Bearing on a great circle

Percentage Accurate: 79.5% → 99.7%

Time: 33.6s

Alternatives: 34

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. *-commutative 81.8%

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

   2. associate-*l* 81.8%

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

3. Simplified 81.8%

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

   1. sin-diff 91.7%

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

   2. sub-neg 91.7%

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

6. Applied egg-rr 91.7%

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

   1. fma-define 91.8%

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

   2. *-commutative 91.8%

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

   3. distribute-lft-neg-in 91.8%

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

8. Simplified 91.8%

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

   1. associate-*r* 91.8%

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

   2. cos-diff 99.7%

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

   3. distribute-lft-in 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. distribute-lft-out 99.7%

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

    2. associate-*l* 99.7%

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

    3. +-commutative 99.7%

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

    4. fma-define 99.7%

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

12. Simplified 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. *-commutative 81.8%

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

   2. associate-*l* 81.8%

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

3. Simplified 81.8%

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

   1. sin-diff 91.7%

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

   2. sub-neg 91.7%

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

6. Applied egg-rr 91.7%

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

   1. fma-define 91.8%

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

   2. *-commutative 91.8%

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

   3. distribute-lft-neg-in 91.8%

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

8. Simplified 91.8%

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

   1. cos-diff 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), ((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[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $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[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 81.8%

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

   1. *-commutative 81.8%

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

   2. associate-*l* 81.8%

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

3. Simplified 81.8%

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

   1. cos-diff 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

6. Applied egg-rr 82.1%

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

   1. sin-diff 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 89.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.8%

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

   1. *-commutative 81.8%

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

   2. associate-*l* 81.8%

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

3. Simplified 81.8%

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

   1. sin-diff 91.7%

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

   2. sub-neg 91.7%

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

6. Applied egg-rr 91.7%

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

   1. fma-define 91.8%

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

   2. *-commutative 91.8%

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

   3. distribute-lft-neg-in 91.8%

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

8. Simplified 91.8%

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

Alternative 5: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -5.1 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 3 \cdot 10^{-19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{t\_0 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= phi1 -5.1e-5) (not (<= phi1 3e-19)))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (-
       t_0
       (*
        (cos phi2)
        (*
         (sin phi1)
         (+
          (* (cos lambda2) (cos lambda1))
          (* (sin lambda1) (sin lambda2)))))))
     (atan2
      (*
       (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
       (cos phi2))
      (- t_0 (* (cos (- lambda2 lambda1)) (* (cos phi2) phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((phi1 <= -5.1e-5) || !(phi1 <= 3e-19)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), (t_0 - (cos((lambda2 - lambda1)) * (cos(phi2) * phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((phi1 <= -5.1e-5) || !(phi1 <= 3e-19))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2)))))));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), Float64(t_0 - Float64(cos(Float64(lambda2 - lambda1)) * Float64(cos(phi2) * phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -5.1e-5], N[Not[LessEqual[phi1, 3e-19]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.09999999999999996e-5 or 2.99999999999999993e-19 < phi1

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*l* 81.4%

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

   3. Simplified 81.4%

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

      1. cos-diff 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

   6. Applied egg-rr 82.0%

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

   if -5.09999999999999996e-5 < phi1 < 2.99999999999999993e-19

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*l* 82.2%

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

   3. Simplified 82.2%

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

      1. sin-diff 99.2%

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

      2. sub-neg 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. fma-define 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-lft-neg-in 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in phi1 around 0 99.3%

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

       1. associate-*r* 99.3%

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

       2. *-commutative 99.3%

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

       3. sub-neg 99.3%

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

       4. neg-mul-1 99.3%

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

       5. remove-double-neg 99.3%

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

       6. mul-1-neg 99.3%

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

       7. neg-mul-1 99.3%

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

       8. distribute-neg-in 99.3%

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

       9. +-commutative 99.3%

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

       10. cos-neg 99.3%

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

       11. mul-1-neg 99.3%

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

       12. unsub-neg 99.3%

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

   11. Simplified 99.3%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.1 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 3 \cdot 10^{-19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{t\_0 - t\_1 \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t\_1\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (cos (- lambda2 lambda1)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -8.2e-5)
     (atan2
      t_2
      (-
       t_0
       (*
        (cos phi2)
        (* (sin phi1) (log1p (expm1 (cos (- lambda1 lambda2))))))))
     (if (<= phi1 3e-19)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
         (cos phi2))
        (- t_0 (* t_1 (* (cos phi2) phi1))))
       (atan2
        t_2
        (- t_0 (* (cos phi2) (* (sin phi1) (expm1 (log1p t_1))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos((lambda2 - lambda1));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -8.2e-5) {
		tmp = atan2(t_2, (t_0 - (cos(phi2) * (sin(phi1) * log1p(expm1(cos((lambda1 - lambda2))))))));
	} else if (phi1 <= 3e-19) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), (t_0 - (t_1 * (cos(phi2) * phi1))));
	} else {
		tmp = atan2(t_2, (t_0 - (cos(phi2) * (sin(phi1) * expm1(log1p(t_1))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda2 - lambda1))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -8.2e-5)
		tmp = atan(t_2, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * log1p(expm1(cos(Float64(lambda1 - lambda2))))))));
	elseif (phi1 <= 3e-19)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), Float64(t_0 - Float64(t_1 * Float64(cos(phi2) * phi1))));
	else
		tmp = atan(t_2, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * expm1(log1p(t_1))))));
	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[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8.2e-5], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Log[1 + N[(Exp[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3e-19], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(Exp[N[Log[1 + t$95$1], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -8.20000000000000009e-5

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l* 74.1%

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

   3. Simplified 74.1%

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

      1. log1p-expm1-u 74.2%

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

   6. Applied egg-rr 74.2%

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

   if -8.20000000000000009e-5 < phi1 < 2.99999999999999993e-19

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*l* 82.2%

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

   3. Simplified 82.2%

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

      1. sin-diff 99.2%

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

      2. sub-neg 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. fma-define 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-lft-neg-in 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in phi1 around 0 99.3%

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

       1. associate-*r* 99.3%

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

       2. *-commutative 99.3%

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

       3. sub-neg 99.3%

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

       4. neg-mul-1 99.3%

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

       5. remove-double-neg 99.3%

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

       6. mul-1-neg 99.3%

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

       7. neg-mul-1 99.3%

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

       8. distribute-neg-in 99.3%

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

       9. +-commutative 99.3%

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

       10. cos-neg 99.3%

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

       11. mul-1-neg 99.3%

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

       12. unsub-neg 99.3%

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

   11. Simplified 99.3%

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

   if 2.99999999999999993e-19 < phi1

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-*l* 89.5%

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

   3. Simplified 89.5%

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

      1. expm1-log1p-u 89.5%

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

      2. expm1-undefine 89.5%

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

   6. Applied egg-rr 89.5%

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

      1. expm1-define 89.5%

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

      2. sub-neg 89.5%

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

      3. neg-mul-1 89.5%

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

      4. remove-double-neg 89.5%

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

      5. mul-1-neg 89.5%

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

      6. neg-mul-1 89.5%

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

      7. distribute-neg-in 89.5%

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

      8. +-commutative 89.5%

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

      9. cos-neg 89.5%

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

      10. mul-1-neg 89.5%

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

      11. unsub-neg 89.5%

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

   8. Simplified 89.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_2 - \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t\_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{t\_0 - \sin \phi_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (cos (- lambda1 lambda2))))
   (if (<= phi1 -3.1e-5)
     (atan2 t_1 (- t_0 (* (cos phi2) (* (sin phi1) (log1p (expm1 t_2))))))
     (if (<= phi1 3e-19)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
         (cos phi2))
        (- t_0 (* (sin phi1) t_2)))
       (atan2
        t_1
        (-
         t_0
         (*
          (cos phi2)
          (* (sin phi1) (expm1 (log1p (cos (- lambda2 lambda1))))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double t_2 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -3.1e-5) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * log1p(expm1(t_2))))));
	} else if (phi1 <= 3e-19) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), (t_0 - (sin(phi1) * t_2)));
	} else {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * expm1(log1p(cos((lambda2 - lambda1))))))));
	}
	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 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -3.1e-5)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * log1p(expm1(t_2))))));
	elseif (phi1 <= 3e-19)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * t_2)));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * expm1(log1p(cos(Float64(lambda2 - lambda1))))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.1e-5], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Log[1 + N[(Exp[t$95$2] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3e-19], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(Exp[N[Log[1 + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.10000000000000014e-5

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l* 74.1%

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

   3. Simplified 74.1%

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

      1. log1p-expm1-u 74.2%

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

   6. Applied egg-rr 74.2%

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

   if -3.10000000000000014e-5 < phi1 < 2.99999999999999993e-19

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*l* 82.2%

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

   3. Simplified 82.2%

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

      1. sin-diff 99.2%

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

      2. sub-neg 99.2%

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

   6. Applied egg-rr 99.2%

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

      1. fma-define 99.3%

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

      2. *-commutative 99.3%

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

      3. distribute-lft-neg-in 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in phi2 around 0 99.3%

