
    if (lambda1 <= (-2.2d+16)) then
        tmp = atan2(t_0, (cos(lambda1) * -sin(phi1)))
    else
        tmp = atan2(t_0, (sin(phi1) * -cos(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(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (lambda1 <= -2.2e+16) {
		tmp = Math.atan2(t_0, (Math.cos(lambda1) * -Math.sin(phi1)));
	} else {
		tmp = Math.atan2(t_0, (Math.sin(phi1) * -Math.cos(lambda2)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if lambda1 <= -2.2e+16:
		tmp = math.atan2(t_0, (math.cos(lambda1) * -math.sin(phi1)))
	else:
		tmp = math.atan2(t_0, (math.sin(phi1) * -math.cos(lambda2)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (lambda1 <= -2.2e+16)
		tmp = atan(t_0, Float64(cos(lambda1) * Float64(-sin(phi1))));
	else
		tmp = atan(t_0, Float64(sin(phi1) * Float64(-cos(lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (lambda1 <= -2.2e+16)
		tmp = atan2(t_0, (cos(lambda1) * -sin(phi1)));
	else
		tmp = atan2(t_0, (sin(phi1) * -cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -2.2e+16], N[ArcTan[t$95$0 / N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[lambda2], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.2e16

   1. Initial program 55.9%

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

      1. add-exp-log 1.6%

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

   4. Applied egg-rr 1.6%

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

   5. Taylor expanded in phi2 around 0 35.4%

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

      1. mul-1-neg 35.4%

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

      2. *-commutative 35.4%

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

      3. distribute-rgt-neg-in 35.4%

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

      4. cos-neg 35.4%

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

      5. sub-neg 35.4%

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

      6. distribute-neg-in 35.4%

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

      7. remove-double-neg 35.4%

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

      8. +-commutative 35.4%

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

      9. sub-neg 35.4%

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

   7. Simplified 35.4%

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

   8. Taylor expanded in lambda2 around 0 35.4%

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

      1. cos-neg 35.4%

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

   10. Simplified 35.4%

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

   if -2.2e16 < lambda1

   1. Initial program 87.6%

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

      1. add-exp-log 40.3%

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

   4. Applied egg-rr 40.3%

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

   5. Taylor expanded in phi2 around 0 53.7%

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

      1. mul-1-neg 53.7%

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

      2. *-commutative 53.7%

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

      3. distribute-rgt-neg-in 53.7%

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

      4. cos-neg 53.7%

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

      5. sub-neg 53.7%

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

      6. distribute-neg-in 53.7%

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

      7. remove-double-neg 53.7%

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

      8. +-commutative 53.7%

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

      9. sub-neg 53.7%

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

   7. Simplified 53.7%

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

   8. Taylor expanded in lambda1 around 0 52.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.2 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 48.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. add-exp-log 30.6%

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

4. Applied egg-rr 30.6%

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

5. Taylor expanded in phi2 around 0 49.1%

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

   1. mul-1-neg 49.1%

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

   2. *-commutative 49.1%

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

   3. distribute-rgt-neg-in 49.1%

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

   4. cos-neg 49.1%

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

   5. sub-neg 49.1%

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

   6. distribute-neg-in 49.1%

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

   7. remove-double-neg 49.1%

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

   8. +-commutative 49.1%

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

   9. sub-neg 49.1%

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

7. Simplified 49.1%

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

8. Final simplification 49.1%

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

Alternative 29: 43.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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(lambda1) * -sin(phi1)))
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(lambda1) * -Math.sin(phi1)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.7%

