Bearing on a great circle

Percentage Accurate: 80.5% → 99.7%

Time: 33.2s

Alternatives: 26

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: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin((lambda1 - lambda2)) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((sin(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. sin-diff 89.7%

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

   2. fma-neg 89.7%

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

4. Applied egg-rr 89.7%

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

   1. cos-diff 99.7%

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

   2. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. fma-define 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. sin-diff 89.7%

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

   2. fma-neg 89.7%

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

4. Applied egg-rr 89.7%

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

   1. cos-diff 81.7%

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

   2. +-commutative 81.7%

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

   3. *-commutative 81.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \mathsf{expm1}\left(t\_0\right)\\ t_2 := \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2\\ t_3 := \cos \phi_2 \cdot \sin \phi_1\\ t_4 := t\_3 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{log1p}\left(t\_1\right) - t\_4}\\ \mathbf{elif}\;\phi_2 \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{t\_0 - t\_3 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\log \left(t\_1 + 1\right) - t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (expm1 t_0))
        (t_2
         (*
          (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
          (cos phi2)))
        (t_3 (* (cos phi2) (sin phi1)))
        (t_4 (* t_3 (cos (- lambda1 lambda2)))))
   (if (<= phi2 -1e-5)
     (atan2 t_2 (- (log1p t_1) t_4))
     (if (<= phi2 4.2e-7)
       (atan2
        (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
        (-
         t_0
         (*
          t_3
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (atan2 t_2 (- (log (+ t_1 1.0)) t_4))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = expm1(t_0);
	double t_2 = fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2);
	double t_3 = cos(phi2) * sin(phi1);
	double t_4 = t_3 * cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -1e-5) {
		tmp = atan2(t_2, (log1p(t_1) - t_4));
	} else if (phi2 <= 4.2e-7) {
		tmp = atan2(((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))), (t_0 - (t_3 * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2(t_2, (log((t_1 + 1.0)) - t_4));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = expm1(t_0)
	t_2 = Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2))
	t_3 = Float64(cos(phi2) * sin(phi1))
	t_4 = Float64(t_3 * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -1e-5)
		tmp = atan(t_2, Float64(log1p(t_1) - t_4));
	elseif (phi2 <= 4.2e-7)
		tmp = atan(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))), Float64(t_0 - Float64(t_3 * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(t_2, Float64(log(Float64(t_1 + 1.0)) - t_4));
	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[(Exp[t$95$0] - 1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1e-5], N[ArcTan[t$95$2 / N[(N[Log[1 + t$95$1], $MachinePrecision] - t$95$4), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 4.2e-7], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$3 * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[Log[N[(t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$4), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.00000000000000008e-5

   1. Initial program 80.2%

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

      1. sin-diff 93.4%

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

      2. fma-neg 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. log1p-expm1-u 93.4%

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

   6. Applied egg-rr 93.4%

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

   if -1.00000000000000008e-5 < phi2 < 4.2e-7

   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. Add Preprocessing
   3. Step-by-step derivation

      1. sin-diff 88.8%

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

      2. fma-neg 88.8%

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

   4. Applied egg-rr 88.8%

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

      1. cos-diff 99.8%

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

      2. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. fma-define 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in phi2 around 0 99.8%

