Bearing on a great circle

Percentage Accurate: 80.0% → 99.7%

Time: 44.3s

Alternatives: 36

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: 80.0% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ \tan^{-1}_* \frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(t_0 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin phi1))))
   (atan2
    (log1p
     (expm1
      (*
       (cos phi2)
       (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))))
    (-
     (* (cos phi1) (sin phi2))
     (+
      (* t_0 (* (cos lambda2) (cos lambda1)))
      (* t_0 (* (sin lambda1) (sin lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin(phi1);
	return atan2(log1p(expm1((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))))), ((cos(phi1) * sin(phi2)) - ((t_0 * (cos(lambda2) * cos(lambda1))) + (t_0 * (sin(lambda1) * sin(lambda2))))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(phi1))
	return atan(log1p(expm1(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(t_0 * Float64(cos(lambda2) * cos(lambda1))) + Float64(t_0 * Float64(sin(lambda1) * sin(lambda2))))))
end

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. sin-diff 92.0%

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

   2. cancel-sign-sub-inv 92.0%

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

   3. fma-def 92.0%

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

3. Applied egg-rr 92.0%

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

   1. cos-diff 99.7%

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

   2. distribute-lft-in 99.7%

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

   3. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

   1. log1p-expm1-u 99.7%

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

   2. *-commutative 99.7%

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

   3. *-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. sin-diff 92.0%

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

   2. cancel-sign-sub-inv 92.0%

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

   3. fma-def 92.0%

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

3. Applied egg-rr 92.0%

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

   1. cos-diff 99.7%

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

   2. distribute-lft-in 99.7%

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

   3. *-commutative 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in phi1 around inf 99.7%

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

   1. associate-*r* 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. sin-diff 92.0%

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

   2. cancel-sign-sub-inv 92.0%

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

   3. fma-def 92.0%

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

3. Applied egg-rr 92.0%

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

   1. cos-diff 82.1%

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

   2. +-commutative 82.1%

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

   3. *-commutative 82.1%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)\\ t_2 := \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right) - t_2}\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right) - t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (*
          (cos phi2)
          (fma
           (sin lambda1)
           (cos lambda2)
           (* (sin lambda2) (- (cos lambda1))))))
        (t_2 (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
   (if (<= phi2 -4e-8)
     (atan2 t_1 (- (expm1 (log1p t_0)) t_2))
     (if (<= phi2 2e-25)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
        (-
         (sin phi2)
         (*
          (sin phi1)
          (+
           (* (cos lambda2) (cos lambda1))
           (* (sin lambda1) (sin lambda2))))))
       (atan2 t_1 (- (log1p (expm1 t_0)) t_2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)));
	double t_2 = (cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -4e-8) {
		tmp = atan2(t_1, (expm1(log1p(t_0)) - t_2));
	} else if (phi2 <= 2e-25) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2(t_1, (log1p(expm1(t_0)) - t_2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1)))))
	t_2 = Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -4e-8)
		tmp = atan(t_1, Float64(expm1(log1p(t_0)) - t_2));
	elseif (phi2 <= 2e-25)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(sin(phi2) - Float64(sin(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(t_1, Float64(log1p(expm1(t_0)) - t_2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4e-8], N[ArcTan[t$95$1 / N[(N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 2e-25], 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[Sin[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], N[ArcTan[t$95$1 / N[(N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.0000000000000001e-8

   1. Initial program 73.4%

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

      1. sin-diff 90.1%

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

      2. cancel-sign-sub-inv 90.1%

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

      3. fma-def 90.1%

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

   3. Applied egg-rr 90.1%

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

      1. expm1-log1p-u 90.1%

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

   5. Applied egg-rr 90.1%

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

   if -4.0000000000000001e-8 < phi2 < 2.00000000000000008e-25

   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. Taylor expanded in phi2 around 0 85.3%

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

   3. Taylor expanded in phi1 around 0 85.3%

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

      1. sin-diff 62.6%

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

      2. sub-neg 62.6%

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

   5. Applied egg-rr 91.4%

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

      1. sub-neg 62.6%

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

   7. Simplified 91.4%

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

      1. cos-diff 85.6%

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

      2. +-commutative 85.6%

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

      3. *-commutative 85.6%

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

   9. Applied egg-rr 99.9%

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

   if 2.00000000000000008e-25 < phi2

   1. Initial program 85.2%

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

      1. sin-diff 95.7%

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

      2. cancel-sign-sub-inv 95.7%

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

      3. fma-def 95.8%

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

   3. Applied egg-rr 95.8%

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

      1. log1p-expm1-u 95.8%

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

   5. Applied egg-rr 95.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-25}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 5: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)\\ t_2 := \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right) - t_2}\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (*
          (cos phi2)
          (fma
           (sin lambda1)
           (cos lambda2)
           (* (sin lambda2) (- (cos lambda1))))))
        (t_2 (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
   (if (<= phi2 -3.5e-8)
     (atan2 t_1 (- (expm1 (log1p t_0)) t_2))
     (if (<= phi2 2e-22)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
        (-
         (sin phi2)
         (*
          (sin phi1)
          (+
           (* (cos lambda2) (cos lambda1))
           (* (sin lambda1) (sin lambda2))))))
       (atan2 t_1 (- t_0 t_2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)));
	double t_2 = (cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -3.5e-8) {
		tmp = atan2(t_1, (expm1(log1p(t_0)) - t_2));
	} else if (phi2 <= 2e-22) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2(t_1, (t_0 - t_2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1)))))
	t_2 = Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -3.5e-8)
		tmp = atan(t_1, Float64(expm1(log1p(t_0)) - t_2));
	elseif (phi2 <= 2e-22)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(sin(phi2) - Float64(sin(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(t_1, Float64(t_0 - t_2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.5e-8], N[ArcTan[t$95$1 / N[(N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 2e-22], 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[Sin[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], N[ArcTan[t$95$1 / N[(t$95$0 - t$95$2), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.50000000000000024e-8

   1. Initial program 73.4%

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

      1. sin-diff 90.1%

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

      2. cancel-sign-sub-inv 90.1%

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

      3. fma-def 90.1%

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

   3. Applied egg-rr 90.1%

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

      1. expm1-log1p-u 90.1%

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

   5. Applied egg-rr 90.1%

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

   if -3.50000000000000024e-8 < phi2 < 2.0000000000000001e-22

   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. Taylor expanded in phi2 around 0 85.3%

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

   3. Taylor expanded in phi1 around 0 85.3%

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

      1. sin-diff 62.6%

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

      2. sub-neg 62.6%

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

   5. Applied egg-rr 91.4%

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

      1. sub-neg 62.6%

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

   7. Simplified 91.4%

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

      1. cos-diff 85.6%

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

      2. +-commutative 85.6%

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

      3. *-commutative 85.6%

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

   9. Applied egg-rr 99.9%

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

   if 2.0000000000000001e-22 < phi2

   1. Initial program 85.2%

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

      1. sin-diff 95.7%

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

      2. cancel-sign-sub-inv 95.7%

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

      3. fma-def 95.8%

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

   3. Applied egg-rr 95.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 6: 94.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-8} \lor \neg \left(\phi_2 \leq 2.8 \cdot 10^{-22}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -3.5e-8) (not (<= phi2 2.8e-22)))
   (atan2
    (*
     (cos phi2)
     (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))
    (-
     (* (cos phi1) (sin phi2))
     (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
   (atan2
    (*
     (cos phi2)
     (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
    (-
     (sin phi2)
     (*
      (sin phi1)
      (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda1) (sin lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -3.5e-8) || !(phi2 <= 2.8e-22)) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - (sin(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -3.5e-8) || !(phi2 <= 2.8e-22))
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(sin(phi2) - Float64(sin(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -3.5e-8], N[Not[LessEqual[phi2, 2.8e-22]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $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[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $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[(N[Sin[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]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.50000000000000024e-8 or 2.79999999999999995e-22 < phi2

   1. Initial program 78.7%

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

      1. sin-diff 92.6%

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

      2. cancel-sign-sub-inv 92.6%

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

      3. fma-def 92.7%

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

   3. Applied egg-rr 92.7%

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

   if -3.50000000000000024e-8 < phi2 < 2.79999999999999995e-22

   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. Taylor expanded in phi2 around 0 85.3%

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

   3. Taylor expanded in phi1 around 0 85.3%

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

      1. sin-diff 62.6%

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

      2. sub-neg 62.6%

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

   5. Applied egg-rr 91.4%

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

      1. sub-neg 62.6%

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

   7. Simplified 91.4%

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

      1. cos-diff 85.6%

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

      2. +-commutative 85.6%

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

      3. *-commutative 85.6%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-8} \lor \neg \left(\phi_2 \leq 2.8 \cdot 10^{-22}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]

Alternative 7: 89.9% accurate, 0.7× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

   1. sin-diff 92.0%

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

   2. cancel-sign-sub-inv 92.0%

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

   3. fma-def 92.0%

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

3. Applied egg-rr 92.0%

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

4. Final simplification 92.0%

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

Alternative 8: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -44000000000 \lor \neg \left(\lambda_1 \leq 410000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - t_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (* (cos phi2) (sin phi1))))
   (if (or (<= lambda1 -44000000000.0) (not (<= lambda1 410000.0)))
     (atan2
      (*
       (cos phi2)
       (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
      (- t_0 (* (cos lambda1) t_1)))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (-
       t_0
       (*
        t_1
        (+
         (* (cos lambda2) (cos lambda1))
         (* (sin lambda1) (sin lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin(phi1);
	double tmp;
	if ((lambda1 <= -44000000000.0) || !(lambda1 <= 410000.0)) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (t_0 - (cos(lambda1) * t_1)));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (t_1 * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin(phi1)
    if ((lambda1 <= (-44000000000.0d0)) .or. (.not. (lambda1 <= 410000.0d0))) then
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (t_0 - (cos(lambda1) * t_1)))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (t_1 * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin(phi1);
	double tmp;
	if ((lambda1 <= -44000000000.0) || !(lambda1 <= 410000.0)) {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1)))), (t_0 - (Math.cos(lambda1) * t_1)));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (t_1 * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos(phi2) * math.sin(phi1)
	tmp = 0
	if (lambda1 <= -44000000000.0) or not (lambda1 <= 410000.0):
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1)))), (t_0 - (math.cos(lambda1) * t_1)))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (t_1 * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(phi1))
	tmp = 0.0
	if ((lambda1 <= -44000000000.0) || !(lambda1 <= 410000.0))
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(t_0 - Float64(cos(lambda1) * t_1)));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(t_1 * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = cos(phi2) * sin(phi1);
	tmp = 0.0;
	if ((lambda1 <= -44000000000.0) || ~((lambda1 <= 410000.0)))
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (t_0 - (cos(lambda1) * t_1)));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (t_1 * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -44000000000.0], N[Not[LessEqual[lambda1, 410000.0]], $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[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.4e10 or 4.1e5 < lambda1