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

   1. add-exp-log 30.6%

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

4. Applied egg-rr 30.6%

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

5. Taylor expanded in phi2 around 0 49.1%

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

   1. mul-1-neg 49.1%

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

   2. *-commutative 49.1%

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

   3. distribute-rgt-neg-in 49.1%

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

   4. cos-neg 49.1%

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

   5. sub-neg 49.1%

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

   6. distribute-neg-in 49.1%

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

   7. remove-double-neg 49.1%

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

   8. +-commutative 49.1%

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

   9. sub-neg 49.1%

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

7. Simplified 49.1%

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

8. Taylor expanded in lambda2 around 0 43.1%

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

   1. cos-neg 43.1%

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

10. Simplified 43.1%

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

11. Final simplification 43.1%

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

Alternative 30: 34.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.25 \cdot 10^{+28}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \lambda_1 \cdot \left(-\phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.25e+28)
     (atan2 t_0 (* phi1 (cos (- lambda2 lambda1))))
     (atan2 t_0 (* (cos lambda1) (- phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.25e+28) {
		tmp = atan2(t_0, (phi1 * cos((lambda2 - lambda1))));
	} else {
		tmp = atan2(t_0, (cos(lambda1) * -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(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= (-2.25d+28)) then
        tmp = atan2(t_0, (phi1 * cos((lambda2 - lambda1))))
    else
        tmp = atan2(t_0, (cos(lambda1) * -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(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.25e+28) {
		tmp = Math.atan2(t_0, (phi1 * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = Math.atan2(t_0, (Math.cos(lambda1) * -phi1));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -2.25e+28:
		tmp = math.atan2(t_0, (phi1 * math.cos((lambda2 - lambda1))))
	else:
		tmp = math.atan2(t_0, (math.cos(lambda1) * -phi1))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.25e+28)
		tmp = atan(t_0, Float64(phi1 * cos(Float64(lambda2 - lambda1))));
	else
		tmp = atan(t_0, Float64(cos(lambda1) * Float64(-phi1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -2.25e+28)
		tmp = atan2(t_0, (phi1 * cos((lambda2 - lambda1))));
	else
		tmp = atan2(t_0, (cos(lambda1) * -phi1));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.25e+28], N[ArcTan[t$95$0 / N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Cos[lambda1], $MachinePrecision] * (-phi1)), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.24999999999999985e28

   1. Initial program 82.3%

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

      1. add-exp-log 24.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(e^{\log \left(\lambda_1 - \lambda_2\right)}\right)}} \]

   4. Applied egg-rr 24.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(e^{\log \left(\lambda_1 - \lambda_2\right)}\right)}} \]

   5. Taylor expanded in phi2 around 0 52.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 52.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

      2. *-commutative 52.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. distribute-rgt-neg-in 52.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. cos-neg 52.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right)} \]

      5. sub-neg 52.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)} \]

      6. distribute-neg-in 52.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\left(-\lambda_1\right) + \left(-\left(-\lambda_2\right)\right)\right)}\right)} \]

      7. remove-double-neg 52.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\left(-\lambda_1\right) + \color{blue}{\lambda_2}\right)\right)} \]

      8. +-commutative 52.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)}\right)} \]

      9. sub-neg 52.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)} \]

   7. Simplified 52.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 9.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 9.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}} \]

      2. distribute-rgt-neg-in 9.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

      3. sub-neg 9.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)}\right)} \]

      4. remove-double-neg 9.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(\color{blue}{\left(-\left(-\lambda_2\right)\right)} + \left(-\lambda_1\right)\right)\right)} \]

      5. distribute-neg-in 9.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(-\left(\left(-\lambda_2\right) + \lambda_1\right)\right)}\right)} \]

      6. +-commutative 9.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)} \]

      7. neg-mul-1 9.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(-\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right)} \]

      8. cos-neg 9.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)} \]

      9. neg-mul-1 9.1%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(\lambda_1 + \color{blue}{\left(-\lambda_2\right)}\right)\right)} \]

      10. sub-neg 9.1%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)} \]

   10. Simplified 9.1%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   11. Step-by-step derivation

       1. add-sqr-sqrt 6.5%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \color{blue}{\left(\sqrt{-\cos \left(\lambda_1 - \lambda_2\right)} \cdot \sqrt{-\cos \left(\lambda_1 - \lambda_2\right)}\right)}} \]

       2. sqrt-unprod 26.7%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \color{blue}{\sqrt{\left(-\cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}}} \]

       3. sqr-neg 26.7%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \sqrt{\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}}} \]

       4. sqrt-unprod 20.2%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \color{blue}{\left(\sqrt{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \sqrt{\cos \left(\lambda_1 - \lambda_2\right)}\right)}} \]

       5. add-sqr-sqrt 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}} \]

       6. pow1 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{{\left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}^{1}}} \]

   12. Applied egg-rr 21.9%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{{\left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}^{1}}} \]
   13. Step-by-step derivation

       1. unpow1 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

       2. sub-neg 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}} \]