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

   if 4.2e-7 < phi2

   1. Initial program 77.4%

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

      1. sin-diff 88.2%

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

      2. fma-neg 88.2%

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

   4. Applied egg-rr 88.2%

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

      1. log1p-expm1-u 88.2%

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

      2. log1p-undefine 88.4%

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

   6. Applied egg-rr 88.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 4.2 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\log \left(\mathsf{expm1}\left(\cos \phi_1 \cdot \sin \phi_2\right) + 1\right) - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \mathsf{expm1}\left(t\_0\right)\\ t_2 := \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2\\ t_3 := \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{log1p}\left(t\_1\right) - t\_3}\\ \mathbf{elif}\;\phi_2 \leq 0.00065:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \sin \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\log \left(t\_1 + 1\right) - t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (expm1 t_0))
        (t_2
         (*
          (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
          (cos phi2)))
        (t_3 (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
   (if (<= phi2 -6.2e-5)
     (atan2 t_2 (- (log1p t_1) t_3))
     (if (<= phi2 0.00065)
       (atan2
        t_2
        (-
         t_0
         (*
          (sin phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda2) (cos lambda1))))))
       (atan2 t_2 (- (log (+ t_1 1.0)) t_3))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = expm1(t_0);
	double t_2 = fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2);
	double t_3 = (cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -6.2e-5) {
		tmp = atan2(t_2, (log1p(t_1) - t_3));
	} else if (phi2 <= 0.00065) {
		tmp = atan2(t_2, (t_0 - (sin(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = atan2(t_2, (log((t_1 + 1.0)) - t_3));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = expm1(t_0)
	t_2 = Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2))
	t_3 = Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -6.2e-5)
		tmp = atan(t_2, Float64(log1p(t_1) - t_3));
	elseif (phi2 <= 0.00065)
		tmp = atan(t_2, Float64(t_0 - Float64(sin(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = atan(t_2, Float64(log(Float64(t_1 + 1.0)) - t_3));
	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[(Exp[t$95$0] - 1), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -6.2e-5], N[ArcTan[t$95$2 / N[(N[Log[1 + t$95$1], $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 0.00065], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[Log[N[(t$95$1 + 1.0), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.20000000000000027e-5

   1. Initial program 80.2%

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

      1. sin-diff 93.4%

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

      2. fma-neg 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. log1p-expm1-u 93.4%

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

   6. Applied egg-rr 93.4%

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

   if -6.20000000000000027e-5 < phi2 < 6.4999999999999997e-4

   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. Add Preprocessing
   3. Step-by-step derivation

      1. sin-diff 88.8%

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

      2. fma-neg 88.8%

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

   4. Applied egg-rr 88.8%

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

      1. cos-diff 99.8%

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

      2. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. fma-define 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in phi2 around 0 99.8%

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

   if 6.4999999999999997e-4 < phi2

   1. Initial program 77.4%

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

      1. sin-diff 88.2%

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

      2. fma-neg 88.2%

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

   4. Applied egg-rr 88.2%

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

      1. log1p-expm1-u 88.2%

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

      2. log1p-undefine 88.4%

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

   6. Applied egg-rr 88.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 0.00065:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\log \left(\mathsf{expm1}\left(\cos \phi_1 \cdot \sin \phi_2\right) + 1\right) - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ t_3 := t\_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right) - t\_3}\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{t\_0 - t\_2 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\log \left(e^{t\_0}\right) - t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (*
          (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
          (cos phi2)))
        (t_2 (* (cos phi2) (sin phi1)))
        (t_3 (* t_2 (cos (- lambda1 lambda2)))))
   (if (<= phi2 -7e-6)
     (atan2 t_1 (- (log1p (expm1 t_0)) t_3))
     (if (<= phi2 3e-6)
       (atan2
        (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
        (-
         t_0
         (*
          t_2
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (atan2 t_1 (- (log (exp t_0)) t_3))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2);
	double t_2 = cos(phi2) * sin(phi1);
	double t_3 = t_2 * cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -7e-6) {
		tmp = atan2(t_1, (log1p(expm1(t_0)) - t_3));
	} else if (phi2 <= 3e-6) {
		tmp = atan2(((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))), (t_0 - (t_2 * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = atan2(t_1, (log(exp(t_0)) - t_3));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2))
	t_2 = Float64(cos(phi2) * sin(phi1))
	t_3 = Float64(t_2 * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -7e-6)
		tmp = atan(t_1, Float64(log1p(expm1(t_0)) - t_3));
	elseif (phi2 <= 3e-6)
		tmp = atan(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))), Float64(t_0 - Float64(t_2 * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = atan(t_1, Float64(log(exp(t_0)) - t_3));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -7e-6], N[ArcTan[t$95$1 / N[(N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 3e-6], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$2 * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.99999999999999989e-6

   1. Initial program 80.2%

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

      1. sin-diff 93.4%

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

      2. fma-neg 93.4%

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

   4. Applied egg-rr 93.4%

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

      1. log1p-expm1-u 93.4%

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

   6. Applied egg-rr 93.4%

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

   if -6.99999999999999989e-6 < phi2 < 3.0000000000000001e-6

   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. Add Preprocessing
   3. Step-by-step derivation

      1. sin-diff 88.8%

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

      2. fma-neg 88.8%

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

   4. Applied egg-rr 88.8%

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

      1. cos-diff 99.8%

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

      2. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. fma-define 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in phi2 around 0 99.8%