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

   if 2.99999999999999993e-19 < phi1

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-*l* 89.5%

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

   3. Simplified 89.5%

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

      1. expm1-log1p-u 89.5%

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

      2. expm1-undefine 89.5%

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

   6. Applied egg-rr 89.5%

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

      1. expm1-define 89.5%

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

      2. sub-neg 89.5%

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

      3. neg-mul-1 89.5%

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

      4. remove-double-neg 89.5%

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

      5. mul-1-neg 89.5%

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

      6. neg-mul-1 89.5%

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

      7. distribute-neg-in 89.5%

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

      8. +-commutative 89.5%

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

      9. cos-neg 89.5%

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

      10. mul-1-neg 89.5%

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

      11. unsub-neg 89.5%

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

   8. Simplified 89.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (cos phi2)
   (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
  (-
   (* (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)) - (sin(lambda2) * cos(lambda1)))), ((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)) - (sin(lambda2) * cos(lambda1)))), ((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.sin(lambda2) * Math.cos(lambda1)))), ((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.sin(lambda2) * math.cos(lambda1)))), ((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(sin(lambda2) * cos(lambda1)))), 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)) - (sin(lambda2) * cos(lambda1)))), ((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[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $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 81.8%

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

   1. *-commutative 81.8%

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

   2. associate-*l* 81.8%

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

3. Simplified 81.8%

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

   1. sin-diff 99.7%

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

6. Applied egg-rr 91.7%

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

7. Final simplification 91.7%

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

Alternative 9: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t\_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t\_0 - \phi_1 \cdot \left(\cos \phi_2 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (cos (- lambda1 lambda2))))
   (if (<= phi1 -3.1e-5)
     (atan2 t_1 (- t_0 (* (cos phi2) (* (sin phi1) (log1p (expm1 t_2))))))
     (if (<= phi1 3e-19)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
        (- t_0 (* phi1 (* (cos phi2) t_2))))
       (atan2
        t_1
        (-
         t_0
         (*
          (cos phi2)
          (* (sin phi1) (expm1 (log1p (cos (- lambda2 lambda1))))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double t_2 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -3.1e-5) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * log1p(expm1(t_2))))));
	} else if (phi1 <= 3e-19) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (t_0 - (phi1 * (cos(phi2) * t_2))));
	} else {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * expm1(log1p(cos((lambda2 - lambda1))))))));
	}
	return tmp;
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double t_2 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -3.1e-5) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.log1p(Math.expm1(t_2))))));
	} else if (phi1 <= 3e-19) {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1)))), (t_0 - (phi1 * (Math.cos(phi2) * t_2))));
	} else {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.expm1(Math.log1p(Math.cos((lambda2 - lambda1))))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	t_2 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -3.1e-5:
		tmp = math.atan2(t_1, (t_0 - (math.cos(phi2) * (math.sin(phi1) * math.log1p(math.expm1(t_2))))))
	elif phi1 <= 3e-19:
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1)))), (t_0 - (phi1 * (math.cos(phi2) * t_2))))
	else:
		tmp = math.atan2(t_1, (t_0 - (math.cos(phi2) * (math.sin(phi1) * math.expm1(math.log1p(math.cos((lambda2 - lambda1))))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_2 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -3.1e-5)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * log1p(expm1(t_2))))));
	elseif (phi1 <= 3e-19)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(t_0 - Float64(phi1 * Float64(cos(phi2) * t_2))));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * expm1(log1p(cos(Float64(lambda2 - lambda1))))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.1e-5], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Log[1 + N[(Exp[t$95$2] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3e-19], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(phi1 * N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(Exp[N[Log[1 + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.10000000000000014e-5

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l* 74.1%

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

   3. Simplified 74.1%

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

      1. log1p-expm1-u 74.2%

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

   6. Applied egg-rr 74.2%

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

   if -3.10000000000000014e-5 < phi1 < 2.99999999999999993e-19

   1. Initial program 82.2%

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

      1. *-commutative 82.2%

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

      2. associate-*l* 82.2%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in phi1 around 0 82.2%

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

      1. *-commutative 82.2%

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

   7. Simplified 82.2%

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

      1. sin-diff 99.8%

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

   9. Applied egg-rr 99.2%

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

   if 2.99999999999999993e-19 < phi1

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-*l* 89.5%

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

   3. Simplified 89.5%

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

      1. expm1-log1p-u 89.5%

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

      2. expm1-undefine 89.5%

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

   6. Applied egg-rr 89.5%

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

      1. expm1-define 89.5%

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

      2. sub-neg 89.5%

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

      3. neg-mul-1 89.5%

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

      4. remove-double-neg 89.5%

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

      5. mul-1-neg 89.5%

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

      6. neg-mul-1 89.5%

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

      7. distribute-neg-in 89.5%

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

      8. +-commutative 89.5%

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

      9. cos-neg 89.5%

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

      10. mul-1-neg 89.5%

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

      11. unsub-neg 89.5%

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

   8. Simplified 89.5%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-19}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 9.5 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.65e-17)
     (atan2
      t_1
      (-
       t_0
       (*
        (cos phi2)
        (* (sin phi1) (log1p (expm1 (cos (- lambda1 lambda2))))))))
     (if (<= phi1 9.5e-47)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
         (cos phi2))
        (sin phi2))
       (atan2
        t_1
        (-
         t_0
         (*
          (cos phi2)
          (* (sin phi1) (expm1 (log1p (cos (- lambda2 lambda1))))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.65e-17) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * log1p(expm1(cos((lambda1 - lambda2))))))));
	} else if (phi1 <= 9.5e-47) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * expm1(log1p(cos((lambda2 - lambda1))))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.65e-17)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * log1p(expm1(cos(Float64(lambda1 - lambda2))))))));
	elseif (phi1 <= 9.5e-47)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * expm1(log1p(cos(Float64(lambda2 - lambda1))))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.65e-17], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Log[1 + N[(Exp[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 9.5e-47], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(Exp[N[Log[1 + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.6499999999999999e-17

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*l* 72.8%

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

   3. Simplified 72.8%

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

      1. log1p-expm1-u 72.8%

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

   6. Applied egg-rr 72.8%

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

   if -2.6499999999999999e-17 < phi1 < 9.4999999999999991e-47

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*l* 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in lambda1 around 0 83.0%

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

      1. cancel-sign-sub-inv 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*l* 83.0%

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

      4. fma-undefine 83.0%

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

      5. *-commutative 83.0%

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

      6. *-commutative 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in phi1 around 0 80.9%

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

      1. sin-diff 99.8%

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

      2. sub-neg 99.8%

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

   10. Applied egg-rr 97.7%

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

       1. fma-define 99.8%

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

       2. *-commutative 99.8%

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

       3. distribute-lft-neg-in 99.8%

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

   12. Simplified 97.7%

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

   if 9.4999999999999991e-47 < phi1

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-*l* 89.3%

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

   3. Simplified 89.3%

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

      1. expm1-log1p-u 89.3%

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

      2. expm1-undefine 89.3%

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

   6. Applied egg-rr 89.3%

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

      1. expm1-define 89.3%

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

      2. sub-neg 89.3%

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

      3. neg-mul-1 89.3%

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

      4. remove-double-neg 89.3%

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

      5. mul-1-neg 89.3%

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

      6. neg-mul-1 89.3%

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

      7. distribute-neg-in 89.3%

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

      8. +-commutative 89.3%

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

      9. cos-neg 89.3%

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

      10. mul-1-neg 89.3%

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

      11. unsub-neg 89.3%

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

   8. Simplified 89.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.65 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 9.5 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.9 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 5.2 \cdot 10^{-55}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -1.9e-17)
     (atan2
      t_1
      (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (if (<= phi1 5.2e-55)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
         (cos phi2))
        (sin phi2))
       (atan2
        t_1
        (-
         t_0
         (*
          (cos phi2)
          (* (sin phi1) (expm1 (log1p (cos (- lambda2 lambda1))))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.9e-17) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else if (phi1 <= 5.2e-55) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (sin(phi1) * expm1(log1p(cos((lambda2 - lambda1))))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -1.9e-17)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	elseif (phi1 <= 5.2e-55)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * expm1(log1p(cos(Float64(lambda2 - lambda1))))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.9e-17], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 5.2e-55], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(Exp[N[Log[1 + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.9000000000000001e-17

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*l* 72.8%

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

   3. Simplified 72.8%

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

   if -1.9000000000000001e-17 < phi1 < 5.1999999999999998e-55

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*l* 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in lambda1 around 0 83.0%

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

      1. cancel-sign-sub-inv 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*l* 83.0%

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

      4. fma-undefine 83.0%

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

      5. *-commutative 83.0%

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

      6. *-commutative 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in phi1 around 0 80.9%