   1. Initial program 62.3%

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

      1. sin-diff 84.7%

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

      2. cancel-sign-sub-inv 84.7%

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

      3. fma-def 84.7%

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

   3. Applied egg-rr 84.7%

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

   4. Taylor expanded in lambda2 around 0 84.7%

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

   5. Taylor expanded in lambda1 around inf 84.7%

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

      1. *-commutative 84.7%

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

      2. mul-1-neg 84.7%

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

      3. distribute-rgt-neg-out 84.7%

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

      4. +-commutative 84.7%

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

      5. *-commutative 84.7%

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

      6. cancel-sign-sub-inv 84.7%

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

      7. *-commutative 84.7%

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

   7. Simplified 84.7%

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

   if -4.4e10 < lambda1 < 4.1e5

   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. cos-diff 98.2%

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

      2. +-commutative 98.2%

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

      3. *-commutative 98.2%

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

   3. Applied egg-rr 98.2%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -44000000000 \lor \neg \left(\lambda_1 \leq 410000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]

Alternative 9: 90.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -1.32 \cdot 10^{-6} \lor \neg \left(\lambda_1 \leq 0.00025\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{t_0 - t_1 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (* (cos phi2) (sin phi1))))
   (if (or (<= lambda1 -1.32e-6) (not (<= lambda1 0.00025)))
     (atan2
      (*
       (cos phi2)
       (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
      (- t_0 (* (cos lambda1) t_1)))
     (atan2
      (* (cos phi2) (- (* lambda1 (cos lambda2)) (sin lambda2)))
      (- t_0 (* t_1 (+ (cos lambda2) (* lambda1 (sin lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin(phi1);
	double tmp;
	if ((lambda1 <= -1.32e-6) || !(lambda1 <= 0.00025)) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (t_0 - (cos(lambda1) * t_1)));
	} else {
		tmp = atan2((cos(phi2) * ((lambda1 * cos(lambda2)) - sin(lambda2))), (t_0 - (t_1 * (cos(lambda2) + (lambda1 * sin(lambda2))))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin(phi1)
    if ((lambda1 <= (-1.32d-6)) .or. (.not. (lambda1 <= 0.00025d0))) then
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (t_0 - (cos(lambda1) * t_1)))
    else
        tmp = atan2((cos(phi2) * ((lambda1 * cos(lambda2)) - sin(lambda2))), (t_0 - (t_1 * (cos(lambda2) + (lambda1 * sin(lambda2))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin(phi1);
	double tmp;
	if ((lambda1 <= -1.32e-6) || !(lambda1 <= 0.00025)) {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1)))), (t_0 - (Math.cos(lambda1) * t_1)));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * ((lambda1 * Math.cos(lambda2)) - Math.sin(lambda2))), (t_0 - (t_1 * (Math.cos(lambda2) + (lambda1 * Math.sin(lambda2))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos(phi2) * math.sin(phi1)
	tmp = 0
	if (lambda1 <= -1.32e-6) or not (lambda1 <= 0.00025):
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1)))), (t_0 - (math.cos(lambda1) * t_1)))
	else:
		tmp = math.atan2((math.cos(phi2) * ((lambda1 * math.cos(lambda2)) - math.sin(lambda2))), (t_0 - (t_1 * (math.cos(lambda2) + (lambda1 * math.sin(lambda2))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(phi1))
	tmp = 0.0
	if ((lambda1 <= -1.32e-6) || !(lambda1 <= 0.00025))
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(t_0 - Float64(cos(lambda1) * t_1)));
	else
		tmp = atan(Float64(cos(phi2) * Float64(Float64(lambda1 * cos(lambda2)) - sin(lambda2))), Float64(t_0 - Float64(t_1 * Float64(cos(lambda2) + Float64(lambda1 * sin(lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = cos(phi2) * sin(phi1);
	tmp = 0.0;
	if ((lambda1 <= -1.32e-6) || ~((lambda1 <= 0.00025)))
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (t_0 - (cos(lambda1) * t_1)));
	else
		tmp = atan2((cos(phi2) * ((lambda1 * cos(lambda2)) - sin(lambda2))), (t_0 - (t_1 * (cos(lambda2) + (lambda1 * sin(lambda2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -1.32e-6], N[Not[LessEqual[lambda1, 0.00025]], $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[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(lambda1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[(N[Cos[lambda2], $MachinePrecision] + N[(lambda1 * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.3200000000000001e-6 or 2.5000000000000001e-4 < lambda1

   1. Initial program 63.4%

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

      1. sin-diff 84.3%

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

      2. cancel-sign-sub-inv 84.3%

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

      3. fma-def 84.3%

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

   3. Applied egg-rr 84.3%

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

   4. Taylor expanded in lambda2 around 0 84.3%

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

   5. Taylor expanded in lambda1 around inf 84.2%

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

      1. *-commutative 84.2%

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

      2. mul-1-neg 84.2%

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

      3. distribute-rgt-neg-out 84.2%

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

      4. +-commutative 84.2%

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

      5. *-commutative 84.2%

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

      6. cancel-sign-sub-inv 84.2%

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

      7. *-commutative 84.2%

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

   7. Simplified 84.2%

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

   if -1.3200000000000001e-6 < lambda1 < 2.5000000000000001e-4

   1. Initial program 99.3%

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

   2. Taylor expanded in lambda1 around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. sin-neg 99.3%

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

      3. unsub-neg 99.3%

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

      4. cos-neg 99.3%

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

      5. *-commutative 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in lambda1 around 0 99.4%

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

      1. cos-neg 99.4%

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

      2. +-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. neg-mul-1 99.4%

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

      5. sin-neg 99.4%

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

      6. remove-double-neg 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.32 \cdot 10^{-6} \lor \neg \left(\lambda_1 \leq 0.00025\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]

Alternative 10: 89.9% accurate, 0.7× speedup

Math

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

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(Float64(cos(phi2) * 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[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 81.9%

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

   1. sin-diff 61.6%

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

   2. sub-neg 61.6%

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

3. Applied egg-rr 92.0%

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

   1. sub-neg 61.6%

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

5. Simplified 92.0%

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

6. Final simplification 92.0%

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

Alternative 11: 88.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{t_0}}}\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2 - \cos \lambda_1 \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 + t_0 \cdot \left(1 - e^{\mathsf{log1p}\left(t_3\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_3 (* (cos phi2) (sin phi1))))
   (if (<= phi1 -5e-11)
     (atan2
      t_2
      (-
       t_1
       (/
        (* (cos (/ (- (- phi2) phi2) 2.0)) (* 2.0 (sin (/ (+ phi1 phi1) 2.0))))
        (/ 2.0 t_0))))
     (if (<= phi1 1.15e-7)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))
        (- (sin phi2) (* (cos lambda1) t_3)))
       (atan2 t_2 (+ t_1 (* t_0 (- 1.0 (exp (log1p t_3))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double t_3 = cos(phi2) * sin(phi1);
	double tmp;
	if (phi1 <= -5e-11) {
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 1.15e-7) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), (sin(phi2) - (cos(lambda1) * t_3)));
	} else {
		tmp = atan2(t_2, (t_1 + (t_0 * (1.0 - exp(log1p(t_3))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_3 = Float64(cos(phi2) * sin(phi1))
	tmp = 0.0
	if (phi1 <= -5e-11)
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(Float64(Float64(Float64(-phi2) - phi2) / 2.0)) * Float64(2.0 * sin(Float64(Float64(phi1 + phi1) / 2.0)))) / Float64(2.0 / t_0))));
	elseif (phi1 <= 1.15e-7)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), Float64(sin(phi2) - Float64(cos(lambda1) * t_3)));
	else
		tmp = atan(t_2, Float64(t_1 + Float64(t_0 * Float64(1.0 - exp(log1p(t_3))))));
	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[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -5e-11], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[N[(N[((-phi2) - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(phi1 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.15e-7], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$1 + N[(t$95$0 * N[(1.0 - N[Exp[N[Log[1 + t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.00000000000000018e-11

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. sin-cos-mult 52.5%

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

      3. associate-*r/ 52.5%

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

   3. Applied egg-rr 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-/l* 52.6%

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

   5. Simplified 52.6%

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

      1. sum-sin 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. associate-*r* 52.6%

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

      2. *-commutative 52.6%

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

      3. associate--r+ 54.7%

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

      4. sub-neg 54.7%

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

      5. +-commutative 54.7%

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

      6. associate--l+ 71.9%

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

      7. +-inverses 71.9%

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

      8. associate-+l- 71.8%

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

      9. +-commutative 71.8%

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

      10. associate--r+ 83.8%

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

      11. +-inverses 83.8%

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

      12. neg-sub0 83.8%

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

      13. mul-1-neg 83.8%

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

      14. unsub-neg 83.8%

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

      15. mul-1-neg 83.8%

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

      16. remove-double-neg 83.8%

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

   9. Simplified 83.8%

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

   if -5.00000000000000018e-11 < phi1 < 1.14999999999999997e-7

   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. sin-diff 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. fma-def 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in lambda2 around 0 99.4%

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

   5. Taylor expanded in phi1 around 0 99.4%

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

   if 1.14999999999999997e-7 < phi1

   1. Initial program 80.1%

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

      1. expm1-log1p-u 80.0%

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

      2. expm1-udef 80.1%

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

   3. Applied egg-rr 80.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{\cos \left(\lambda_1 - \lambda_2\right)}}}\\ \mathbf{elif}\;\phi_1 \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(1 - e^{\mathsf{log1p}\left(\cos \phi_2 \cdot \sin \phi_1\right)}\right)}\\ \end{array} \]