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

   if 3.0000000000000001e-6 < phi2

   1. Initial program 77.4%

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

      1. sin-diff 88.2%

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

      2. fma-neg 88.2%

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

   4. Applied egg-rr 88.2%

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

      1. add-log-exp 88.3%

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

   6. Applied egg-rr 88.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -7 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \phi_1 \cdot \sin \phi_2\right)\right) - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\log \left(e^{\cos \phi_1 \cdot \sin \phi_2}\right) - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. cos-diff 81.7%

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

   2. +-commutative 81.7%

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

   3. *-commutative 81.7%

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

4. Applied egg-rr 81.7%

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

   1. sin-diff 54.4%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 7: 89.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), 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[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 81.3%

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

   1. sin-diff 89.7%

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

   2. fma-neg 89.7%

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

4. Applied egg-rr 89.7%

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

5. Final simplification 89.7%

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

Alternative 8: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.2 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-48}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -8.2e-14) (not (<= phi1 4.5e-48)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (*
      (* (cos phi2) (sin phi1))
      (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda2) (cos lambda1))))))
   (atan2
    (*
     (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
     (cos phi2))
    (- (sin phi2) (* phi1 (cos (- lambda2 lambda1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -8.2e-14) || !(phi1 <= 4.5e-48)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2)), (sin(phi2) - (phi1 * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -8.2e-14) || !(phi1 <= 4.5e-48))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), Float64(sin(phi2) - Float64(phi1 * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -8.2e-14], N[Not[LessEqual[phi1, 4.5e-48]], $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[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.2000000000000004e-14 or 4.49999999999999988e-48 < phi1

   1. Initial program 80.2%

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

      1. cos-diff 80.9%

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

      2. +-commutative 80.9%

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

      3. *-commutative 80.9%

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

   4. Applied egg-rr 80.9%

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

   if -8.2000000000000004e-14 < phi1 < 4.49999999999999988e-48

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in phi1 around 0 82.8%

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

      1. sub-neg 82.8%

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

      2. remove-double-neg 82.8%

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

      3. mul-1-neg 82.8%

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

      4. distribute-neg-in 82.8%

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

      5. +-commutative 82.8%

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

      6. *-commutative 82.8%

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

      7. cos-neg 82.8%

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

      8. mul-1-neg 82.8%

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

      9. unsub-neg 82.8%

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

   7. Simplified 82.8%

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

   8. Taylor expanded in phi1 around 0 82.8%

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

   9. Taylor expanded in phi2 around 0 82.8%

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

       1. sin-diff 99.8%

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

       2. fma-neg 99.8%

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

   11. Applied egg-rr 99.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.2 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-48}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t\_2\\ \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - \cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot t\_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - t\_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_0 - t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (cos (- lambda1 lambda2)))
        (t_3 (* (* (cos phi2) (sin phi1)) t_2)))
   (if (<= phi1 -1.85e+18)
     (atan2 t_1 (- t_0 (* (cos phi2) (log1p (expm1 (* (sin phi1) t_2))))))
     (if (<= phi1 4e-48)
       (atan2
        (*
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
         (cos phi2))
        (- (sin phi2) t_3))
       (atan2 t_1 (- t_0 t_3))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double t_2 = cos((lambda1 - lambda2));
	double t_3 = (cos(phi2) * sin(phi1)) * t_2;
	double tmp;
	if (phi1 <= -1.85e+18) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * log1p(expm1((sin(phi1) * t_2))))));
	} else if (phi1 <= 4e-48) {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2)), (sin(phi2) - t_3));
	} else {
		tmp = atan2(t_1, (t_0 - t_3));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_2 = cos(Float64(lambda1 - lambda2))
	t_3 = Float64(Float64(cos(phi2) * sin(phi1)) * t_2)
	tmp = 0.0
	if (phi1 <= -1.85e+18)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * log1p(expm1(Float64(sin(phi1) * t_2))))));
	elseif (phi1 <= 4e-48)
		tmp = atan(Float64(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), Float64(sin(phi2) - t_3));
	else
		tmp = atan(t_1, Float64(t_0 - t_3));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[phi1, -1.85e+18], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[Log[1 + N[(Exp[N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 4e-48], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - t$95$3), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.85e18