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

      1. sin-diff 99.8%

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

      2. sub-neg 99.8%

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

   10. Applied egg-rr 97.7%

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

       1. fma-define 99.8%

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

       2. *-commutative 99.8%

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

       3. distribute-lft-neg-in 99.8%

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

   12. Simplified 97.7%

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

   if 5.1999999999999998e-55 < phi1

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-*l* 89.3%

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

   3. Simplified 89.3%

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

      1. expm1-log1p-u 89.3%

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

      2. expm1-undefine 89.3%

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

   6. Applied egg-rr 89.3%

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

      1. expm1-define 89.3%

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

      2. sub-neg 89.3%

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

      3. neg-mul-1 89.3%

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

      4. remove-double-neg 89.3%

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

      5. mul-1-neg 89.3%

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

      6. neg-mul-1 89.3%

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

      7. distribute-neg-in 89.3%

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

      8. +-commutative 89.3%

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

      9. cos-neg 89.3%

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

      10. mul-1-neg 89.3%

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

      11. unsub-neg 89.3%

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

   8. Simplified 89.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.9 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 5.2 \cdot 10^{-55}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t\_0\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.1 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(-t\_0\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.5e-17)
     (atan2
      t_1
      (- (* (cos phi1) (sin phi2)) (* (cos phi2) (* (sin phi1) t_0))))
     (if (<= phi1 1.1e-46)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
         (cos phi2))
        (sin phi2))
       (atan2
        t_1
        (fma (sin phi2) (cos phi1) (* (cos phi2) (* (sin phi1) (- t_0)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.5e-17) {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * t_0))));
	} else if (phi1 <= 1.1e-46) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_1, fma(sin(phi2), cos(phi1), (cos(phi2) * (sin(phi1) * -t_0))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.5e-17)
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(sin(phi1) * t_0))));
	elseif (phi1 <= 1.1e-46)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_1, fma(sin(phi2), cos(phi1), Float64(cos(phi2) * Float64(sin(phi1) * Float64(-t_0)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.5e-17], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.1e-46], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * (-t$95$0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.4999999999999999e-17

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*l* 72.8%

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

   3. Simplified 72.8%

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

   if -2.4999999999999999e-17 < phi1 < 1.1e-46

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*l* 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in lambda1 around 0 83.0%

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

      1. cancel-sign-sub-inv 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*l* 83.0%

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

      4. fma-undefine 83.0%

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

      5. *-commutative 83.0%

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

      6. *-commutative 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in phi1 around 0 80.9%

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

      1. sin-diff 99.8%

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

      2. sub-neg 99.8%

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

   10. Applied egg-rr 97.7%

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

       1. fma-define 99.8%

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

       2. *-commutative 99.8%

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

       3. distribute-lft-neg-in 99.8%

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

   12. Simplified 97.7%

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

   if 1.1e-46 < phi1

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-*l* 89.3%

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

   3. Simplified 89.3%

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

      1. log1p-expm1-u 89.2%

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

   6. Applied egg-rr 89.2%

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

      1. cancel-sign-sub-inv 89.2%

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

      2. *-commutative 89.2%

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

      3. log1p-expm1-u 89.3%

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

      4. fma-define 89.3%

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

      5. *-commutative 89.3%

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

   8. Applied egg-rr 89.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.1 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\sin \phi_2, \cos \phi_1, \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot t\_0\right)}\\ \mathbf{elif}\;\phi_1 \leq 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, t\_0 \cdot \left(-\cos \phi_2 \cdot \sin \phi_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2e-17)
     (atan2
      t_1
      (- (* (cos phi1) (sin phi2)) (* (cos phi2) (* (sin phi1) t_0))))
     (if (<= phi1 1e-47)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
         (cos phi2))
        (sin phi2))
       (atan2
        t_1
        (fma (cos phi1) (sin phi2) (* t_0 (- (* (cos phi2) (sin phi1))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2e-17) {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * t_0))));
	} else if (phi1 <= 1e-47) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_1, fma(cos(phi1), sin(phi2), (t_0 * -(cos(phi2) * sin(phi1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2e-17)
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(sin(phi1) * t_0))));
	elseif (phi1 <= 1e-47)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_1, fma(cos(phi1), sin(phi2), Float64(t_0 * Float64(-Float64(cos(phi2) * sin(phi1))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2e-17], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1e-47], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(t$95$0 * (-N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.00000000000000014e-17

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*l* 72.8%

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

   3. Simplified 72.8%

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

   if -2.00000000000000014e-17 < phi1 < 9.9999999999999997e-48

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*l* 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in lambda1 around 0 83.0%

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

      1. cancel-sign-sub-inv 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*l* 83.0%

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

      4. fma-undefine 83.0%

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

      5. *-commutative 83.0%

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

      6. *-commutative 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in phi1 around 0 80.9%

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

      1. sin-diff 99.8%

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

      2. sub-neg 99.8%

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

   10. Applied egg-rr 97.7%

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

       1. fma-define 99.8%

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

       2. *-commutative 99.8%

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

       3. distribute-lft-neg-in 99.8%

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

   12. Simplified 97.7%

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

   if 9.9999999999999997e-48 < phi1

   1. Initial program 89.3%

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

      1. *-commutative 89.3%

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

      2. associate-*l* 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in lambda1 around 0 89.3%

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

      1. cancel-sign-sub-inv 89.3%

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

      2. *-commutative 89.3%

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

      3. associate-*l* 89.3%

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

      4. fma-undefine 89.3%

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

      5. *-commutative 89.3%

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

      6. *-commutative 89.3%

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

   7. Simplified 89.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\cos \phi_2 \cdot \sin \phi_1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 4.6 \cdot 10^{-49}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -1.3e-17) (not (<= phi1 4.6e-49)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
   (atan2
    (*
     (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
     (cos phi2))
    (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -1.3e-17) || !(phi1 <= 4.6e-49)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -1.3e-17) || !(phi1 <= 4.6e-49))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -1.3e-17], N[Not[LessEqual[phi1, 4.6e-49]], $MachinePrecision]], 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], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.30000000000000002e-17 or 4.5999999999999998e-49 < phi1

   1. Initial program 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-*l* 80.8%

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

   3. Simplified 80.8%

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

   if -1.30000000000000002e-17 < phi1 < 4.5999999999999998e-49

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*l* 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in lambda1 around 0 83.0%

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

      1. cancel-sign-sub-inv 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*l* 83.0%

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

      4. fma-undefine 83.0%

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

      5. *-commutative 83.0%

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

      6. *-commutative 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in phi1 around 0 80.9%

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

      1. sin-diff 99.8%

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

      2. sub-neg 99.8%

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

   10. Applied egg-rr 97.7%

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

       1. fma-define 99.8%

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

       2. *-commutative 99.8%

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

       3. distribute-lft-neg-in 99.8%

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

   12. Simplified 97.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 4.6 \cdot 10^{-49}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.9 \cdot 10^{+16} \lor \neg \left(\lambda_2 \leq 265000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -1.9e+16) (not (<= lambda2 265000.0)))
   (atan2
    (*
     (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
     (cos phi2))
    (sin phi2))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (cos phi2) (* (cos lambda1) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -1.9e+16) || !(lambda2 <= 265000.0)) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -1.9e+16) || !(lambda2 <= 265000.0))
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(cos(lambda1) * sin(phi1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -1.9e+16], N[Not[LessEqual[lambda2, 265000.0]], $MachinePrecision]], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], 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[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.9e16 or 265000 < lambda2

   1. Initial program 64.3%

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

      1. *-commutative 64.3%

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

      2. associate-*l* 64.3%

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

   3. Simplified 64.3%

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

   5. Taylor expanded in lambda1 around 0 64.3%

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

      1. cancel-sign-sub-inv 64.3%

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

      2. *-commutative 64.3%

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

      3. associate-*l* 64.3%

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

      4. fma-undefine 64.3%

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

      5. *-commutative 64.3%

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

      6. *-commutative 64.3%

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

   7. Simplified 64.3%

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

   8. Taylor expanded in phi1 around 0 47.2%

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

      1. sin-diff 84.8%

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

      2. sub-neg 84.8%

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

   10. Applied egg-rr 67.4%

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

       1. fma-define 84.9%

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

       2. *-commutative 84.9%

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

       3. distribute-lft-neg-in 84.9%

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

   12. Simplified 67.4%

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

   if -1.9e16 < lambda2 < 265000

   1. Initial program 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*l* 98.1%

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

   3. Simplified 98.1%

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

      1. cos-diff 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in lambda2 around 0 97.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.9 \cdot 10^{+16} \lor \neg \left(\lambda_2 \leq 265000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -0.00082:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 265000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= lambda2 -0.00082)
     (atan2 t_1 (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1)))))
     (if (<= lambda2 265000.0)
       (atan2 t_1 (- t_0 (* (cos phi2) (* (cos lambda1) (sin phi1)))))
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
         (cos phi2))
        (sin phi2))))))

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 (lambda2 <= -0.00082) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	} else if (lambda2 <= 265000.0) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	}
	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 (lambda2 <= -0.00082)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	elseif (lambda2 <= 265000.0)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda1) * sin(phi1)))));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), sin(phi2));
	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[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -0.00082], 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, 265000.0], 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[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -8.1999999999999998e-4