Alternative 12: 88.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{t_0}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{t_1 - t_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 + t_0 \cdot \left(1 - e^{\mathsf{log1p}\left(t_3\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_3 (* (cos phi2) (sin phi1))))
   (if (<= phi1 -6e-11)
     (atan2
      t_2
      (-
       t_1
       (/
        (* (cos (/ (- (- phi2) phi2) 2.0)) (* 2.0 (sin (/ (+ phi1 phi1) 2.0))))
        (/ 2.0 t_0))))
     (if (<= phi1 3.2e-8)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))
        (- t_1 t_3))
       (atan2 t_2 (+ t_1 (* t_0 (- 1.0 (exp (log1p t_3))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double t_3 = cos(phi2) * sin(phi1);
	double tmp;
	if (phi1 <= -6e-11) {
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 3.2e-8) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), (t_1 - t_3));
	} else {
		tmp = atan2(t_2, (t_1 + (t_0 * (1.0 - exp(log1p(t_3))))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_3 = Float64(cos(phi2) * sin(phi1))
	tmp = 0.0
	if (phi1 <= -6e-11)
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(Float64(Float64(Float64(-phi2) - phi2) / 2.0)) * Float64(2.0 * sin(Float64(Float64(phi1 + phi1) / 2.0)))) / Float64(2.0 / t_0))));
	elseif (phi1 <= 3.2e-8)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), Float64(t_1 - t_3));
	else
		tmp = atan(t_2, Float64(t_1 + Float64(t_0 * Float64(1.0 - exp(log1p(t_3))))));
	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[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -6e-11], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[N[(N[((-phi2) - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(phi1 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.2e-8], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - t$95$3), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$1 + N[(t$95$0 * N[(1.0 - N[Exp[N[Log[1 + t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -6e-11

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. sin-cos-mult 52.5%

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

      3. associate-*r/ 52.5%

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

   3. Applied egg-rr 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-/l* 52.6%

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

   5. Simplified 52.6%

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

      1. sum-sin 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. associate-*r* 52.6%

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

      2. *-commutative 52.6%

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

      3. associate--r+ 54.7%

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

      4. sub-neg 54.7%

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

      5. +-commutative 54.7%

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

      6. associate--l+ 71.9%

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

      7. +-inverses 71.9%

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

      8. associate-+l- 71.8%

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

      9. +-commutative 71.8%

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

      10. associate--r+ 83.8%

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

      11. +-inverses 83.8%

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

      12. neg-sub0 83.8%

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

      13. mul-1-neg 83.8%

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

      14. unsub-neg 83.8%

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

      15. mul-1-neg 83.8%

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

      16. remove-double-neg 83.8%

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

   9. Simplified 83.8%

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

   if -6e-11 < phi1 < 3.2000000000000002e-8

   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. sin-diff 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. fma-def 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in lambda2 around 0 99.4%

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

   5. Taylor expanded in lambda1 around 0 99.4%

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

   if 3.2000000000000002e-8 < phi1

   1. Initial program 80.1%

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

      1. expm1-log1p-u 80.0%

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

      2. expm1-udef 80.1%

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

   3. Applied egg-rr 80.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{-11}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{\cos \left(\lambda_1 - \lambda_2\right)}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(1 - e^{\mathsf{log1p}\left(\cos \phi_2 \cdot \sin \phi_1\right)}\right)}\\ \end{array} \]

Alternative 13: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{t_0}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \frac{\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 + t_0 \cdot \left(1 - e^{\mathsf{log1p}\left(\cos \phi_2 \cdot \sin \phi_1\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -3.8e-13)
     (atan2
      t_2
      (-
       t_1
       (/
        (* (cos (/ (- (- phi2) phi2) 2.0)) (* 2.0 (sin (/ (+ phi1 phi1) 2.0))))
        (/ 2.0 t_0))))
     (if (<= phi1 3.2e-8)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
        (-
         (sin phi2)
         (/
          (+
           (sin (+ lambda2 (- phi1 lambda1)))
           (sin (+ phi1 (- lambda1 lambda2))))
          2.0)))
       (atan2
        t_2
        (+ t_1 (* t_0 (- 1.0 (exp (log1p (* (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(phi1) * sin(phi2);
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -3.8e-13) {
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 3.2e-8) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - ((sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2)))) / 2.0)));
	} else {
		tmp = atan2(t_2, (t_1 + (t_0 * (1.0 - exp(log1p((cos(phi2) * sin(phi1))))))));
	}
	return tmp;
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double t_2 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -3.8e-13) {
		tmp = Math.atan2(t_2, (t_1 - ((Math.cos(((-phi2 - phi2) / 2.0)) * (2.0 * Math.sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 3.2e-8) {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1)))), (Math.sin(phi2) - ((Math.sin((lambda2 + (phi1 - lambda1))) + Math.sin((phi1 + (lambda1 - lambda2)))) / 2.0)));
	} else {
		tmp = Math.atan2(t_2, (t_1 + (t_0 * (1.0 - Math.exp(Math.log1p((Math.cos(phi2) * Math.sin(phi1))))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi1) * math.sin(phi2)
	t_2 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -3.8e-13:
		tmp = math.atan2(t_2, (t_1 - ((math.cos(((-phi2 - phi2) / 2.0)) * (2.0 * math.sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))))
	elif phi1 <= 3.2e-8:
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1)))), (math.sin(phi2) - ((math.sin((lambda2 + (phi1 - lambda1))) + math.sin((phi1 + (lambda1 - lambda2)))) / 2.0)))
	else:
		tmp = math.atan2(t_2, (t_1 + (t_0 * (1.0 - math.exp(math.log1p((math.cos(phi2) * math.sin(phi1))))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -3.8e-13)
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(Float64(Float64(Float64(-phi2) - phi2) / 2.0)) * Float64(2.0 * sin(Float64(Float64(phi1 + phi1) / 2.0)))) / Float64(2.0 / t_0))));
	elseif (phi1 <= 3.2e-8)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(sin(phi2) - Float64(Float64(sin(Float64(lambda2 + Float64(phi1 - lambda1))) + sin(Float64(phi1 + Float64(lambda1 - lambda2)))) / 2.0)));
	else
		tmp = atan(t_2, Float64(t_1 + Float64(t_0 * Float64(1.0 - exp(log1p(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[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.8e-13], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[N[(N[((-phi2) - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(phi1 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.2e-8], 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[Sin[phi2], $MachinePrecision] - N[(N[(N[Sin[N[(lambda2 + N[(phi1 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(phi1 + N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$1 + N[(t$95$0 * N[(1.0 - N[Exp[N[Log[1 + N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.8e-13

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. sin-cos-mult 52.5%

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

      3. associate-*r/ 52.5%

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

   3. Applied egg-rr 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-/l* 52.6%

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

   5. Simplified 52.6%

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

      1. sum-sin 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. associate-*r* 52.6%

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

      2. *-commutative 52.6%

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

      3. associate--r+ 54.7%

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

      4. sub-neg 54.7%

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

      5. +-commutative 54.7%

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

      6. associate--l+ 71.9%

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

      7. +-inverses 71.9%

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

      8. associate-+l- 71.8%

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

      9. +-commutative 71.8%

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

      10. associate--r+ 83.8%

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

      11. +-inverses 83.8%

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

      12. neg-sub0 83.8%

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

      13. mul-1-neg 83.8%

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

      14. unsub-neg 83.8%

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

      15. mul-1-neg 83.8%

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

      16. remove-double-neg 83.8%

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

   9. Simplified 83.8%

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

   if -3.8e-13 < phi1 < 3.2000000000000002e-8

   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. Taylor expanded in phi2 around 0 81.8%

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

   3. Taylor expanded in phi1 around 0 81.8%

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

      1. sin-diff 97.5%

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

      2. sub-neg 97.5%

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

   5. Applied egg-rr 99.3%

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

      1. sub-neg 97.5%

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

   7. Simplified 99.3%

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

      1. sin-cos-mult 99.4%

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

      2. associate--r- 99.4%

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

   9. Applied egg-rr 99.4%

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

   if 3.2000000000000002e-8 < phi1

   1. Initial program 80.1%

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

      1. expm1-log1p-u 80.0%

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

      2. expm1-udef 80.1%

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

   3. Applied egg-rr 80.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{\cos \left(\lambda_1 - \lambda_2\right)}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \frac{\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 + \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(1 - e^{\mathsf{log1p}\left(\cos \phi_2 \cdot \sin \phi_1\right)}\right)}\\ \end{array} \]

Alternative 14: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{t_0}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \frac{\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.9e-15)
     (atan2
      t_2
      (-
       t_1
       (/
        (* (cos (/ (- (- phi2) phi2) 2.0)) (* 2.0 (sin (/ (+ phi1 phi1) 2.0))))
        (/ 2.0 t_0))))
     (if (<= phi1 3.2e-8)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
        (-
         (sin phi2)
         (/
          (+
           (sin (+ lambda2 (- phi1 lambda1)))
           (sin (+ phi1 (- lambda1 lambda2))))
          2.0)))
       (atan2 t_2 (- t_1 (* (* (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(phi1) * sin(phi2);
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.9e-15) {
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 3.2e-8) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - ((sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2)))) / 2.0)));
	} else {
		tmp = atan2(t_2, (t_1 - ((cos(phi2) * sin(phi1)) * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi1) * sin(phi2)
    t_2 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= (-2.9d-15)) then
        tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0d0)) * (2.0d0 * sin(((phi1 + phi1) / 2.0d0)))) / (2.0d0 / t_0))))
    else if (phi1 <= 3.2d-8) then
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - ((sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2)))) / 2.0d0)))
    else
        tmp = atan2(t_2, (t_1 - ((cos(phi2) * sin(phi1)) * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double t_2 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.9e-15) {
		tmp = Math.atan2(t_2, (t_1 - ((Math.cos(((-phi2 - phi2) / 2.0)) * (2.0 * Math.sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 3.2e-8) {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1)))), (Math.sin(phi2) - ((Math.sin((lambda2 + (phi1 - lambda1))) + Math.sin((phi1 + (lambda1 - lambda2)))) / 2.0)));
	} else {
		tmp = Math.atan2(t_2, (t_1 - ((Math.cos(phi2) * Math.sin(phi1)) * t_0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi1) * math.sin(phi2)
	t_2 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -2.9e-15:
		tmp = math.atan2(t_2, (t_1 - ((math.cos(((-phi2 - phi2) / 2.0)) * (2.0 * math.sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))))
	elif phi1 <= 3.2e-8:
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1)))), (math.sin(phi2) - ((math.sin((lambda2 + (phi1 - lambda1))) + math.sin((phi1 + (lambda1 - lambda2)))) / 2.0)))
	else:
		tmp = math.atan2(t_2, (t_1 - ((math.cos(phi2) * math.sin(phi1)) * t_0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.9e-15)
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(Float64(Float64(Float64(-phi2) - phi2) / 2.0)) * Float64(2.0 * sin(Float64(Float64(phi1 + phi1) / 2.0)))) / Float64(2.0 / t_0))));
	elseif (phi1 <= 3.2e-8)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(sin(phi2) - Float64(Float64(sin(Float64(lambda2 + Float64(phi1 - lambda1))) + sin(Float64(phi1 + Float64(lambda1 - lambda2)))) / 2.0)));
	else
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(phi2) * sin(phi1)) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi1) * sin(phi2);
	t_2 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -2.9e-15)
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	elseif (phi1 <= 3.2e-8)
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - ((sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2)))) / 2.0)));
	else
		tmp = atan2(t_2, (t_1 - ((cos(phi2) * sin(phi1)) * t_0)));
	end
	tmp_2 = 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[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.9e-15], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[N[(N[((-phi2) - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(phi1 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.2e-8], 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[Sin[phi2], $MachinePrecision] - N[(N[(N[Sin[N[(lambda2 + N[(phi1 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(phi1 + N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.90000000000000019e-15