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. associate-*l* 80.4%

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

   3. Simplified 80.4%

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

      1. log1p-expm1-u 80.5%

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

      2. *-commutative 80.5%

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

   6. Applied egg-rr 80.5%

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

   if -1.85e18 < phi1 < 3.9999999999999999e-48

   1. Initial program 82.6%

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

      1. sin-diff 99.0%

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

      2. fma-neg 99.0%

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

   4. Applied egg-rr 99.0%

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

   5. Taylor expanded in phi1 around 0 99.0%

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

   if 3.9999999999999999e-48 < phi1

   1. Initial program 80.2%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-48}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. sin-diff 54.4%

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

4. Applied egg-rr 89.7%

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

5. Final simplification 89.7%

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

Alternative 11: 87.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.15 \cdot 10^{-10} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-48}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -1.15e-10) (not (<= phi1 4.5e-48)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
   (atan2
    (*
     (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2))))
     (cos phi2))
    (- (sin phi2) (* phi1 (cos (- lambda2 lambda1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -1.15e-10) || !(phi1 <= 4.5e-48)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2))) * cos(phi2)), (sin(phi2) - (phi1 * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -1.15e-10) || !(phi1 <= 4.5e-48))
		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(fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2)))) * cos(phi2)), Float64(sin(phi2) - Float64(phi1 * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -1.15e-10], N[Not[LessEqual[phi1, 4.5e-48]], $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[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.15000000000000004e-10 or 4.49999999999999988e-48 < 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. Add Preprocessing

   if -1.15000000000000004e-10 < phi1 < 4.49999999999999988e-48

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in phi1 around 0 82.8%

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

      1. sub-neg 82.8%

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

      2. remove-double-neg 82.8%

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

      3. mul-1-neg 82.8%

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

      4. distribute-neg-in 82.8%

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

      5. +-commutative 82.8%

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

      6. *-commutative 82.8%

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

      7. cos-neg 82.8%

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

      8. mul-1-neg 82.8%

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

      9. unsub-neg 82.8%

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

   7. Simplified 82.8%

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

   8. Taylor expanded in phi1 around 0 82.8%

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

   9. Taylor expanded in phi2 around 0 82.8%

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

       1. sin-diff 99.6%

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

       2. fma-neg 99.7%

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

   11. Applied egg-rr 99.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.15 \cdot 10^{-10} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-48}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-10} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-48}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -1.85e-10) (not (<= phi1 4.5e-48)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
   (atan2
    (*
     (cos phi2)
     (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2))))
    (- (sin phi2) (* phi1 (cos (- lambda2 lambda1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -1.85e-10) || !(phi1 <= 4.5e-48)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (sin(phi2) - (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) :: tmp
    if ((phi1 <= (-1.85d-10)) .or. (.not. (phi1 <= 4.5d-48))) then
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))))
    else
        tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (sin(phi2) - (phi1 * cos((lambda2 - lambda1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -1.85e-10) || !(phi1 <= 4.5e-48)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos(phi2) * Math.sin(phi1)) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2)))), (Math.sin(phi2) - (phi1 * Math.cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -1.85e-10) or not (phi1 <= 4.5e-48):
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), ((math.cos(phi1) * math.sin(phi2)) - ((math.cos(phi2) * math.sin(phi1)) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2((math.cos(phi2) * ((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2)))), (math.sin(phi2) - (phi1 * math.cos((lambda2 - lambda1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -1.85e-10) || !(phi1 <= 4.5e-48))
		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) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2)))), Float64(sin(phi2) - Float64(phi1 * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -1.85e-10) || ~((phi1 <= 4.5e-48)))
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	else
		tmp = atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (sin(phi2) - (phi1 * cos((lambda2 - lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -1.85e-10], N[Not[LessEqual[phi1, 4.5e-48]], $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[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.85000000000000007e-10 or 4.49999999999999988e-48 < 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. Add Preprocessing

   if -1.85000000000000007e-10 < phi1 < 4.49999999999999988e-48

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l* 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in phi1 around 0 82.8%