   1. Initial program 69.6%

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

      1. *-commutative 69.6%

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

      2. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in lambda1 around 0 69.7%

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

      1. *-commutative 69.7%

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

      2. cos-neg 69.7%

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

   7. Simplified 69.7%

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

   if -8.1999999999999998e-4 < lambda2 < 265000

   1. Initial program 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*l* 98.7%

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

   3. Simplified 98.7%

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

      1. cos-diff 99.6%

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

      2. +-commutative 99.6%

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

      3. *-commutative 99.6%

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

   6. Applied egg-rr 98.7%

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

   7. Taylor expanded in lambda2 around 0 98.3%

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

   if 265000 < lambda2

   1. Initial program 58.9%

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

      1. *-commutative 58.9%

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

      2. associate-*l* 58.9%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in lambda1 around 0 58.9%

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

      1. cancel-sign-sub-inv 58.9%

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

      2. *-commutative 58.9%

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

      3. associate-*l* 58.9%

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

      4. fma-undefine 58.9%

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

      5. *-commutative 58.9%

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

      6. *-commutative 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in phi1 around 0 40.0%

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

      1. sin-diff 84.5%

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

      2. sub-neg 84.5%

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

   10. Applied egg-rr 65.5%

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

       1. fma-define 84.5%

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

       2. *-commutative 84.5%

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

       3. distribute-lft-neg-in 84.5%

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

   12. Simplified 65.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.00082:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_2 \leq 265000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_2 \leq -2.7 \cdot 10^{-137}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t\_0 - t\_1}\\ \mathbf{elif}\;\lambda_2 \leq 4.1 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_0 - \cos \phi_2 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (sin phi1) (cos (- lambda1 lambda2)))))
   (if (<= lambda2 -2.7e-137)
     (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (- t_0 t_1))
     (if (<= lambda2 4.1e-17)
       (atan2 (* (sin lambda1) (cos phi2)) (- t_0 (* (cos phi2) t_1)))
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
         (cos phi2))
        (sin phi2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin(phi1) * cos((lambda1 - lambda2));
	double tmp;
	if (lambda2 <= -2.7e-137) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - t_1));
	} else if (lambda2 <= 4.1e-17) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * t_1)));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (lambda2 <= -2.7e-137)
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - t_1));
	elseif (lambda2 <= 4.1e-17)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(cos(phi2) * t_1)));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -2.7e-137], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 4.1e-17], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -2.69999999999999993e-137

   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. Step-by-step derivation

      1. *-commutative 78.8%

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

      2. associate-*l* 78.8%

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

   3. Simplified 78.8%

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

      1. log1p-expm1-u 78.8%

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

   6. Applied egg-rr 78.8%

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

   7. Taylor expanded in phi2 around 0 70.8%

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

   if -2.69999999999999993e-137 < lambda2 < 4.1000000000000001e-17

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda2 around 0 93.5%

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

   if 4.1000000000000001e-17 < lambda2

   1. Initial program 61.0%

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

      1. *-commutative 61.0%

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

      2. associate-*l* 61.0%

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

   3. Simplified 61.0%

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

   5. Taylor expanded in lambda1 around 0 61.0%

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

      1. cancel-sign-sub-inv 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-*l* 61.0%

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

      4. fma-undefine 61.0%

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

      5. *-commutative 61.0%

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

      6. *-commutative 61.0%

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

   7. Simplified 61.0%

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

   8. Taylor expanded in phi1 around 0 42.4%

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

      1. sin-diff 85.1%

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

      2. sub-neg 85.1%

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

   10. Applied egg-rr 66.3%

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

       1. fma-define 85.2%

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

       2. *-commutative 85.2%

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

       3. distribute-lft-neg-in 85.2%

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

   12. Simplified 66.4%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.7 \cdot 10^{-137}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 4.1 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.4 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 9 \cdot 10^{-49}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -5.4e-17) (not (<= phi1 9e-49)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (- (* (cos phi1) (sin phi2)) (* (sin phi1) (cos (- lambda1 lambda2)))))
   (atan2
    (*
     (fma (sin lambda1) (cos lambda2) (* (- (sin lambda2)) (cos lambda1)))
     (cos phi2))
    (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -5.4e-17) || !(phi1 <= 9e-49)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (-sin(lambda2) * cos(lambda1))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -5.4e-17) || !(phi1 <= 9e-49))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(Float64(-sin(lambda2)) * cos(lambda1))) * cos(phi2)), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -5.4e-17], N[Not[LessEqual[phi1, 9e-49]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[((-N[Sin[lambda2], $MachinePrecision]) * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.4000000000000002e-17 or 9.0000000000000004e-49 < phi1

   1. Initial program 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-*l* 80.8%

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

   3. Simplified 80.8%

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

      1. log1p-expm1-u 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in phi2 around 0 59.1%

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

   if -5.4000000000000002e-17 < phi1 < 9.0000000000000004e-49

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*l* 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in lambda1 around 0 83.0%

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

      1. cancel-sign-sub-inv 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*l* 83.0%

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

      4. fma-undefine 83.0%

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

      5. *-commutative 83.0%

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

      6. *-commutative 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in phi1 around 0 80.9%

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

      1. sin-diff 99.8%

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

      2. sub-neg 99.8%

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

   10. Applied egg-rr 97.7%

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

       1. fma-define 99.8%

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

       2. *-commutative 99.8%

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

       3. distribute-lft-neg-in 99.8%

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

   12. Simplified 97.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.4 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 9 \cdot 10^{-49}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \left(-\sin \lambda_2\right) \cdot \cos \lambda_1\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 73.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 1.2 \cdot 10^{-49}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -8.5e-17) (not (<= phi1 1.2e-49)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (- (* (cos phi1) (sin phi2)) (* (sin phi1) (cos (- lambda1 lambda2)))))
   (atan2
    (*
     (cos phi2)
     (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
    (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -8.5e-17) || !(phi1 <= 1.2e-49)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2));
	}
	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 <= (-8.5d-17)) .or. (.not. (phi1 <= 1.2d-49))) then
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))))
    else
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -8.5e-17) || !(phi1 <= 1.2e-49)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), ((Math.cos(phi1) * Math.sin(phi2)) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1)))), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -8.5e-17) or not (phi1 <= 1.2e-49):
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), ((math.cos(phi1) * math.sin(phi2)) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1)))), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -8.5e-17) || !(phi1 <= 1.2e-49))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -8.5e-17) || ~((phi1 <= 1.2e-49)))
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	else
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -8.5e-17], N[Not[LessEqual[phi1, 1.2e-49]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.5e-17 or 1.19999999999999996e-49 < phi1

   1. Initial program 80.8%

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

      1. *-commutative 80.8%

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

      2. associate-*l* 80.8%

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

   3. Simplified 80.8%

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

      1. log1p-expm1-u 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in phi2 around 0 59.1%

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

   if -8.5e-17 < phi1 < 1.19999999999999996e-49

   1. Initial program 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*l* 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in lambda1 around 0 83.0%

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

      1. cancel-sign-sub-inv 83.0%

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

      2. *-commutative 83.0%

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

      3. associate-*l* 83.0%

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

      4. fma-undefine 83.0%

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

      5. *-commutative 83.0%

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

      6. *-commutative 83.0%

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

   7. Simplified 83.0%

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

   8. Taylor expanded in phi1 around 0 80.9%

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

      1. sin-diff 99.8%

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

   10. Applied egg-rr 97.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 1.2 \cdot 10^{-49}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 71.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.3 \cdot 10^{-8} \lor \neg \left(\phi_2 \leq 0.016\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -4.3e-8) (not (<= phi2 0.016)))
   (atan2
    (*
     (cos phi2)
     (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
    (sin phi2))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda1 lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -4.3e-8) || !(phi2 <= 0.016)) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi2 <= (-4.3d-8)) .or. (.not. (phi2 <= 0.016d0))) then
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -4.3e-8) || !(phi2 <= 0.016)) {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1)))), Math.sin(phi2));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), ((phi2 * Math.cos(phi1)) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi2 <= -4.3e-8) or not (phi2 <= 0.016):
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1)))), math.sin(phi2))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), ((phi2 * math.cos(phi1)) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -4.3e-8) || !(phi2 <= 0.016))
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), sin(phi2));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi2 <= -4.3e-8) || ~((phi2 <= 0.016)))
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -4.3e-8], N[Not[LessEqual[phi2, 0.016]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -4.3000000000000001e-8 or 0.016 < phi2

   1. Initial program 74.7%

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

      1. *-commutative 74.7%

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

      2. associate-*l* 74.7%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in lambda1 around 0 74.7%

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

      1. cancel-sign-sub-inv 74.7%

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

      2. *-commutative 74.7%

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

      3. associate-*l* 74.7%

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

      4. fma-undefine 74.7%

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

      5. *-commutative 74.7%

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

      6. *-commutative 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in phi1 around 0 49.0%

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

      1. sin-diff 99.5%

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

   10. Applied egg-rr 63.7%

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

   if -4.3000000000000001e-8 < phi2 < 0.016

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

      2. associate-*l* 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in lambda1 around 0 89.5%