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. sin-cos-mult 52.5%

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

      3. associate-*r/ 52.5%

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

   3. Applied egg-rr 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-/l* 52.6%

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

   5. Simplified 52.6%

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

      1. sum-sin 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. associate-*r* 52.6%

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

      2. *-commutative 52.6%

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

      3. associate--r+ 54.7%

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

      4. sub-neg 54.7%

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

      5. +-commutative 54.7%

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

      6. associate--l+ 71.9%

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

      7. +-inverses 71.9%

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

      8. associate-+l- 71.8%

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

      9. +-commutative 71.8%

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

      10. associate--r+ 83.8%

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

      11. +-inverses 83.8%

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

      12. neg-sub0 83.8%

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

      13. mul-1-neg 83.8%

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

      14. unsub-neg 83.8%

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

      15. mul-1-neg 83.8%

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

      16. remove-double-neg 83.8%

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

   9. Simplified 83.8%

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

   if -2.90000000000000019e-15 < phi1 < 3.2000000000000002e-8

   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. Taylor expanded in phi2 around 0 81.8%

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

   3. Taylor expanded in phi1 around 0 81.8%

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

      1. sin-diff 97.5%

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

      2. sub-neg 97.5%

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

   5. Applied egg-rr 99.3%

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

      1. sub-neg 97.5%

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

   7. Simplified 99.3%

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

      1. sin-cos-mult 99.4%

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

      2. associate--r- 99.4%

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

   9. Applied egg-rr 99.4%

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

   if 3.2000000000000002e-8 < phi1

   1. Initial program 80.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-15}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{\cos \left(\lambda_1 - \lambda_2\right)}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \frac{\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 15: 88.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{t_0}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.1e-13)
     (atan2
      t_2
      (-
       t_1
       (/
        (* (cos (/ (- (- phi2) phi2) 2.0)) (* 2.0 (sin (/ (+ phi1 phi1) 2.0))))
        (/ 2.0 t_0))))
     (if (<= phi1 3.2e-8)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
        (- (sin phi2) (* (cos lambda1) (sin phi1))))
       (atan2 t_2 (- t_1 (* (* (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(phi1) * sin(phi2);
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.1e-13) {
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 3.2e-8) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - (cos(lambda1) * sin(phi1))));
	} else {
		tmp = atan2(t_2, (t_1 - ((cos(phi2) * sin(phi1)) * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi1) * sin(phi2)
    t_2 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= (-2.1d-13)) then
        tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0d0)) * (2.0d0 * sin(((phi1 + phi1) / 2.0d0)))) / (2.0d0 / t_0))))
    else if (phi1 <= 3.2d-8) then
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - (cos(lambda1) * sin(phi1))))
    else
        tmp = atan2(t_2, (t_1 - ((cos(phi2) * sin(phi1)) * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double t_2 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.1e-13) {
		tmp = Math.atan2(t_2, (t_1 - ((Math.cos(((-phi2 - phi2) / 2.0)) * (2.0 * Math.sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 3.2e-8) {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1)))), (Math.sin(phi2) - (Math.cos(lambda1) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_2, (t_1 - ((Math.cos(phi2) * Math.sin(phi1)) * t_0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi1) * math.sin(phi2)
	t_2 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -2.1e-13:
		tmp = math.atan2(t_2, (t_1 - ((math.cos(((-phi2 - phi2) / 2.0)) * (2.0 * math.sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))))
	elif phi1 <= 3.2e-8:
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1)))), (math.sin(phi2) - (math.cos(lambda1) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_2, (t_1 - ((math.cos(phi2) * math.sin(phi1)) * t_0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.1e-13)
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(Float64(Float64(Float64(-phi2) - phi2) / 2.0)) * Float64(2.0 * sin(Float64(Float64(phi1 + phi1) / 2.0)))) / Float64(2.0 / t_0))));
	elseif (phi1 <= 3.2e-8)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(sin(phi2) - Float64(cos(lambda1) * sin(phi1))));
	else
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(phi2) * sin(phi1)) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi1) * sin(phi2);
	t_2 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -2.1e-13)
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	elseif (phi1 <= 3.2e-8)
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - (cos(lambda1) * sin(phi1))));
	else
		tmp = atan2(t_2, (t_1 - ((cos(phi2) * sin(phi1)) * t_0)));
	end
	tmp_2 = 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[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.1e-13], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[N[(N[((-phi2) - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(phi1 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.2e-8], 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[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.09999999999999989e-13

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. sin-cos-mult 52.5%

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

      3. associate-*r/ 52.5%

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

   3. Applied egg-rr 52.5%

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

      1. *-commutative 52.5%

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

      2. associate-/l* 52.6%

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

   5. Simplified 52.6%

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

      1. sum-sin 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. associate-*r* 52.6%

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

      2. *-commutative 52.6%

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

      3. associate--r+ 54.7%

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

      4. sub-neg 54.7%

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

      5. +-commutative 54.7%

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

      6. associate--l+ 71.9%

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

      7. +-inverses 71.9%

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

      8. associate-+l- 71.8%

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

      9. +-commutative 71.8%

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

      10. associate--r+ 83.8%

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

      11. +-inverses 83.8%

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

      12. neg-sub0 83.8%

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

      13. mul-1-neg 83.8%

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

      14. unsub-neg 83.8%

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

      15. mul-1-neg 83.8%

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

      16. remove-double-neg 83.8%

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

   9. Simplified 83.8%

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

   if -2.09999999999999989e-13 < phi1 < 3.2000000000000002e-8

   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. Taylor expanded in phi2 around 0 81.8%

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

   3. Taylor expanded in phi1 around 0 81.8%

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

      1. sin-diff 97.5%

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

      2. sub-neg 97.5%

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

   5. Applied egg-rr 99.3%

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

      1. sub-neg 97.5%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in lambda2 around 0 99.3%

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

   if 3.2000000000000002e-8 < phi1

   1. Initial program 80.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{\cos \left(\lambda_1 - \lambda_2\right)}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 16: 88.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{t_0}}}\\ \mathbf{elif}\;\phi_1 \leq 9.6 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -1.4e-9)
     (atan2
      t_2
      (-
       t_1
       (/
        (* (cos (/ (- (- phi2) phi2) 2.0)) (* 2.0 (sin (/ (+ phi1 phi1) 2.0))))
        (/ 2.0 t_0))))
     (if (<= phi1 9.6e-8)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
        (- (sin phi2) (* (cos lambda2) (sin phi1))))
       (atan2 t_2 (- t_1 (* (* (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(phi1) * sin(phi2);
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.4e-9) {
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 9.6e-8) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - (cos(lambda2) * sin(phi1))));
	} else {
		tmp = atan2(t_2, (t_1 - ((cos(phi2) * sin(phi1)) * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi1) * sin(phi2)
    t_2 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= (-1.4d-9)) then
        tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0d0)) * (2.0d0 * sin(((phi1 + phi1) / 2.0d0)))) / (2.0d0 / t_0))))
    else if (phi1 <= 9.6d-8) then
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - (cos(lambda2) * sin(phi1))))
    else
        tmp = atan2(t_2, (t_1 - ((cos(phi2) * sin(phi1)) * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double t_2 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.4e-9) {
		tmp = Math.atan2(t_2, (t_1 - ((Math.cos(((-phi2 - phi2) / 2.0)) * (2.0 * Math.sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 9.6e-8) {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.sin(lambda2) * Math.cos(lambda1)))), (Math.sin(phi2) - (Math.cos(lambda2) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_2, (t_1 - ((Math.cos(phi2) * Math.sin(phi1)) * t_0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi1) * math.sin(phi2)
	t_2 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -1.4e-9:
		tmp = math.atan2(t_2, (t_1 - ((math.cos(((-phi2 - phi2) / 2.0)) * (2.0 * math.sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))))
	elif phi1 <= 9.6e-8:
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.sin(lambda2) * math.cos(lambda1)))), (math.sin(phi2) - (math.cos(lambda2) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_2, (t_1 - ((math.cos(phi2) * math.sin(phi1)) * t_0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -1.4e-9)
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(Float64(Float64(Float64(-phi2) - phi2) / 2.0)) * Float64(2.0 * sin(Float64(Float64(phi1 + phi1) / 2.0)))) / Float64(2.0 / t_0))));
	elseif (phi1 <= 9.6e-8)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), Float64(sin(phi2) - Float64(cos(lambda2) * sin(phi1))));
	else
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(phi2) * sin(phi1)) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi1) * sin(phi2);
	t_2 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -1.4e-9)
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	elseif (phi1 <= 9.6e-8)
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), (sin(phi2) - (cos(lambda2) * sin(phi1))));
	else
		tmp = atan2(t_2, (t_1 - ((cos(phi2) * sin(phi1)) * t_0)));
	end
	tmp_2 = 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[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.4e-9], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[N[(N[((-phi2) - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(phi1 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 9.6e-8], 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[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.39999999999999992e-9