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

      1. sub-neg 82.8%

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

      2. remove-double-neg 82.8%

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

      3. mul-1-neg 82.8%

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

      4. distribute-neg-in 82.8%

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

      5. +-commutative 82.8%

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

      6. *-commutative 82.8%

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

      7. cos-neg 82.8%

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

      8. mul-1-neg 82.8%

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

      9. unsub-neg 82.8%

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

   7. Simplified 82.8%

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

   8. Taylor expanded in phi1 around 0 82.8%

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

   9. Taylor expanded in phi2 around 0 82.8%

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

       1. sin-diff 99.6%

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

   11. Applied egg-rr 99.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-10} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-48}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.28 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 3200000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_2}{t\_0 - \cos \lambda_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin phi1)))
        (t_2 (sin (- lambda1 lambda2))))
   (if (or (<= phi2 -1.28e-5) (not (<= phi2 3200000000.0)))
     (atan2 (* (cos phi2) t_2) (- t_0 (* (cos lambda1) t_1)))
     (atan2 t_2 (- t_0 (* t_1 (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin(phi1);
	double t_2 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -1.28e-5) || !(phi2 <= 3200000000.0)) {
		tmp = atan2((cos(phi2) * t_2), (t_0 - (cos(lambda1) * t_1)));
	} else {
		tmp = atan2(t_2, (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin(phi1)
    t_2 = sin((lambda1 - lambda2))
    if ((phi2 <= (-1.28d-5)) .or. (.not. (phi2 <= 3200000000.0d0))) then
        tmp = atan2((cos(phi2) * t_2), (t_0 - (cos(lambda1) * t_1)))
    else
        tmp = atan2(t_2, (t_0 - (t_1 * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin(phi1);
	double t_2 = Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -1.28e-5) || !(phi2 <= 3200000000.0)) {
		tmp = Math.atan2((Math.cos(phi2) * t_2), (t_0 - (Math.cos(lambda1) * t_1)));
	} else {
		tmp = Math.atan2(t_2, (t_0 - (t_1 * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos(phi2) * math.sin(phi1)
	t_2 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -1.28e-5) or not (phi2 <= 3200000000.0):
		tmp = math.atan2((math.cos(phi2) * t_2), (t_0 - (math.cos(lambda1) * t_1)))
	else:
		tmp = math.atan2(t_2, (t_0 - (t_1 * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(phi1))
	t_2 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi2 <= -1.28e-5) || !(phi2 <= 3200000000.0))
		tmp = atan(Float64(cos(phi2) * t_2), Float64(t_0 - Float64(cos(lambda1) * t_1)));
	else
		tmp = atan(t_2, Float64(t_0 - Float64(t_1 * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = cos(phi2) * sin(phi1);
	t_2 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -1.28e-5) || ~((phi2 <= 3200000000.0)))
		tmp = atan2((cos(phi2) * t_2), (t_0 - (cos(lambda1) * t_1)));
	else
		tmp = atan2(t_2, (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -1.28e-5], N[Not[LessEqual[phi2, 3200000000.0]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$0 - N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.2799999999999999e-5 or 3.2e9 < phi2

   1. Initial program 78.6%

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

   3. Taylor expanded in lambda2 around 0 67.6%

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

   if -1.2799999999999999e-5 < phi2 < 3.2e9

   1. Initial program 83.9%

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

      1. add-sqr-sqrt 43.9%

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

      2. pow2 43.9%

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

   4. Applied egg-rr 43.9%

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

   5. Taylor expanded in phi2 around 0 83.2%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + -0.5 \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 83.2%

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

      2. distribute-rgt1-in 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in phi2 around 0 83.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.28 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 3200000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -6 \cdot 10^{-13} \lor \neg \left(\lambda_1 \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \lambda_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{t\_0 - \cos \lambda_2 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin phi1)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= lambda1 -6e-13) (not (<= lambda1 3.8e-5)))
     (atan2 t_2 (- t_0 (* (cos lambda1) t_1)))
     (atan2 t_2 (- t_0 (* (cos lambda2) t_1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin(phi1);
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if ((lambda1 <= -6e-13) || !(lambda1 <= 3.8e-5)) {
		tmp = atan2(t_2, (t_0 - (cos(lambda1) * t_1)));
	} else {
		tmp = atan2(t_2, (t_0 - (cos(lambda2) * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin(phi1)
    t_2 = cos(phi2) * sin((lambda1 - lambda2))
    if ((lambda1 <= (-6d-13)) .or. (.not. (lambda1 <= 3.8d-5))) then
        tmp = atan2(t_2, (t_0 - (cos(lambda1) * t_1)))
    else
        tmp = atan2(t_2, (t_0 - (cos(lambda2) * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin(phi1);
	double t_2 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if ((lambda1 <= -6e-13) || !(lambda1 <= 3.8e-5)) {
		tmp = Math.atan2(t_2, (t_0 - (Math.cos(lambda1) * t_1)));
	} else {
		tmp = Math.atan2(t_2, (t_0 - (Math.cos(lambda2) * t_1)));
	}
	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)
	t_2 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if (lambda1 <= -6e-13) or not (lambda1 <= 3.8e-5):
		tmp = math.atan2(t_2, (t_0 - (math.cos(lambda1) * t_1)))
	else:
		tmp = math.atan2(t_2, (t_0 - (math.cos(lambda2) * t_1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(phi1))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if ((lambda1 <= -6e-13) || !(lambda1 <= 3.8e-5))
		tmp = atan(t_2, Float64(t_0 - Float64(cos(lambda1) * t_1)));
	else
		tmp = atan(t_2, Float64(t_0 - Float64(cos(lambda2) * t_1)));
	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);
	t_2 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((lambda1 <= -6e-13) || ~((lambda1 <= 3.8e-5)))
		tmp = atan2(t_2, (t_0 - (cos(lambda1) * t_1)));
	else
		tmp = atan2(t_2, (t_0 - (cos(lambda2) * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -6e-13], N[Not[LessEqual[lambda1, 3.8e-5]], $MachinePrecision]], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Cos[lambda2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -5.99999999999999968e-13 or 3.8000000000000002e-5 < lambda1