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

      1. cancel-sign-sub-inv 89.5%

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

      2. *-commutative 89.5%

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

      3. associate-*l* 89.5%

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

      4. fma-undefine 89.5%

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

      5. *-commutative 89.5%

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

      6. *-commutative 89.5%

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

   7. Simplified 89.5%

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

   8. Taylor expanded in phi2 around 0 88.9%

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

      1. neg-mul-1 88.9%

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

      2. +-commutative 88.9%

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

      3. sub-neg 88.9%

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

   10. Simplified 88.9%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.3 \cdot 10^{-8} \lor \neg \left(\phi_2 \leq 0.016\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1 \lor \neg \left(\phi_1 \leq 3 \cdot 10^{-19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - \cos \phi_2 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi1 -1.0) (not (<= phi1 3e-19)))
     (atan2 t_0 (* (sin phi1) (- (cos (- lambda2 lambda1)))))
     (atan2
      t_0
      (- (sin phi2) (* (cos phi2) (* phi1 (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 <= -1.0) || !(phi1 <= 3e-19)) {
		tmp = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))));
	} else {
		tmp = atan2(t_0, (sin(phi2) - (cos(phi2) * (phi1 * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if ((phi1 <= (-1.0d0)) .or. (.not. (phi1 <= 3d-19))) then
        tmp = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))))
    else
        tmp = atan2(t_0, (sin(phi2) - (cos(phi2) * (phi1 * cos((lambda1 - lambda2))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -1.0) || !(phi1 <= 3e-19)) {
		tmp = Math.atan2(t_0, (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.cos(phi2) * (phi1 * Math.cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi1 <= -1.0) or not (phi1 <= 3e-19):
		tmp = math.atan2(t_0, (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	else:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.cos(phi2) * (phi1 * math.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 <= -1.0) || !(phi1 <= 3e-19))
		tmp = atan(t_0, Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(t_0, Float64(sin(phi2) - Float64(cos(phi2) * Float64(phi1 * cos(Float64(lambda1 - 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 ((phi1 <= -1.0) || ~((phi1 <= 3e-19)))
		tmp = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))));
	else
		tmp = atan2(t_0, (sin(phi2) - (cos(phi2) * (phi1 * cos((lambda1 - 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[Or[LessEqual[phi1, -1.0], N[Not[LessEqual[phi1, 3e-19]], $MachinePrecision]], N[ArcTan[t$95$0 / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(phi1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1 or 2.99999999999999993e-19 < phi1

   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. Step-by-step derivation

      1. log1p-expm1-u 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. log1p-expm1-u 81.6%

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

      2. associate-*r* 81.6%

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

      3. add-log-exp 81.4%

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

   8. Applied egg-rr 81.4%

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

   9. Taylor expanded in phi2 around 0 53.5%

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

       1. *-commutative 53.5%

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

       2. neg-mul-1 53.5%

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

       3. distribute-rgt-neg-in 53.5%

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

       4. sub-neg 53.5%

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

       5. remove-double-neg 53.5%

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

       6. mul-1-neg 53.5%

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

       7. distribute-neg-in 53.5%

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

       8. +-commutative 53.5%

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

       9. cos-neg 53.5%

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

       10. mul-1-neg 53.5%

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

       11. sub-neg 53.5%

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

   11. Simplified 53.5%

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

   if -1 < phi1 < 2.99999999999999993e-19

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*l* 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in lambda1 around 0 82.1%

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

      1. cancel-sign-sub-inv 82.1%

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

      2. *-commutative 82.1%

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

      3. associate-*l* 82.1%

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

      4. fma-undefine 82.1%

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

      5. *-commutative 82.1%

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

      6. *-commutative 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in phi1 around 0 81.5%

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

      1. mul-1-neg 81.5%

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

      2. *-commutative 81.5%

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

      3. associate-*l* 81.5%

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

      4. unsub-neg 81.5%

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

      5. *-commutative 81.5%

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

   10. Simplified 81.5%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \lor \neg \left(\phi_1 \leq 3 \cdot 10^{-19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \phi_2 \cdot \left(\phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t\_0\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.003:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 1800:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (* (cos phi2) t_0))
        (t_2 (cos (- lambda1 lambda2))))
   (if (<= phi2 -0.003)
     (atan2 (* t_0 (log1p (expm1 (cos phi2)))) (sin phi2))
     (if (<= phi2 1800.0)
       (atan2 t_1 (- (* phi2 (cos phi1)) (* (sin phi1) t_2)))
       (atan2 t_1 (- (* (cos phi1) (sin phi2)) (* phi1 t_2)))))))

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 t_2 = cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -0.003) {
		tmp = atan2((t_0 * log1p(expm1(cos(phi2)))), sin(phi2));
	} else if (phi2 <= 1800.0) {
		tmp = atan2(t_1, ((phi2 * cos(phi1)) - (sin(phi1) * t_2)));
	} else {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (phi1 * t_2)));
	}
	return tmp;
}

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 t_2 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -0.003) {
		tmp = Math.atan2((t_0 * Math.log1p(Math.expm1(Math.cos(phi2)))), Math.sin(phi2));
	} else if (phi2 <= 1800.0) {
		tmp = Math.atan2(t_1, ((phi2 * Math.cos(phi1)) - (Math.sin(phi1) * t_2)));
	} else {
		tmp = Math.atan2(t_1, ((Math.cos(phi1) * Math.sin(phi2)) - (phi1 * t_2)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.cos(phi2) * t_0
	t_2 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi2 <= -0.003:
		tmp = math.atan2((t_0 * math.log1p(math.expm1(math.cos(phi2)))), math.sin(phi2))
	elif phi2 <= 1800.0:
		tmp = math.atan2(t_1, ((phi2 * math.cos(phi1)) - (math.sin(phi1) * t_2)))
	else:
		tmp = math.atan2(t_1, ((math.cos(phi1) * math.sin(phi2)) - (phi1 * t_2)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * t_0)
	t_2 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= -0.003)
		tmp = atan(Float64(t_0 * log1p(expm1(cos(phi2)))), sin(phi2));
	elseif (phi2 <= 1800.0)
		tmp = atan(t_1, Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * t_2)));
	else
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(phi1 * t_2)));
	end
	return 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]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.003], N[ArcTan[N[(t$95$0 * N[Log[1 + N[(Exp[N[Cos[phi2], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 1800.0], N[ArcTan[t$95$1 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(phi1 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0030000000000000001

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l* 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in lambda1 around 0 77.6%

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

      1. cancel-sign-sub-inv 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*l* 77.6%

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

      4. fma-undefine 77.6%

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

      5. *-commutative 77.6%

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

      6. *-commutative 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in phi1 around 0 54.3%

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

      1. log1p-expm1-u 54.4%

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

   10. Applied egg-rr 54.4%

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

   if -0.0030000000000000001 < phi2 < 1800

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*l* 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in lambda1 around 0 88.6%

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

      1. cancel-sign-sub-inv 88.6%

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

      2. *-commutative 88.6%

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

      3. associate-*l* 88.6%

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

      4. fma-undefine 88.6%

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

      5. *-commutative 88.6%

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

      6. *-commutative 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in phi2 around 0 87.5%

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

      1. neg-mul-1 87.5%

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

      2. +-commutative 87.5%

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

      3. sub-neg 87.5%

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

   10. Simplified 87.5%

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

   if 1800 < phi2

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*l* 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in phi1 around 0 42.2%

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

      1. *-commutative 42.2%

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

   7. Simplified 42.2%

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

   8. Taylor expanded in phi2 around 0 45.8%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.003:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 1800:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 64.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.0019:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 1800:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi2 -0.0019)
     (atan2 (* t_0 (log1p (expm1 (cos phi2)))) (sin phi2))
     (if (<= phi2 1800.0)
       (atan2
        (* (cos phi2) t_0)
        (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda1 lambda2)))))
       (atan2 (* (cos phi2) (log1p (expm1 t_0))) (sin phi2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -0.0019) {
		tmp = atan2((t_0 * log1p(expm1(cos(phi2)))), sin(phi2));
	} else if (phi2 <= 1800.0) {
		tmp = atan2((cos(phi2) * t_0), ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((cos(phi2) * log1p(expm1(t_0))), sin(phi2));
	}
	return tmp;
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -0.0019) {
		tmp = Math.atan2((t_0 * Math.log1p(Math.expm1(Math.cos(phi2)))), Math.sin(phi2));
	} else if (phi2 <= 1800.0) {
		tmp = Math.atan2((Math.cos(phi2) * t_0), ((phi2 * Math.cos(phi1)) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.log1p(Math.expm1(t_0))), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= -0.0019:
		tmp = math.atan2((t_0 * math.log1p(math.expm1(math.cos(phi2)))), math.sin(phi2))
	elif phi2 <= 1800.0:
		tmp = math.atan2((math.cos(phi2) * t_0), ((phi2 * math.cos(phi1)) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.log1p(math.expm1(t_0))), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= -0.0019)
		tmp = atan(Float64(t_0 * log1p(expm1(cos(phi2)))), sin(phi2));
	elseif (phi2 <= 1800.0)
		tmp = atan(Float64(cos(phi2) * t_0), Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(cos(phi2) * log1p(expm1(t_0))), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -0.0019], N[ArcTan[N[(t$95$0 * N[Log[1 + N[(Exp[N[Cos[phi2], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 1800.0], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0019