   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%

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

      2. sin-cos-mult 51.8%

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

      3. associate-*r/ 51.8%

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

   3. Applied egg-rr 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-/l* 51.8%

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

   5. Simplified 51.8%

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

      1. sum-sin 51.8%

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

   7. Applied egg-rr 51.8%

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

      1. associate-*r* 51.8%

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

      2. *-commutative 51.8%

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

      3. associate--r+ 54.0%

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

      4. sub-neg 54.0%

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

      5. +-commutative 54.0%

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

      6. associate--l+ 71.5%

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

      7. +-inverses 71.5%

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

      8. associate-+l- 71.3%

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

      9. +-commutative 71.3%

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

      10. associate--r+ 83.6%

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

      11. +-inverses 83.6%

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

      12. neg-sub0 83.6%

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

      13. mul-1-neg 83.6%

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

      14. unsub-neg 83.6%

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

      15. mul-1-neg 83.6%

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

      16. remove-double-neg 83.6%

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

   9. Simplified 83.6%

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

   if -1.39999999999999992e-9 < phi1 < 9.59999999999999994e-8

   1. Initial program 81.9%

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

   2. Taylor expanded in phi2 around 0 81.9%

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

   3. Taylor expanded in phi1 around 0 81.9%

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

      1. sin-diff 97.3%

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

      2. sub-neg 97.3%

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

   5. Applied egg-rr 99.3%

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

      1. sub-neg 97.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in lambda1 around 0 99.4%

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

      1. cos-neg 81.9%

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

   10. Simplified 99.4%

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

   if 9.59999999999999994e-8 < phi1

   1. Initial program 80.1%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{\cos \left(\lambda_1 - \lambda_2\right)}}}\\ \mathbf{elif}\;\phi_1 \leq 9.6 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 17: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{t_0}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -1.3e-46)
     (atan2
      t_2
      (-
       t_1
       (/
        (* (cos (/ (- (- phi2) phi2) 2.0)) (* 2.0 (sin (/ (+ phi1 phi1) 2.0))))
        (/ 2.0 t_0))))
     (if (<= phi1 3.2e-8)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))
        (sin phi2))
       (atan2 t_2 (- t_1 (* (* (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(phi1) * sin(phi2);
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.3e-46) {
		tmp = atan2(t_2, (t_1 - ((cos(((-phi2 - phi2) / 2.0)) * (2.0 * sin(((phi1 + phi1) / 2.0)))) / (2.0 / t_0))));
	} else if (phi1 <= 3.2e-8) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), sin(phi2));
	} else {
		tmp = atan2(t_2, (t_1 - ((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(phi1) * sin(phi2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -1.3e-46)
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(Float64(Float64(Float64(-phi2) - phi2) / 2.0)) * Float64(2.0 * sin(Float64(Float64(phi1 + phi1) / 2.0)))) / Float64(2.0 / t_0))));
	elseif (phi1 <= 3.2e-8)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), sin(phi2));
	else
		tmp = atan(t_2, Float64(t_1 - Float64(Float64(cos(phi2) * sin(phi1)) * 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[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.3e-46], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[N[(N[((-phi2) - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 * N[Sin[N[(N[(phi1 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.2e-8], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$1 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.3000000000000001e-46

   1. Initial program 83.8%

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

      1. *-commutative 83.8%

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

      2. sin-cos-mult 54.8%

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

      3. associate-*r/ 54.8%

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

   3. Applied egg-rr 54.8%

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

      1. *-commutative 54.8%

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

      2. associate-/l* 54.9%

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

   5. Simplified 54.9%

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

      1. sum-sin 54.9%

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

   7. Applied egg-rr 54.9%

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

      1. associate-*r* 54.9%

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

      2. *-commutative 54.9%

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

      3. associate--r+ 56.9%

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

      4. sub-neg 56.9%

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

      5. +-commutative 56.9%

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

      6. associate--l+ 72.8%

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

      7. +-inverses 72.8%

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

      8. associate-+l- 72.7%

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

      9. +-commutative 72.7%

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

      10. associate--r+ 83.9%

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

      11. +-inverses 83.9%

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

      12. neg-sub0 83.9%

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

      13. mul-1-neg 83.9%

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

      14. unsub-neg 83.9%

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

      15. mul-1-neg 83.9%

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

      16. remove-double-neg 83.9%

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

   9. Simplified 83.9%

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

   if -1.3000000000000001e-46 < phi1 < 3.2000000000000002e-8

   1. Initial program 81.7%

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

   2. Taylor expanded in phi2 around 0 81.7%

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

   3. Taylor expanded in phi1 around 0 81.7%

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

   4. Taylor expanded in phi1 around 0 81.2%

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

      1. sin-diff 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. fma-def 99.4%

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

   6. Applied egg-rr 98.9%

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

   if 3.2000000000000002e-8 < phi1

   1. Initial program 80.1%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \frac{\cos \left(\frac{\left(-\phi_2\right) - \phi_2}{2}\right) \cdot \left(2 \cdot \sin \left(\frac{\phi_1 + \phi_1}{2}\right)\right)}{\frac{2}{\cos \left(\lambda_1 - \lambda_2\right)}}}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 18: 87.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-46} \lor \neg \left(\phi_1 \leq 3.2 \cdot 10^{-8}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -1.85e-46) (not (<= phi1 3.2e-8)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
   (atan2
    (*
     (cos phi2)
     (fma (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 <= -1.85e-46) || !(phi1 <= 3.2e-8)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -1.85e-46) || !(phi1 <= 3.2e-8))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -1.85e-46], N[Not[LessEqual[phi1, 3.2e-8]], $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[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $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 < -1.84999999999999992e-46 or 3.2000000000000002e-8 < phi1

   1. Initial program 82.1%

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

   if -1.84999999999999992e-46 < phi1 < 3.2000000000000002e-8

   1. Initial program 81.7%

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

   2. Taylor expanded in phi2 around 0 81.7%

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

   3. Taylor expanded in phi1 around 0 81.7%

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

   4. Taylor expanded in phi1 around 0 81.2%

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

      1. sin-diff 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. fma-def 99.4%

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

   6. Applied egg-rr 98.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-46} \lor \neg \left(\phi_1 \leq 3.2 \cdot 10^{-8}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 19: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -44000000000 \lor \neg \left(\lambda_1 \leq 4.3 \cdot 10^{+20}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda1 -44000000000.0) (not (<= lambda1 4.3e+20)))
   (atan2
    (*
     (cos phi2)
     (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))
    (sin phi2))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (cos lambda2) (* (cos phi2) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 <= -44000000000.0) || !(lambda1 <= 4.3e+20)) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), sin(phi2));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda1 <= -44000000000.0) || !(lambda1 <= 4.3e+20))
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), sin(phi2));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda2) * Float64(cos(phi2) * sin(phi1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda1, -44000000000.0], N[Not[LessEqual[lambda1, 4.3e+20]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $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[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.4e10 or 4.3e20 < lambda1

   1. Initial program 61.6%

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

   2. Taylor expanded in phi2 around 0 52.7%

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

   3. Taylor expanded in phi1 around 0 51.6%

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

   4. Taylor expanded in phi1 around 0 44.2%

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

      1. sin-diff 84.4%

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

      2. cancel-sign-sub-inv 84.4%

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

      3. fma-def 84.4%

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

   6. Applied egg-rr 66.4%

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

   if -4.4e10 < lambda1 < 4.3e20

   1. Initial program 98.2%

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

   2. Taylor expanded in lambda1 around 0 96.5%

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

      1. cos-neg 79.0%

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

   4. Simplified 96.5%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -44000000000 \lor \neg \left(\lambda_1 \leq 4.3 \cdot 10^{+20}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 20: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1200:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 0.0045:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -1200.0)
   (atan2
    (*
     (cos phi2)
     (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
    (sin phi2))
   (if (<= lambda2 0.0045)
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (-
       (* (cos phi1) (sin phi2))
       (* (cos lambda1) (* (cos phi2) (sin phi1)))))
     (atan2
      (*
       (cos phi2)
       (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))
      (sin phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -1200.0) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2));
	} else if (lambda2 <= 0.0045) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -1200.0)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), sin(phi2));
	elseif (lambda2 <= 0.0045)
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -1200.0], 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], If[LessEqual[lambda2, 0.0045], 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[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1200

   1. Initial program 62.4%

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

   2. Taylor expanded in phi2 around 0 49.9%

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

   3. Taylor expanded in phi1 around 0 49.0%

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

   4. Taylor expanded in phi1 around 0 38.1%

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

      1. sin-diff 60.9%

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

      2. sub-neg 60.9%

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

   6. Applied egg-rr 60.9%

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

      1. sub-neg 60.9%

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

   8. Simplified 60.9%

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

   if -1200 < lambda2 < 0.00449999999999999966

   1. Initial program 98.8%

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

   2. Taylor expanded in lambda2 around 0 98.2%

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

   if 0.00449999999999999966 < lambda2

   1. Initial program 63.1%

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

   2. Taylor expanded in phi2 around 0 54.1%

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

   3. Taylor expanded in phi1 around 0 54.1%

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

   4. Taylor expanded in phi1 around 0 48.7%

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

      1. sin-diff 81.8%

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

      2. cancel-sign-sub-inv 81.8%

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

      3. fma-def 81.8%

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

   6. Applied egg-rr 66.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1200:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 0.0045:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 21: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -6.6:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_1 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 0.004:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_1 - \cos \lambda_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin phi1))) (t_1 (* (cos phi1) (sin phi2))))
   (if (<= lambda2 -6.6)
     (atan2
      (* (cos phi2) (sin (- lambda2)))
      (- t_1 (* t_0 (cos (- lambda1 lambda2)))))
     (if (<= lambda2 0.004)
       (atan2
        (* (cos phi2) (sin (- lambda1 lambda2)))
        (- t_1 (* (cos lambda1) t_0)))
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))
        (sin phi2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin(phi1);
	double t_1 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -6.6) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_1 - (t_0 * cos((lambda1 - lambda2)))));
	} else if (lambda2 <= 0.004) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_1 - (cos(lambda1) * t_0)));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(phi1))
	t_1 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -6.6)
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_1 - Float64(t_0 * cos(Float64(lambda1 - lambda2)))));
	elseif (lambda2 <= 0.004)
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_1 - Float64(cos(lambda1) * t_0)));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -6.6], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 0.004], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -6.5999999999999996