       3. remove-double-neg 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \cos \left(\color{blue}{\left(-\left(-\lambda_1\right)\right)} + \left(-\lambda_2\right)\right)} \]

       4. mul-1-neg 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \cos \left(\left(-\color{blue}{-1 \cdot \lambda_1}\right) + \left(-\lambda_2\right)\right)} \]

       5. distribute-neg-in 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \cos \color{blue}{\left(-\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)}} \]

       6. +-commutative 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \cos \left(-\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)} \]

       7. cos-neg 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}} \]

       8. mul-1-neg 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \cos \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right)} \]

       9. sub-neg 21.9%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}} \]

   14. Simplified 21.9%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}} \]

   if -2.24999999999999985e28 < phi1

   1. Initial program 78.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-exp-log 32.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(e^{\log \left(\lambda_1 - \lambda_2\right)}\right)}} \]

   4. Applied egg-rr 32.5%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(e^{\log \left(\lambda_1 - \lambda_2\right)}\right)}} \]

   5. Taylor expanded in phi2 around 0 48.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. mul-1-neg 48.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

      2. *-commutative 48.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. distribute-rgt-neg-in 48.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. cos-neg 48.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right)} \]

      5. sub-neg 48.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)} \]

      6. distribute-neg-in 48.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\left(-\lambda_1\right) + \left(-\left(-\lambda_2\right)\right)\right)}\right)} \]

      7. remove-double-neg 48.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\left(-\lambda_1\right) + \color{blue}{\lambda_2}\right)\right)} \]

      8. +-commutative 48.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)}\right)} \]

      9. sub-neg 48.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)} \]

   7. Simplified 48.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 39.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}} \]
   9. Step-by-step derivation

      1. mul-1-neg 39.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}} \]

      2. distribute-rgt-neg-in 39.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

      3. sub-neg 39.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)}\right)} \]

      4. remove-double-neg 39.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(\color{blue}{\left(-\left(-\lambda_2\right)\right)} + \left(-\lambda_1\right)\right)\right)} \]

      5. distribute-neg-in 39.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(-\left(\left(-\lambda_2\right) + \lambda_1\right)\right)}\right)} \]

      6. +-commutative 39.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)} \]

      7. neg-mul-1 39.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(-\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right)} \]

      8. cos-neg 39.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)} \]

      9. neg-mul-1 39.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(\lambda_1 + \color{blue}{\left(-\lambda_2\right)}\right)\right)} \]

      10. sub-neg 39.2%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)} \]

   10. Simplified 39.2%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   11. Taylor expanded in lambda2 around 0 39.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\phi_1 \cdot \cos \lambda_1\right)}} \]
   12. Step-by-step derivation

       1. mul-1-neg 39.8%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\phi_1 \cdot \cos \lambda_1}} \]

       2. *-commutative 39.8%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\color{blue}{\cos \lambda_1 \cdot \phi_1}} \]

       3. distribute-rgt-neg-in 39.8%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \lambda_1 \cdot \left(-\phi_1\right)}} \]

   13. Simplified 39.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \lambda_1 \cdot \left(-\phi_1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.25 \cdot 10^{+28}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 \cdot \left(-\phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 31.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 \cdot \left(-\phi_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (* (cos lambda1) (- phi1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda1) * -phi1));
}

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(lambda1) * -phi1))
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(lambda1) * -phi1));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(lambda1) * -phi1))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(lambda1) * Float64(-phi1)))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda1) * -phi1));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] * (-phi1)), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 30.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(e^{\log \left(\lambda_1 - \lambda_2\right)}\right)}} \]

4. Applied egg-rr 30.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(e^{\log \left(\lambda_1 - \lambda_2\right)}\right)}} \]

5. Taylor expanded in phi2 around 0 49.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

   2. *-commutative 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. distribute-rgt-neg-in 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   4. cos-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right)} \]

   5. sub-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)} \]

   6. distribute-neg-in 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\left(-\lambda_1\right) + \left(-\left(-\lambda_2\right)\right)\right)}\right)} \]

   7. remove-double-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\left(-\lambda_1\right) + \color{blue}{\lambda_2}\right)\right)} \]

   8. +-commutative 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)}\right)} \]

   9. sub-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)} \]

7. Simplified 49.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

8. Taylor expanded in phi1 around 0 31.8%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}} \]
9. Step-by-step derivation