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l* 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in lambda1 around 0 77.6%

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

      1. cancel-sign-sub-inv 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*l* 77.6%

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

      4. fma-undefine 77.6%

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

      5. *-commutative 77.6%

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

      6. *-commutative 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in phi1 around 0 54.3%

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

      1. log1p-expm1-u 54.4%

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

   10. Applied egg-rr 54.4%

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

   if -0.0019 < phi2 < 1800

   1. Initial program 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*l* 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in lambda1 around 0 88.6%

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

      1. cancel-sign-sub-inv 88.6%

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

      2. *-commutative 88.6%

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

      3. associate-*l* 88.6%

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

      4. fma-undefine 88.6%

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

      5. *-commutative 88.6%

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

      6. *-commutative 88.6%

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

   7. Simplified 88.6%

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

   8. Taylor expanded in phi2 around 0 87.5%

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

      1. neg-mul-1 87.5%

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

      2. +-commutative 87.5%

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

      3. sub-neg 87.5%

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

   10. Simplified 87.5%

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

   if 1800 < phi2

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*l* 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in lambda1 around 0 72.6%

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

      1. cancel-sign-sub-inv 72.6%

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

      2. *-commutative 72.6%

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

      3. associate-*l* 72.6%

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

      4. fma-undefine 72.6%

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

      5. *-commutative 72.6%

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

      6. *-commutative 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in phi1 around 0 44.2%

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

      1. log1p-expm1-u 44.2%

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

   10. Applied egg-rr 44.2%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.0019:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\phi_2 \leq 1800:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 64.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.95 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 3 \cdot 10^{-19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi1 -1.95e-7) (not (<= phi1 3e-19)))
     (atan2 t_0 (* (sin phi1) (- (cos (- lambda2 lambda1)))))
     (atan2 t_0 (- (sin phi2) (* (cos phi2) (* (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 <= -1.95e-7) || !(phi1 <= 3e-19)) {
		tmp = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))));
	} else {
		tmp = atan2(t_0, (sin(phi2) - (cos(phi2) * (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 <= (-1.95d-7)) .or. (.not. (phi1 <= 3d-19))) then
        tmp = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))))
    else
        tmp = atan2(t_0, (sin(phi2) - (cos(phi2) * (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 <= -1.95e-7) || !(phi1 <= 3e-19)) {
		tmp = Math.atan2(t_0, (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.cos(phi2) * (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 <= -1.95e-7) or not (phi1 <= 3e-19):
		tmp = math.atan2(t_0, (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	else:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.cos(phi2) * (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 <= -1.95e-7) || !(phi1 <= 3e-19))
		tmp = atan(t_0, Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(t_0, Float64(sin(phi2) - Float64(cos(phi2) * Float64(cos(lambda1) * 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 <= -1.95e-7) || ~((phi1 <= 3e-19)))
		tmp = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))));
	else
		tmp = atan2(t_0, (sin(phi2) - (cos(phi2) * (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[Or[LessEqual[phi1, -1.95e-7], N[Not[LessEqual[phi1, 3e-19]], $MachinePrecision]], N[ArcTan[t$95$0 / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.95000000000000012e-7 or 2.99999999999999993e-19 < phi1

   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. Step-by-step derivation

      1. *-commutative 80.9%

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

      2. associate-*l* 80.9%

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

   3. Simplified 80.9%

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

      1. log1p-expm1-u 80.9%

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

   6. Applied egg-rr 80.9%

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

      1. log1p-expm1-u 80.9%

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

      2. associate-*r* 80.9%

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

      3. add-log-exp 80.7%

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

   8. Applied egg-rr 80.7%

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

   9. Taylor expanded in phi2 around 0 53.0%

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

       1. *-commutative 53.0%

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

       2. neg-mul-1 53.0%

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

       3. distribute-rgt-neg-in 53.0%

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

       4. sub-neg 53.0%

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

       5. remove-double-neg 53.0%

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

       6. mul-1-neg 53.0%

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

       7. distribute-neg-in 53.0%

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

       8. +-commutative 53.0%

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

       9. cos-neg 53.0%

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

       10. mul-1-neg 53.0%

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

       11. sub-neg 53.0%

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

   11. Simplified 53.0%

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

   if -1.95000000000000012e-7 < phi1 < 2.99999999999999993e-19

   1. Initial program 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*l* 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in lambda1 around 0 82.7%

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

      1. cancel-sign-sub-inv 82.7%

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

      2. *-commutative 82.7%

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

      3. associate-*l* 82.7%

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

      4. fma-undefine 82.7%

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

      5. *-commutative 82.7%

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

      6. *-commutative 82.7%

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

   7. Simplified 82.7%

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

   8. Taylor expanded in lambda2 around 0 82.7%

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

      1. associate-*r* 82.7%

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

      2. mul-1-neg 82.7%

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

   10. Simplified 82.7%

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

   11. Taylor expanded in phi1 around 0 82.7%

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

       1. mul-1-neg 82.7%

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

       2. unsub-neg 82.7%

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

       3. associate-*r* 82.7%

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

   13. Simplified 82.7%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.95 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 3 \cdot 10^{-19}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-29}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2\right)\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (or (<= phi1 -8.5e-17) (not (<= phi1 4.5e-29)))
     (atan2 (* (cos phi2) t_0) (* (sin phi1) (- (cos (- lambda2 lambda1)))))
     (atan2 (* t_0 (log1p (expm1 (cos phi2)))) (sin phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -8.5e-17) || !(phi1 <= 4.5e-29)) {
		tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))));
	} else {
		tmp = atan2((t_0 * log1p(expm1(cos(phi2)))), sin(phi2));
	}
	return tmp;
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -8.5e-17) || !(phi1 <= 4.5e-29)) {
		tmp = Math.atan2((Math.cos(phi2) * t_0), (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else {
		tmp = Math.atan2((t_0 * Math.log1p(Math.expm1(Math.cos(phi2)))), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi1 <= -8.5e-17) or not (phi1 <= 4.5e-29):
		tmp = math.atan2((math.cos(phi2) * t_0), (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	else:
		tmp = math.atan2((t_0 * math.log1p(math.expm1(math.cos(phi2)))), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi1 <= -8.5e-17) || !(phi1 <= 4.5e-29))
		tmp = atan(Float64(cos(phi2) * t_0), Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(Float64(t_0 * log1p(expm1(cos(phi2)))), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi1, -8.5e-17], N[Not[LessEqual[phi1, 4.5e-29]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(t$95$0 * N[Log[1 + N[(Exp[N[Cos[phi2], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.5e-17 or 4.4999999999999998e-29 < phi1

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-*l* 80.2%

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

   3. Simplified 80.2%

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

      1. log1p-expm1-u 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. log1p-expm1-u 80.2%

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

      2. associate-*r* 80.2%

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

      3. add-log-exp 78.7%

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

   8. Applied egg-rr 78.7%

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

   9. Taylor expanded in phi2 around 0 53.0%

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

       1. *-commutative 53.0%

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

       2. neg-mul-1 53.0%

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

       3. distribute-rgt-neg-in 53.0%

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

       4. sub-neg 53.0%

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

       5. remove-double-neg 53.0%

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

       6. mul-1-neg 53.0%

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

       7. distribute-neg-in 53.0%

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

       8. +-commutative 53.0%

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

       9. cos-neg 53.0%

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

       10. mul-1-neg 53.0%

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

       11. sub-neg 53.0%

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

   11. Simplified 53.0%

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

   if -8.5e-17 < phi1 < 4.4999999999999998e-29

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*l* 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in lambda1 around 0 83.5%

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

      1. cancel-sign-sub-inv 83.5%

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

      2. *-commutative 83.5%

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

      3. associate-*l* 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 83.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 81.5%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. log1p-expm1-u 81.5%

         \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2\right)\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]

   10. Applied egg-rr 81.5%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2\right)\right)} \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-29}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2\right)\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 8 \cdot 10^{-31}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (or (<= phi1 -8.5e-17) (not (<= phi1 8e-31)))
     (atan2 (* (cos phi2) t_0) (* (sin phi1) (- (cos (- lambda2 lambda1)))))
     (atan2 (* (cos phi2) (log1p (expm1 t_0))) (sin phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -8.5e-17) || !(phi1 <= 8e-31)) {
		tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))));
	} else {
		tmp = atan2((cos(phi2) * log1p(expm1(t_0))), sin(phi2));
	}
	return tmp;
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -8.5e-17) || !(phi1 <= 8e-31)) {
		tmp = Math.atan2((Math.cos(phi2) * t_0), (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.log1p(Math.expm1(t_0))), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi1 <= -8.5e-17) or not (phi1 <= 8e-31):
		tmp = math.atan2((math.cos(phi2) * t_0), (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.log1p(math.expm1(t_0))), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi1 <= -8.5e-17) || !(phi1 <= 8e-31))
		tmp = atan(Float64(cos(phi2) * t_0), Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(Float64(cos(phi2) * log1p(expm1(t_0))), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi1, -8.5e-17], N[Not[LessEqual[phi1, 8e-31]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.5e-17 or 8.000000000000001e-31 < phi1