   1. Initial program 62.9%

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

   2. Taylor expanded in lambda1 around 0 65.9%

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

   if -6.5999999999999996 < lambda2 < 0.0040000000000000001

   1. Initial program 98.8%

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

   2. Taylor expanded in lambda2 around 0 98.7%

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

   if 0.0040000000000000001 < lambda2

   1. Initial program 63.1%

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

   2. Taylor expanded in phi2 around 0 54.1%

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

   3. Taylor expanded in phi1 around 0 54.1%

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

   4. Taylor expanded in phi1 around 0 48.7%

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

      1. sin-diff 81.8%

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

      2. cancel-sign-sub-inv 81.8%

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

      3. fma-def 81.8%

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

   6. Applied egg-rr 66.8%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -6.6:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 0.004:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 22: 71.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{t_0 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 0.00043:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (<= phi1 -8.6e-5)
     (atan2
      (* (cos phi2) (sin lambda1))
      (- t_0 (* (cos lambda1) (* (cos phi2) (sin phi1)))))
     (if (<= phi1 0.00043)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))
        (sin phi2))
       (atan2
        (* (cos phi2) (sin (- lambda1 lambda2)))
        (- t_0 (* (sin phi1) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -8.6e-5) {
		tmp = atan2((cos(phi2) * sin(lambda1)), (t_0 - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	} else if (phi1 <= 0.00043) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), sin(phi2));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (sin(phi1) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -8.6e-5)
		tmp = atan(Float64(cos(phi2) * sin(lambda1)), Float64(t_0 - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	elseif (phi1 <= 0.00043)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), sin(phi2));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8.6e-5], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 0.00043], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $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[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -8.6000000000000003e-5

   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. sin-diff 85.3%

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

      2. cancel-sign-sub-inv 85.3%

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

      3. fma-def 85.3%

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

   3. Applied egg-rr 85.3%

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

   4. Taylor expanded in lambda2 around 0 67.7%

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

   5. Taylor expanded in lambda2 around 0 54.9%

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

   if -8.6000000000000003e-5 < phi1 < 4.29999999999999989e-4

   1. Initial program 82.0%

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

   2. Taylor expanded in phi2 around 0 81.6%

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

   3. Taylor expanded in phi1 around 0 81.6%

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

   4. Taylor expanded in phi1 around 0 79.6%

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

      1. sin-diff 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. fma-def 99.4%

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

   6. Applied egg-rr 97.0%

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

   if 4.29999999999999989e-4 < phi1

   1. Initial program 79.8%

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

   2. Taylor expanded in phi2 around 0 53.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 0.00043:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 23: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - t_0}\\ \mathbf{elif}\;\phi_1 \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \sin \phi_2 - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (cos (- lambda1 lambda2))))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -8.6e-47)
     (atan2 t_1 (- (sin phi2) t_0))
     (if (<= phi1 1.7e-6)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (sin lambda2) (- (cos lambda1)))))
        (sin phi2))
       (atan2 t_1 (- (* (cos phi1) (sin phi2)) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -8.6e-47) {
		tmp = atan2(t_1, (sin(phi2) - t_0));
	} else if (phi1 <= 1.7e-6) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (sin(lambda2) * -cos(lambda1)))), sin(phi2));
	} else {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - t_0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -8.6e-47)
		tmp = atan(t_1, Float64(sin(phi2) - t_0));
	elseif (phi1 <= 1.7e-6)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(sin(lambda2) * Float64(-cos(lambda1))))), sin(phi2));
	else
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - t_0));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8.6e-47], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.7e-6], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * (-N[Cos[lambda1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -8.5999999999999995e-47

   1. Initial program 83.8%

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

   2. Taylor expanded in phi2 around 0 53.5%

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

   3. Taylor expanded in phi1 around 0 53.7%

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

   if -8.5999999999999995e-47 < phi1 < 1.70000000000000003e-6

   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. Taylor expanded in phi2 around 0 81.4%

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

   3. Taylor expanded in phi1 around 0 81.4%

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

   4. Taylor expanded in phi1 around 0 81.0%

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

      1. sin-diff 99.4%

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

      2. cancel-sign-sub-inv 99.4%

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

      3. fma-def 99.4%

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

   6. Applied egg-rr 98.5%

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

   if 1.70000000000000003e-6 < phi1

   1. Initial program 79.8%

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

   2. Taylor expanded in phi2 around 0 53.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \sin \lambda_2 \cdot \left(-\cos \lambda_1\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 24: 72.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - t_0}\\ \mathbf{elif}\;\phi_1 \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\cos \phi_1 \cdot \sin \phi_2 - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (cos (- lambda1 lambda2))))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -9e-46)
     (atan2 t_1 (- (sin phi2) t_0))
     (if (<= phi1 1.7e-6)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
        (sin phi2))
       (atan2 t_1 (- (* (cos phi1) (sin phi2)) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -9e-46) {
		tmp = atan2(t_1, (sin(phi2) - t_0));
	} else if (phi1 <= 1.7e-6) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2));
	} else {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(phi1) * cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= (-9d-46)) then
        tmp = atan2(t_1, (sin(phi2) - t_0))
    else if (phi1 <= 1.7d-6) then
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2))
    else
        tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -9e-46) {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - t_0));
	} else if (phi1 <= 1.7e-6) {
		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(t_1, ((Math.cos(phi1) * Math.sin(phi2)) - t_0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -9e-46:
		tmp = math.atan2(t_1, (math.sin(phi2) - t_0))
	elif phi1 <= 1.7e-6:
		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(t_1, ((math.cos(phi1) * math.sin(phi2)) - t_0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -9e-46)
		tmp = atan(t_1, Float64(sin(phi2) - t_0));
	elseif (phi1 <= 1.7e-6)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), sin(phi2));
	else
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -9e-46)
		tmp = atan2(t_1, (sin(phi2) - t_0));
	elseif (phi1 <= 1.7e-6)
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2));
	else
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -9e-46], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.7e-6], 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[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -9.00000000000000001e-46

   1. Initial program 83.8%

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

   2. Taylor expanded in phi2 around 0 53.5%

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

   3. Taylor expanded in phi1 around 0 53.7%

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

   if -9.00000000000000001e-46 < phi1 < 1.70000000000000003e-6

   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. Taylor expanded in phi2 around 0 81.4%

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

   3. Taylor expanded in phi1 around 0 81.4%

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

   4. Taylor expanded in phi1 around 0 81.0%

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

      1. sin-diff 98.5%

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

      2. sub-neg 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. sub-neg 98.5%

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

   8. Simplified 98.5%

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

   if 1.70000000000000003e-6 < phi1

   1. Initial program 79.8%

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

   2. Taylor expanded in phi2 around 0 53.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 25: 72.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \left(-\sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -8.6e-46)
     (atan2 t_0 (- (sin phi2) (* (sin phi1) (cos (- lambda1 lambda2)))))
     (if (<= phi1 2.8e-6)
       (atan2
        (*
         (cos phi2)
         (- (* (sin lambda1) (cos lambda2)) (* (sin lambda2) (cos lambda1))))
        (sin phi2))
       (atan2 t_0 (* (cos (- lambda2 lambda1)) (- (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -8.6e-46) {
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else if (phi1 <= 2.8e-6) {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2));
	} else {
		tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= (-8.6d-46)) then
        tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))))
    else if (phi1 <= 2.8d-6) then
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2))
    else
        tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -8.6e-46) {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	} else if (phi1 <= 2.8e-6) {
		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(t_0, (Math.cos((lambda2 - lambda1)) * -Math.sin(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -8.6e-46:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	elif phi1 <= 2.8e-6:
		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(t_0, (math.cos((lambda2 - lambda1)) * -math.sin(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -8.6e-46)
		tmp = atan(t_0, Float64(sin(phi2) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	elseif (phi1 <= 2.8e-6)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(sin(lambda2) * cos(lambda1)))), sin(phi2));
	else
		tmp = atan(t_0, Float64(cos(Float64(lambda2 - lambda1)) * Float64(-sin(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 <= -8.6e-46)
		tmp = atan2(t_0, (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	elseif (phi1 <= 2.8e-6)
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (sin(lambda2) * cos(lambda1)))), sin(phi2));
	else
		tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8.6e-46], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 2.8e-6], 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[t$95$0 / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -8.6000000000000007e-46

   1. Initial program 83.8%

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

   2. Taylor expanded in phi2 around 0 53.5%

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

   3. Taylor expanded in phi1 around 0 53.7%

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

   if -8.6000000000000007e-46 < phi1 < 2.79999999999999987e-6

   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. Taylor expanded in phi2 around 0 81.4%

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

   3. Taylor expanded in phi1 around 0 81.4%

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

   4. Taylor expanded in phi1 around 0 81.0%

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

      1. sin-diff 98.5%

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

      2. sub-neg 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. sub-neg 98.5%

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

   8. Simplified 98.5%

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

   if 2.79999999999999987e-6 < phi1

   1. Initial program 79.8%

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

   2. Taylor expanded in phi2 around 0 53.4%

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

   3. Taylor expanded in phi1 around 0 49.5%

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

   4. Taylor expanded in phi2 around 0 51.0%

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

      1. associate-*r* 51.0%

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

      2. *-commutative 51.0%

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

      3. sub-neg 51.0%

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

      4. +-commutative 51.0%

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

      5. neg-mul-1 51.0%

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

      6. neg-mul-1 51.0%

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

      7. remove-double-neg 51.0%

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

      8. mul-1-neg 51.0%

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

      9. distribute-neg-in 51.0%

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

      10. +-commutative 51.0%

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

      11. neg-mul-1 51.0%

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

      12. cos-neg 51.0%

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

      13. +-commutative 51.0%

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

      14. mul-1-neg 51.0%

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

      15. unsub-neg 51.0%

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

   6. Simplified 51.0%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{-46}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2 \cdot \cos \lambda_1\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \left(-\sin \phi_1\right)}\\ \end{array} \]

Alternative 26: 62.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -22500000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -22500000.0)
   (atan2
    (* (cos phi2) (sin (- lambda2)))
    (- (sin phi2) (* (sin phi1) (cos (- lambda1 lambda2)))))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (- (sin phi2) (* (cos lambda1) (sin phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -22500000.0) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda1) * sin(phi1))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= (-22500000.0d0)) then
        tmp = atan2((cos(phi2) * sin(-lambda2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda1) * sin(phi1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -22500000.0) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (Math.sin(phi2) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.sin(phi2) - (Math.cos(lambda1) * Math.sin(phi1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -22500000.0:
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), (math.sin(phi2) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.sin(phi2) - (math.cos(lambda1) * math.sin(phi1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -22500000.0)
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(sin(phi2) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(sin(phi2) - Float64(cos(lambda1) * sin(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -22500000.0)
		tmp = atan2((cos(phi2) * sin(-lambda2)), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda1) * sin(phi1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -22500000.0], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -2.25e7

   1. Initial program 62.4%

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

   2. Taylor expanded in phi2 around 0 49.9%

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

   3. Taylor expanded in phi1 around 0 49.0%

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

   4. Taylor expanded in lambda1 around 0 52.0%

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

   if -2.25e7 < lambda2

   1. Initial program 88.9%

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

   2. Taylor expanded in phi2 around 0 74.1%

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

   3. Taylor expanded in phi1 around 0 73.3%

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

   4. Taylor expanded in lambda2 around 0 71.8%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -22500000:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]