   1. mul-1-neg 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}} \]

   2. distribute-rgt-neg-in 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

   3. sub-neg 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)}\right)} \]

   4. remove-double-neg 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(\color{blue}{\left(-\left(-\lambda_2\right)\right)} + \left(-\lambda_1\right)\right)\right)} \]

   5. distribute-neg-in 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(-\left(\left(-\lambda_2\right) + \lambda_1\right)\right)}\right)} \]

   6. +-commutative 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)} \]

   7. neg-mul-1 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(-\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right)} \]

   8. cos-neg 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)} \]

   9. neg-mul-1 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(\lambda_1 + \color{blue}{\left(-\lambda_2\right)}\right)\right)} \]

   10. sub-neg 31.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)} \]

10. Simplified 31.8%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

11. Taylor expanded in lambda2 around 0 31.5%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\phi_1 \cdot \cos \lambda_1\right)}} \]
12. Step-by-step derivation

    1. mul-1-neg 31.5%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\phi_1 \cdot \cos \lambda_1}} \]

    2. *-commutative 31.5%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\color{blue}{\cos \lambda_1 \cdot \phi_1}} \]

    3. distribute-rgt-neg-in 31.5%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \lambda_1 \cdot \left(-\phi_1\right)}} \]

13. Simplified 31.5%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\cos \lambda_1 \cdot \left(-\phi_1\right)}} \]

14. Final simplification 31.5%

    \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 \cdot \left(-\phi_1\right)} \]
15. Add Preprocessing

Alternative 32: 28.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\phi_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (sin (- lambda1 lambda2)) (* (cos (- lambda1 lambda2)) (- phi1))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin((lambda1 - lambda2)), (cos((lambda1 - lambda2)) * -phi1));
}

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((lambda1 - lambda2)) * -phi1))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos((lambda1 - lambda2)) * -phi1));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2(math.sin((lambda1 - lambda2)), (math.cos((lambda1 - lambda2)) * -phi1))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(Float64(lambda1 - lambda2)), Float64(cos(Float64(lambda1 - lambda2)) * Float64(-phi1)))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2(sin((lambda1 - lambda2)), (cos((lambda1 - lambda2)) * -phi1));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * (-phi1)), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 79.7%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-exp-log 30.6%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(e^{\log \left(\lambda_1 - \lambda_2\right)}\right)}} \]

4. Applied egg-rr 30.6%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(e^{\log \left(\lambda_1 - \lambda_2\right)}\right)}} \]

5. Taylor expanded in phi2 around 0 49.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
6. Step-by-step derivation

   1. mul-1-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1}} \]

   2. *-commutative 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-\color{blue}{\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   3. distribute-rgt-neg-in 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   4. cos-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right)} \]

   5. sub-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)} \]

   6. distribute-neg-in 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\left(-\lambda_1\right) + \left(-\left(-\lambda_2\right)\right)\right)}\right)} \]

   7. remove-double-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\left(-\lambda_1\right) + \color{blue}{\lambda_2}\right)\right)} \]

   8. +-commutative 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)}\right)} \]

   9. sub-neg 49.1%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)} \]

7. Simplified 49.1%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

8. Taylor expanded in phi1 around 0 31.8%

   \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}} \]
9. Step-by-step derivation

   1. mul-1-neg 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}} \]

   2. distribute-rgt-neg-in 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

   3. sub-neg 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 + \left(-\lambda_1\right)\right)}\right)} \]

   4. remove-double-neg 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(\color{blue}{\left(-\left(-\lambda_2\right)\right)} + \left(-\lambda_1\right)\right)\right)} \]

   5. distribute-neg-in 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(-\left(\left(-\lambda_2\right) + \lambda_1\right)\right)}\right)} \]

   6. +-commutative 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right)} \]

   7. neg-mul-1 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(-\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right)} \]

   8. cos-neg 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}\right)} \]

   9. neg-mul-1 31.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \left(\lambda_1 + \color{blue}{\left(-\lambda_2\right)}\right)\right)} \]

   10. sub-neg 31.8%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right)} \]

10. Simplified 31.8%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

11. Taylor expanded in phi2 around 0 28.9%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{1}}{\phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)} \]

12. Final simplification 28.9%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\phi_1\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(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))))))