   1. Initial program 80.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 80.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. log1p-expm1-u 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

   6. Applied egg-rr 80.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

      2. associate-*r* 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. add-log-exp 78.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\log \left(e^{\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}} \]

   8. Applied egg-rr 78.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\log \left(e^{\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}} \]

   9. Taylor expanded in phi2 around 0 53.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

       2. neg-mul-1 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

       3. distribute-rgt-neg-in 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

       4. sub-neg 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)} \]

       5. remove-double-neg 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\color{blue}{\left(-\left(-\lambda_1\right)\right)} + \left(-\lambda_2\right)\right)\right)} \]

       6. mul-1-neg 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\left(-\color{blue}{-1 \cdot \lambda_1}\right) + \left(-\lambda_2\right)\right)\right)} \]

       7. distribute-neg-in 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(-\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)}\right)} \]

       8. +-commutative 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)} \]

       9. cos-neg 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)} \]

       10. mul-1-neg 53.0%

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right)\right)} \]

       11. sub-neg 53.0%

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)} \]

   11. Simplified 53.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

   if -8.5e-17 < phi1 < 8.000000000000001e-31

   1. Initial program 83.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 83.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 83.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 83.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 81.5%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
   9. Step-by-step derivation

      1. log1p-expm1-u 81.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sin \phi_2} \]

   10. Applied egg-rr 81.5%

       \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 8 \cdot 10^{-31}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 62.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-29}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi1 -8.5e-17) (not (<= phi1 4.5e-29)))
     (atan2 t_0 (* (sin phi1) (- (cos (- lambda2 lambda1)))))
     (atan2 t_0 (sin phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -8.5e-17) || !(phi1 <= 4.5e-29)) {
		tmp = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))));
	} else {
		tmp = atan2(t_0, sin(phi2));
	}
	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 <= (-8.5d-17)) .or. (.not. (phi1 <= 4.5d-29))) then
        tmp = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))))
    else
        tmp = atan2(t_0, sin(phi2))
    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 <= -8.5e-17) || !(phi1 <= 4.5e-29)) {
		tmp = Math.atan2(t_0, (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else {
		tmp = Math.atan2(t_0, Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi1 <= -8.5e-17) or not (phi1 <= 4.5e-29):
		tmp = math.atan2(t_0, (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	else:
		tmp = math.atan2(t_0, math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if ((phi1 <= -8.5e-17) || !(phi1 <= 4.5e-29))
		tmp = atan(t_0, Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(t_0, sin(phi2));
	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 <= -8.5e-17) || ~((phi1 <= 4.5e-29)))
		tmp = atan2(t_0, (sin(phi1) * -cos((lambda2 - lambda1))));
	else
		tmp = atan2(t_0, sin(phi2));
	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[phi1, -8.5e-17], N[Not[LessEqual[phi1, 4.5e-29]], $MachinePrecision]], N[ArcTan[t$95$0 / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.5e-17 or 4.4999999999999998e-29 < phi1

   1. Initial program 80.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 80.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. log1p-expm1-u 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]

   6. Applied egg-rr 80.2%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)} \]
   7. Step-by-step derivation

      1. log1p-expm1-u 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)} \]

      2. associate-*r* 80.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      3. add-log-exp 78.7%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\log \left(e^{\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}} \]

   8. Applied egg-rr 78.7%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\log \left(e^{\left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}} \]

   9. Taylor expanded in phi2 around 0 53.0%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-1 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   10. Step-by-step derivation

       1. *-commutative 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{-1 \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

       2. neg-mul-1 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{-\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

       3. distribute-rgt-neg-in 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

       4. sub-neg 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)} \]

       5. remove-double-neg 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\color{blue}{\left(-\left(-\lambda_1\right)\right)} + \left(-\lambda_2\right)\right)\right)} \]

       6. mul-1-neg 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\left(-\color{blue}{-1 \cdot \lambda_1}\right) + \left(-\lambda_2\right)\right)\right)} \]

       7. distribute-neg-in 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(-\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)}\right)} \]

       8. +-commutative 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(-\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right)} \]

       9. cos-neg 53.0%

          \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)} \]

       10. mul-1-neg 53.0%

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right)\right)} \]

       11. sub-neg 53.0%

           \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_1 \cdot \left(-\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)} \]

   11. Simplified 53.0%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}} \]

   if -8.5e-17 < phi1 < 4.4999999999999998e-29

   1. Initial program 83.5%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 83.5%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 83.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 83.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 83.5%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 83.5%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 81.5%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-17} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-29}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 59.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -9.2 \cdot 10^{-20} \lor \neg \left(\phi_2 \leq 2.95\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2 \cdot \cos \phi_1 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (or (<= phi2 -9.2e-20) (not (<= phi2 2.95)))
     (atan2 (* (cos phi2) t_0) (sin phi2))
     (atan2 t_0 (- (* phi2 (cos phi1)) (* (cos lambda1) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -9.2e-20) || !(phi2 <= 2.95)) {
		tmp = atan2((cos(phi2) * t_0), sin(phi2));
	} else {
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (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 = sin((lambda1 - lambda2))
    if ((phi2 <= (-9.2d-20)) .or. (.not. (phi2 <= 2.95d0))) then
        tmp = atan2((cos(phi2) * t_0), sin(phi2))
    else
        tmp = atan2(t_0, ((phi2 * cos(phi1)) - (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.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -9.2e-20) || !(phi2 <= 2.95)) {
		tmp = Math.atan2((Math.cos(phi2) * t_0), Math.sin(phi2));
	} else {
		tmp = Math.atan2(t_0, ((phi2 * Math.cos(phi1)) - (Math.cos(lambda1) * Math.sin(phi1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -9.2e-20) or not (phi2 <= 2.95):
		tmp = math.atan2((math.cos(phi2) * t_0), math.sin(phi2))
	else:
		tmp = math.atan2(t_0, ((phi2 * math.cos(phi1)) - (math.cos(lambda1) * math.sin(phi1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi2 <= -9.2e-20) || !(phi2 <= 2.95))
		tmp = atan(Float64(cos(phi2) * t_0), sin(phi2));
	else
		tmp = atan(t_0, Float64(Float64(phi2 * cos(phi1)) - Float64(cos(lambda1) * sin(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -9.2e-20) || ~((phi2 <= 2.95)))
		tmp = atan2((cos(phi2) * t_0), sin(phi2));
	else
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (cos(lambda1) * sin(phi1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -9.2e-20], N[Not[LessEqual[phi2, 2.95]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -9.1999999999999997e-20 or 2.9500000000000002 < phi2

   1. Initial program 76.0%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 76.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 76.0%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 76.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 76.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 76.0%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 76.0%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 76.0%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 76.0%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 76.0%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 76.0%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 76.0%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 50.3%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   if -9.1999999999999997e-20 < phi2 < 2.9500000000000002

   1. Initial program 88.4%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 88.4%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 88.4%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 88.4%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 88.4%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 88.4%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 88.4%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 88.4%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 88.4%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 88.4%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 88.4%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 88.4%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in lambda2 around 0 77.7%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{-1 \cdot \left(\cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 77.7%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\left(-1 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\right)} \]

      2. mul-1-neg 77.7%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\left(-\cos \lambda_1\right)} \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)\right)} \]

   10. Simplified 77.7%

       \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\left(-\cos \lambda_1\right) \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\right)} \]

   11. Taylor expanded in phi2 around 0 77.1%

       \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{-1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right) + \phi_2 \cdot \cos \phi_1}} \]
   12. Step-by-step derivation

       1. +-commutative 77.1%

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 + -1 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}} \]

       2. mul-1-neg 77.1%

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 + \color{blue}{\left(-\cos \lambda_1 \cdot \sin \phi_1\right)}} \]

       3. unsub-neg 77.1%

          \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \lambda_1 \cdot \sin \phi_1}} \]

   13. Simplified 77.1%

       \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2 \cdot \cos \phi_1 - \cos \lambda_1 \cdot \sin \phi_1}} \]