Alternative 27: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 5.2 \cdot 10^{+33}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \left(-\sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 5.2e+33)
     (atan2 t_0 (- (sin phi2) (* (cos lambda1) (sin phi1))))
     (atan2 t_0 (* (cos (- lambda2 lambda1)) (- (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 5.2e+33) {
		tmp = atan2(t_0, (sin(phi2) - (cos(lambda1) * sin(phi1))));
	} else {
		tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= 5.2d+33) then
        tmp = atan2(t_0, (sin(phi2) - (cos(lambda1) * sin(phi1))))
    else
        tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 5.2e+33) {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.cos(lambda1) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_0, (Math.cos((lambda2 - lambda1)) * -Math.sin(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= 5.2e+33:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.cos(lambda1) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_0, (math.cos((lambda2 - lambda1)) * -math.sin(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= 5.2e+33)
		tmp = atan(t_0, Float64(sin(phi2) - Float64(cos(lambda1) * sin(phi1))));
	else
		tmp = atan(t_0, Float64(cos(Float64(lambda2 - lambda1)) * Float64(-sin(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 <= 5.2e+33)
		tmp = atan2(t_0, (sin(phi2) - (cos(lambda1) * sin(phi1))));
	else
		tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 5.2e+33], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 5.1999999999999995e33

   1. Initial program 82.5%

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

   2. Taylor expanded in phi2 around 0 70.8%

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

   3. Taylor expanded in phi1 around 0 70.8%

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

   4. Taylor expanded in lambda2 around 0 68.7%

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

   if 5.1999999999999995e33 < phi1

   1. Initial program 79.4%

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

   2. Taylor expanded in phi2 around 0 55.4%

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

   3. Taylor expanded in phi1 around 0 51.1%

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

   4. Taylor expanded in phi2 around 0 52.8%

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

      1. associate-*r* 52.8%

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

      2. *-commutative 52.8%

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

      3. sub-neg 52.8%

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

      4. +-commutative 52.8%

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

      5. neg-mul-1 52.8%

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

      6. neg-mul-1 52.8%

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

      7. remove-double-neg 52.8%

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

      8. mul-1-neg 52.8%

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

      9. distribute-neg-in 52.8%

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

      10. +-commutative 52.8%

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

      11. neg-mul-1 52.8%

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

      12. cos-neg 52.8%

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

      13. +-commutative 52.8%

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

      14. mul-1-neg 52.8%

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

      15. unsub-neg 52.8%

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

   6. Simplified 52.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 5.2 \cdot 10^{+33}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \left(-\sin \phi_1\right)}\\ \end{array} \]

Alternative 28: 62.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 0.22:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \left(-\sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 0.22)
     (atan2 t_0 (- (sin phi2) (* (cos lambda2) (sin phi1))))
     (atan2 t_0 (* (cos (- lambda2 lambda1)) (- (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 0.22) {
		tmp = atan2(t_0, (sin(phi2) - (cos(lambda2) * sin(phi1))));
	} else {
		tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= 0.22d0) then
        tmp = atan2(t_0, (sin(phi2) - (cos(lambda2) * sin(phi1))))
    else
        tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 0.22) {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.cos(lambda2) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_0, (Math.cos((lambda2 - lambda1)) * -Math.sin(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= 0.22:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.cos(lambda2) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_0, (math.cos((lambda2 - lambda1)) * -math.sin(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= 0.22)
		tmp = atan(t_0, Float64(sin(phi2) - Float64(cos(lambda2) * sin(phi1))));
	else
		tmp = atan(t_0, Float64(cos(Float64(lambda2 - lambda1)) * Float64(-sin(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 <= 0.22)
		tmp = atan2(t_0, (sin(phi2) - (cos(lambda2) * sin(phi1))));
	else
		tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 0.22], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 0.220000000000000001

   1. Initial program 82.5%

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

   2. Taylor expanded in phi2 around 0 71.7%

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

   3. Taylor expanded in phi1 around 0 71.8%

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

   4. Taylor expanded in lambda1 around 0 70.3%

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

      1. cos-neg 70.3%

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

   6. Simplified 70.3%

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

   if 0.220000000000000001 < phi1

   1. Initial program 79.8%

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

   2. Taylor expanded in phi2 around 0 53.4%

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

   3. Taylor expanded in phi1 around 0 49.5%

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

   4. Taylor expanded in phi2 around 0 51.0%

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

      1. associate-*r* 51.0%

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

      2. *-commutative 51.0%

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

      3. sub-neg 51.0%

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

      4. +-commutative 51.0%

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

      5. neg-mul-1 51.0%

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

      6. neg-mul-1 51.0%

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

      7. remove-double-neg 51.0%

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

      8. mul-1-neg 51.0%

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

      9. distribute-neg-in 51.0%

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

      10. +-commutative 51.0%

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

      11. neg-mul-1 51.0%

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

      12. cos-neg 51.0%

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

      13. +-commutative 51.0%

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

      14. mul-1-neg 51.0%

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

      15. unsub-neg 51.0%

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

   6. Simplified 51.0%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 0.22:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \left(-\sin \phi_1\right)}\\ \end{array} \]

Alternative 29: 64.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (sin(phi1) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.9%

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

2. Taylor expanded in phi2 around 0 67.6%

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

3. Taylor expanded in phi1 around 0 66.8%

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

4. Final simplification 66.8%

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

Alternative 30: 64.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.00038 \lor \neg \left(\phi_2 \leq 0.7\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi2 -0.00038) (not (<= phi2 0.7)))
     (atan2 t_0 (sin phi2))
     (atan2 t_0 (- phi2 (* (sin phi1) (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -0.00038) || !(phi2 <= 0.7)) {
		tmp = atan2(t_0, sin(phi2));
	} else {
		tmp = atan2(t_0, (phi2 - (sin(phi1) * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if ((phi2 <= (-0.00038d0)) .or. (.not. (phi2 <= 0.7d0))) then
        tmp = atan2(t_0, sin(phi2))
    else
        tmp = atan2(t_0, (phi2 - (sin(phi1) * cos((lambda2 - lambda1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -0.00038) || !(phi2 <= 0.7)) {
		tmp = Math.atan2(t_0, Math.sin(phi2));
	} else {
		tmp = Math.atan2(t_0, (phi2 - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -0.00038) or not (phi2 <= 0.7):
		tmp = math.atan2(t_0, math.sin(phi2))
	else:
		tmp = math.atan2(t_0, (phi2 - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if ((phi2 <= -0.00038) || !(phi2 <= 0.7))
		tmp = atan(t_0, sin(phi2));
	else
		tmp = atan(t_0, Float64(phi2 - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -0.00038) || ~((phi2 <= 0.7)))
		tmp = atan2(t_0, sin(phi2));
	else
		tmp = atan2(t_0, (phi2 - (sin(phi1) * cos((lambda2 - lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -0.00038], N[Not[LessEqual[phi2, 0.7]], $MachinePrecision]], N[ArcTan[t$95$0 / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(phi2 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.8000000000000002e-4 or 0.69999999999999996 < phi2

   1. Initial program 77.5%

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

   2. Taylor expanded in phi2 around 0 49.4%

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

   3. Taylor expanded in phi1 around 0 48.6%

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

   4. Taylor expanded in phi1 around 0 47.0%

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

   if -3.8000000000000002e-4 < phi2 < 0.69999999999999996

   1. Initial program 86.0%

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

   2. Taylor expanded in phi2 around 0 84.8%

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

   3. Taylor expanded in phi1 around 0 84.0%

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

   4. Taylor expanded in phi2 around 0 84.0%

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

      1. +-commutative 84.0%

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

      2. mul-1-neg 84.0%

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

      3. *-commutative 84.0%

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

      4. unsub-neg 84.0%

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

      5. sub-neg 84.0%

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

      6. +-commutative 84.0%

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

      7. neg-mul-1 84.0%

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

      8. neg-mul-1 84.0%

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

      9. remove-double-neg 84.0%

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

      10. mul-1-neg 84.0%

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

      11. distribute-neg-in 84.0%

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

      12. +-commutative 84.0%

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

      13. cos-neg 84.0%

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

      14. +-commutative 84.0%

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

      15. mul-1-neg 84.0%

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

      16. unsub-neg 84.0%

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

   6. Simplified 84.0%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00038 \lor \neg \left(\phi_2 \leq 0.7\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]

Alternative 31: 63.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.95 \cdot 10^{+28}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\phi_2 - \sin \phi_1 \cdot t_0}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 \cdot \left(-\sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -1.95e+28)
     (atan2 t_1 (- phi2 (* (sin phi1) t_0)))
     (if (<= phi1 3.2e+14)
       (atan2 t_1 (- (sin phi2) (* phi1 (cos (- lambda1 lambda2)))))
       (atan2 t_1 (* t_0 (- (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.95e+28) {
		tmp = atan2(t_1, (phi2 - (sin(phi1) * t_0)));
	} else if (phi1 <= 3.2e+14) {
		tmp = atan2(t_1, (sin(phi2) - (phi1 * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2(t_1, (t_0 * -sin(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= (-1.95d+28)) then
        tmp = atan2(t_1, (phi2 - (sin(phi1) * t_0)))
    else if (phi1 <= 3.2d+14) then
        tmp = atan2(t_1, (sin(phi2) - (phi1 * cos((lambda1 - lambda2)))))
    else
        tmp = atan2(t_1, (t_0 * -sin(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.95e+28) {
		tmp = Math.atan2(t_1, (phi2 - (Math.sin(phi1) * t_0)));
	} else if (phi1 <= 3.2e+14) {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - (phi1 * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2(t_1, (t_0 * -Math.sin(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -1.95e+28:
		tmp = math.atan2(t_1, (phi2 - (math.sin(phi1) * t_0)))
	elif phi1 <= 3.2e+14:
		tmp = math.atan2(t_1, (math.sin(phi2) - (phi1 * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2(t_1, (t_0 * -math.sin(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -1.95e+28)
		tmp = atan(t_1, Float64(phi2 - Float64(sin(phi1) * t_0)));
	elseif (phi1 <= 3.2e+14)
		tmp = atan(t_1, Float64(sin(phi2) - Float64(phi1 * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(t_1, Float64(t_0 * Float64(-sin(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -1.95e+28)
		tmp = atan2(t_1, (phi2 - (sin(phi1) * t_0)));
	elseif (phi1 <= 3.2e+14)
		tmp = atan2(t_1, (sin(phi2) - (phi1 * cos((lambda1 - lambda2)))));
	else
		tmp = atan2(t_1, (t_0 * -sin(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.95e+28], N[ArcTan[t$95$1 / N[(phi2 - N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.2e+14], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(phi1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.9499999999999999e28

   1. Initial program 83.2%

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

   2. Taylor expanded in phi2 around 0 50.7%

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

   3. Taylor expanded in phi1 around 0 51.0%

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

   4. Taylor expanded in phi2 around 0 48.5%

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

      1. +-commutative 48.5%

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

      2. mul-1-neg 48.5%

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

      3. *-commutative 48.5%

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

      4. unsub-neg 48.5%

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

      5. sub-neg 48.5%

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

      6. +-commutative 48.5%

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

      7. neg-mul-1 48.5%

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

      8. neg-mul-1 48.5%

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

      9. remove-double-neg 48.5%

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

      10. mul-1-neg 48.5%

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

      11. distribute-neg-in 48.5%

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

      12. +-commutative 48.5%

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

      13. cos-neg 48.5%

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

      14. +-commutative 48.5%

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

      15. mul-1-neg 48.5%

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

      16. unsub-neg 48.5%

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

   6. Simplified 48.5%

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

   if -1.9499999999999999e28 < phi1 < 3.2e14

   1. Initial program 81.9%

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

   2. Taylor expanded in phi2 around 0 80.3%

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

   3. Taylor expanded in phi1 around 0 80.2%

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

   4. Taylor expanded in phi1 around 0 80.7%

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

   if 3.2e14 < phi1

   1. Initial program 80.5%

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

   2. Taylor expanded in phi2 around 0 54.8%

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

   3. Taylor expanded in phi1 around 0 50.8%

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

   4. Taylor expanded in phi2 around 0 52.3%

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

      1. associate-*r* 52.3%

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

      2. *-commutative 52.3%

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

      3. sub-neg 52.3%

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

      4. +-commutative 52.3%

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

      5. neg-mul-1 52.3%

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

      6. neg-mul-1 52.3%

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

      7. remove-double-neg 52.3%

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

      8. mul-1-neg 52.3%

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

      9. distribute-neg-in 52.3%

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

      10. +-commutative 52.3%

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

      11. neg-mul-1 52.3%

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

      12. cos-neg 52.3%

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

      13. +-commutative 52.3%

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

      14. mul-1-neg 52.3%

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

      15. unsub-neg 52.3%

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

   6. Simplified 52.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.95 \cdot 10^{+28}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \left(-\sin \phi_1\right)}\\ \end{array} \]

Alternative 32: 62.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{-45} \lor \neg \left(\phi_1 \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi1 -1.1e-45) (not (<= phi1 2.7e-5)))
     (atan2 t_0 (* (cos (- lambda2 lambda1)) (- (sin phi1))))
     (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 <= -1.1e-45) || !(phi1 <= 2.7e-5)) {
		tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)));
	} 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 <= (-1.1d-45)) .or. (.not. (phi1 <= 2.7d-5))) then
        tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)))
    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 <= -1.1e-45) || !(phi1 <= 2.7e-5)) {
		tmp = Math.atan2(t_0, (Math.cos((lambda2 - lambda1)) * -Math.sin(phi1)));
	} 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 <= -1.1e-45) or not (phi1 <= 2.7e-5):
		tmp = math.atan2(t_0, (math.cos((lambda2 - lambda1)) * -math.sin(phi1)))
	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 <= -1.1e-45) || !(phi1 <= 2.7e-5))
		tmp = atan(t_0, Float64(cos(Float64(lambda2 - lambda1)) * Float64(-sin(phi1))));
	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 <= -1.1e-45) || ~((phi1 <= 2.7e-5)))
		tmp = atan2(t_0, (cos((lambda2 - lambda1)) * -sin(phi1)));
	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, -1.1e-45], N[Not[LessEqual[phi1, 2.7e-5]], $MachinePrecision]], N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.09999999999999997e-45 or 2.6999999999999999e-5 < phi1

   1. Initial program 82.0%

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

   2. Taylor expanded in phi2 around 0 53.5%

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

   3. Taylor expanded in phi1 around 0 51.8%

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

   4. Taylor expanded in phi2 around 0 50.1%

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

      1. associate-*r* 50.1%

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

      2. *-commutative 50.1%

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

      3. sub-neg 50.1%

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

      4. +-commutative 50.1%

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

      5. neg-mul-1 50.1%

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

      6. neg-mul-1 50.1%

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

      7. remove-double-neg 50.1%

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

      8. mul-1-neg 50.1%

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

      9. distribute-neg-in 50.1%

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

      10. +-commutative 50.1%

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

      11. neg-mul-1 50.1%

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

      12. cos-neg 50.1%

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

      13. +-commutative 50.1%

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

      14. mul-1-neg 50.1%

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

      15. unsub-neg 50.1%

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

   6. Simplified 50.1%

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

   if -1.09999999999999997e-45 < phi1 < 2.6999999999999999e-5

   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. Taylor expanded in phi2 around 0 81.4%

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

   3. Taylor expanded in phi1 around 0 81.4%

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

   4. Taylor expanded in phi1 around 0 81.0%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{-45} \lor \neg \left(\phi_1 \leq 2.7 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_2 - \lambda_1\right) \cdot \left(-\sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 33: 44.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.02 \cdot 10^{-16} \lor \neg \left(\lambda_2 \leq 1.15 \cdot 10^{-64}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -1.02e-16) (not (<= lambda2 1.15e-64)))
   (atan2 (* (cos phi2) (sin (- lambda2))) (sin phi2))
   (atan2 (* (cos phi2) (sin lambda1)) (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -1.02e-16) || !(lambda2 <= 1.15e-64)) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), sin(phi2));
	} else {
		tmp = atan2((cos(phi2) * 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 ((lambda2 <= (-1.02d-16)) .or. (.not. (lambda2 <= 1.15d-64))) then
        tmp = atan2((cos(phi2) * sin(-lambda2)), sin(phi2))
    else
        tmp = atan2((cos(phi2) * 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 ((lambda2 <= -1.02e-16) || !(lambda2 <= 1.15e-64)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(lambda1)), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda2 <= -1.02e-16) or not (lambda2 <= 1.15e-64):
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), math.sin(phi2))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin(lambda1)), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -1.02e-16) || !(lambda2 <= 1.15e-64))
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), sin(phi2));
	else
		tmp = atan(Float64(cos(phi2) * sin(lambda1)), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda2 <= -1.02e-16) || ~((lambda2 <= 1.15e-64)))
		tmp = atan2((cos(phi2) * sin(-lambda2)), sin(phi2));
	else
		tmp = atan2((cos(phi2) * sin(lambda1)), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -1.02e-16], N[Not[LessEqual[lambda2, 1.15e-64]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.0200000000000001e-16 or 1.1500000000000001e-64 < lambda2

   1. Initial program 66.9%

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

   2. Taylor expanded in phi2 around 0 55.8%

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

   3. Taylor expanded in phi1 around 0 55.1%

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

   4. Taylor expanded in phi1 around 0 45.7%

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

   5. Taylor expanded in lambda1 around 0 44.4%

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

   if -1.0200000000000001e-16 < lambda2 < 1.1500000000000001e-64

   1. Initial program 99.7%

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

   2. Taylor expanded in phi2 around 0 81.7%

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

   3. Taylor expanded in phi1 around 0 80.8%

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

   4. Taylor expanded in phi1 around 0 59.0%

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

   5. Taylor expanded in lambda2 around 0 55.0%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.02 \cdot 10^{-16} \lor \neg \left(\lambda_2 \leq 1.15 \cdot 10^{-64}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\sin \phi_2}\\ \end{array} \]

Alternative 34: 39.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -660 \lor \neg \left(\phi_2 \leq 2.2 \cdot 10^{+55}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -660.0) (not (<= phi2 2.2e+55)))
   (atan2 (* (cos phi2) (sin lambda1)) (sin phi2))
   (atan2 (sin (- lambda1 lambda2)) (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -660.0) || !(phi2 <= 2.2e+55)) {
		tmp = atan2((cos(phi2) * sin(lambda1)), 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 <= (-660.0d0)) .or. (.not. (phi2 <= 2.2d+55))) then
        tmp = atan2((cos(phi2) * sin(lambda1)), 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 <= -660.0) || !(phi2 <= 2.2e+55)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(lambda1)), 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 <= -660.0) or not (phi2 <= 2.2e+55):
		tmp = math.atan2((math.cos(phi2) * math.sin(lambda1)), 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 <= -660.0) || !(phi2 <= 2.2e+55))
		tmp = atan(Float64(cos(phi2) * sin(lambda1)), 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 <= -660.0) || ~((phi2 <= 2.2e+55)))
		tmp = atan2((cos(phi2) * sin(lambda1)), 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, -660.0], N[Not[LessEqual[phi2, 2.2e+55]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $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 < -660 or 2.2000000000000001e55 < phi2

   1. Initial program 77.0%

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

   2. Taylor expanded in phi2 around 0 50.0%

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

   3. Taylor expanded in phi1 around 0 49.3%

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

   4. Taylor expanded in phi1 around 0 48.6%

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

   5. Taylor expanded in lambda2 around 0 33.3%

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

   if -660 < phi2 < 2.2000000000000001e55

   1. Initial program 86.1%

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

   2. Taylor expanded in phi2 around 0 82.7%

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

   3. Taylor expanded in phi1 around 0 81.8%

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

   4. Taylor expanded in phi1 around 0 54.5%

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

   5. Taylor expanded in phi2 around 0 54.4%

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

4. Final simplification 44.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -660 \lor \neg \left(\phi_2 \leq 2.2 \cdot 10^{+55}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \lambda_1}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 35: 49.1% 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 81.9%

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

2. Taylor expanded in phi2 around 0 67.6%

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

3. Taylor expanded in phi1 around 0 66.8%

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

4. Taylor expanded in phi1 around 0 51.7%

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

5. Final simplification 51.7%

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

Alternative 36: 32.4% 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 81.9%

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

2. Taylor expanded in phi2 around 0 67.6%

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

3. Taylor expanded in phi1 around 0 66.8%

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

4. Taylor expanded in phi1 around 0 51.7%

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

5. Taylor expanded in phi2 around 0 34.8%

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

6. Final simplification 34.8%

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

Reproduce

herbie shell --seed 2023214 
(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))))))