   14. Taylor expanded in phi2 around 0 77.2%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\phi_2 \cdot \cos \phi_1 - \cos \lambda_1 \cdot \sin \phi_1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.2 \cdot 10^{-20} \lor \neg \left(\phi_2 \leq 2.95\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 43.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4.3 \cdot 10^{+32} \lor \neg \left(\lambda_2 \leq 3.8 \cdot 10^{-21}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -4.3e+32) (not (<= lambda2 3.8e-21)))
   (atan2 (* (cos phi2) (sin (- lambda2))) (sin phi2))
   (atan2 (* (sin lambda1) (cos phi2)) (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -4.3e+32) || !(lambda2 <= 3.8e-21)) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), sin(phi2));
	} else {
		tmp = atan2((sin(lambda1) * cos(phi2)), sin(phi2));
	}
	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 ((lambda2 <= (-4.3d+32)) .or. (.not. (lambda2 <= 3.8d-21))) then
        tmp = atan2((cos(phi2) * sin(-lambda2)), sin(phi2))
    else
        tmp = atan2((sin(lambda1) * cos(phi2)), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -4.3e+32) || !(lambda2 <= 3.8e-21)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda2 <= -4.3e+32) or not (lambda2 <= 3.8e-21):
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), math.sin(phi2))
	else:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -4.3e+32) || !(lambda2 <= 3.8e-21))
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), sin(phi2));
	else
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda2 <= -4.3e+32) || ~((lambda2 <= 3.8e-21)))
		tmp = atan2((cos(phi2) * sin(-lambda2)), sin(phi2));
	else
		tmp = atan2((sin(lambda1) * cos(phi2)), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -4.3e+32], N[Not[LessEqual[lambda2, 3.8e-21]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -4.2999999999999997e32 or 3.7999999999999998e-21 < lambda2

   1. Initial program 65.6%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 65.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 65.6%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 65.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 65.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 65.6%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 65.6%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 65.6%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 65.6%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 65.6%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 65.6%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 65.6%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 47.2%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in lambda1 around 0 46.4%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\sin \left(-\lambda_2\right)}}{\sin \phi_2} \]

   if -4.2999999999999997e32 < lambda2 < 3.7999999999999998e-21

   1. Initial program 96.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 96.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 96.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 96.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 96.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 96.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 96.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 96.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 96.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 96.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 96.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 56.6%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in lambda2 around 0 50.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\sin \lambda_1}}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4.3 \cdot 10^{+32} \lor \neg \left(\lambda_2 \leq 3.8 \cdot 10^{-21}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 38.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.38 \lor \neg \left(\phi_2 \leq 2.45 \cdot 10^{+57}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -0.38) (not (<= phi2 2.45e+57)))
   (atan2 (* (sin lambda1) (cos phi2)) (sin phi2))
   (atan2 (sin (- lambda1 lambda2)) (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -0.38) || !(phi2 <= 2.45e+57)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(sin((lambda1 - lambda2)), sin(phi2));
	}
	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 ((phi2 <= (-0.38d0)) .or. (.not. (phi2 <= 2.45d+57))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), sin(phi2))
    else
        tmp = atan2(sin((lambda1 - lambda2)), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -0.38) || !(phi2 <= 2.45e+57)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2(Math.sin((lambda1 - lambda2)), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi2 <= -0.38) or not (phi2 <= 2.45e+57):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = math.atan2(math.sin((lambda1 - lambda2)), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -0.38) || !(phi2 <= 2.45e+57))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), sin(phi2));
	else
		tmp = atan(sin(Float64(lambda1 - lambda2)), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi2 <= -0.38) || ~((phi2 <= 2.45e+57)))
		tmp = atan2((sin(lambda1) * cos(phi2)), sin(phi2));
	else
		tmp = atan2(sin((lambda1 - lambda2)), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -0.38], N[Not[LessEqual[phi2, 2.45e+57]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.38 or 2.45e57 < phi2

   1. Initial program 74.2%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 74.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 74.2%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 74.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 74.2%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 74.2%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 74.2%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 74.2%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 74.2%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 74.2%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 74.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 48.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in lambda2 around 0 31.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \color{blue}{\sin \lambda_1}}{\sin \phi_2} \]

   if -0.38 < phi2 < 2.45e57

   1. Initial program 88.8%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 88.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 88.8%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 88.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 88.8%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 88.8%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 88.8%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 88.8%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 88.8%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 88.8%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 88.8%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 55.1%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in phi2 around 0 53.0%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.38 \lor \neg \left(\phi_2 \leq 2.45 \cdot 10^{+57}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 48.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (sin phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2((cos(phi2) * sin((lambda1 - lambda2))), sin(phi2));
}

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(phi2))
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(phi2));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), math.sin(phi2))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), sin(phi2))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), sin(phi2));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 81.8%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 81.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. associate-*l* 81.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

3. Simplified 81.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in lambda1 around 0 81.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 81.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. *-commutative 81.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. associate-*l* 81.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   4. fma-undefine 81.9%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   5. *-commutative 81.9%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

   6. *-commutative 81.9%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

7. Simplified 81.9%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

8. Taylor expanded in phi1 around 0 52.1%

   \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]
9. Add Preprocessing

Alternative 32: 31.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.0)
   (atan2 (sin (- lambda1 lambda2)) phi2)
   (atan2 (sin lambda1) (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.0) {
		tmp = atan2(sin((lambda1 - lambda2)), phi2);
	} else {
		tmp = atan2(sin(lambda1), sin(phi2));
	}
	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 (phi2 <= 2.0d0) then
        tmp = atan2(sin((lambda1 - lambda2)), phi2)
    else
        tmp = atan2(sin(lambda1), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.0) {
		tmp = Math.atan2(Math.sin((lambda1 - lambda2)), phi2);
	} else {
		tmp = Math.atan2(Math.sin(lambda1), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.0:
		tmp = math.atan2(math.sin((lambda1 - lambda2)), phi2)
	else:
		tmp = math.atan2(math.sin(lambda1), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.0)
		tmp = atan(sin(Float64(lambda1 - lambda2)), phi2);
	else
		tmp = atan(sin(lambda1), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.0)
		tmp = atan2(sin((lambda1 - lambda2)), phi2);
	else
		tmp = atan2(sin(lambda1), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.0], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2

   1. Initial program 85.3%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 85.3%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 85.3%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 85.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 85.3%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 85.3%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 85.3%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 85.3%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 85.3%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 85.3%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 85.3%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 55.3%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in phi2 around 0 41.5%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]

   10. Taylor expanded in phi2 around 0 42.4%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]

   if 2 < phi2

   1. Initial program 71.9%

      \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   2. Step-by-step derivation

      1. *-commutative 71.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. associate-*l* 71.9%

         \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. Simplified 71.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in lambda1 around 0 71.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 71.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

      2. *-commutative 71.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      3. associate-*l* 71.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

      4. fma-undefine 71.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. *-commutative 71.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

      6. *-commutative 71.9%

         \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

   7. Simplified 71.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

   8. Taylor expanded in phi1 around 0 43.1%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

   9. Taylor expanded in phi2 around 0 13.1%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]

   10. Taylor expanded in lambda2 around 0 11.6%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 33: 32.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (sin (- lambda1 lambda2)) (sin phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin((lambda1 - lambda2)), sin(phi2));
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2(sin((lambda1 - lambda2)), sin(phi2))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2(Math.sin((lambda1 - lambda2)), Math.sin(phi2));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2(math.sin((lambda1 - lambda2)), math.sin(phi2))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(Float64(lambda1 - lambda2)), sin(phi2))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2(sin((lambda1 - lambda2)), sin(phi2));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 81.8%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 81.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. associate-*l* 81.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

3. Simplified 81.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in lambda1 around 0 81.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 81.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. *-commutative 81.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. associate-*l* 81.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   4. fma-undefine 81.9%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   5. *-commutative 81.9%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

   6. *-commutative 81.9%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

7. Simplified 81.9%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

8. Taylor expanded in phi1 around 0 52.1%

   \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

9. Taylor expanded in phi2 around 0 34.2%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]
10. Add Preprocessing

Alternative 34: 29.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2 (sin (- lambda1 lambda2)) phi2))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return atan2(sin((lambda1 - lambda2)), phi2);
}

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)), phi2)
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2(Math.sin((lambda1 - lambda2)), phi2);
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return math.atan2(math.sin((lambda1 - lambda2)), phi2)

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(Float64(lambda1 - lambda2)), phi2)
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = atan2(sin((lambda1 - lambda2)), phi2);
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / phi2], $MachinePrecision]

Derivation

1. Initial program 81.8%

   \[\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
2. Step-by-step derivation

   1. *-commutative 81.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\left(\cos \phi_2 \cdot \sin \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   2. associate-*l* 81.8%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \color{blue}{\cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

3. Simplified 81.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in lambda1 around 0 81.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 81.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \sin \phi_1\right)}} \]

   2. *-commutative 81.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \left(-\cos \phi_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   3. associate-*l* 81.8%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}} \]

   4. fma-undefine 81.9%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}} \]

   5. *-commutative 81.9%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(-\cos \phi_2\right) \cdot \sin \phi_1\right)}\right)} \]

   6. *-commutative 81.9%

      \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)}\right)} \]

7. Simplified 81.9%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\sin \phi_1 \cdot \left(-\cos \phi_2\right)\right)\right)}} \]

8. Taylor expanded in phi1 around 0 52.1%

   \[\leadsto \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\sin \phi_2}} \]

9. Taylor expanded in phi2 around 0 34.2%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right)}}{\sin \phi_2} \]

10. Taylor expanded in phi2 around 0 32.4%

    \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\color{blue}{\phi_2}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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))))))