   1. Initial program 64.6%

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

   3. Taylor expanded in lambda2 around 0 64.4%

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

   if -5.99999999999999968e-13 < lambda1 < 3.8000000000000002e-5

   1. Initial program 99.6%

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

   3. Taylor expanded in lambda1 around 0 99.6%

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

      1. cos-neg 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6 \cdot 10^{-13} \lor \neg \left(\lambda_1 \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot t\_1\\ t_3 := \cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{+197}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\sin \phi_2 - \sqrt{{t\_0}^{2}}}\\ \mathbf{elif}\;\phi_2 \leq -7 \cdot 10^{-25}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t\_3}\\ \mathbf{elif}\;\phi_2 \leq -1.46 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \left(\sin \lambda_2 \cdot 0.5\right)\right) - \sin \lambda_2\right)}{\sin \phi_2 - t\_0}\\ \mathbf{elif}\;\phi_2 \leq 3200000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* phi1 (cos (- lambda2 lambda1))))
        (t_1 (sin (- lambda1 lambda2)))
        (t_2 (* (cos phi2) t_1))
        (t_3
         (-
          (* (cos phi1) (sin phi2))
          (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2))))))
   (if (<= phi2 -9.5e+197)
     (atan2 t_2 (- (sin phi2) (sqrt (pow t_0 2.0))))
     (if (<= phi2 -7e-25)
       (atan2 (* (sin lambda1) (cos phi2)) t_3)
       (if (<= phi2 -1.46e-47)
         (atan2
          (*
           (cos phi2)
           (-
            (* lambda1 (+ (cos lambda2) (* lambda1 (* (sin lambda2) 0.5))))
            (sin lambda2)))
          (- (sin phi2) t_0))
         (if (<= phi2 3200000000.0)
           (atan2 t_1 t_3)
           (atan2
            t_2
            (-
             (sin phi2)
             (/
              (*
               (cos phi2)
               (+
                (sin (+ lambda2 (- phi1 lambda1)))
                (sin (+ phi1 (- lambda1 lambda2)))))
              2.0)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * cos((lambda2 - lambda1));
	double t_1 = sin((lambda1 - lambda2));
	double t_2 = cos(phi2) * t_1;
	double t_3 = (cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)));
	double tmp;
	if (phi2 <= -9.5e+197) {
		tmp = atan2(t_2, (sin(phi2) - sqrt(pow(t_0, 2.0))));
	} else if (phi2 <= -7e-25) {
		tmp = atan2((sin(lambda1) * cos(phi2)), t_3);
	} else if (phi2 <= -1.46e-47) {
		tmp = atan2((cos(phi2) * ((lambda1 * (cos(lambda2) + (lambda1 * (sin(lambda2) * 0.5)))) - sin(lambda2))), (sin(phi2) - t_0));
	} else if (phi2 <= 3200000000.0) {
		tmp = atan2(t_1, t_3);
	} else {
		tmp = atan2(t_2, (sin(phi2) - ((cos(phi2) * (sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2))))) / 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = phi1 * cos((lambda2 - lambda1))
    t_1 = sin((lambda1 - lambda2))
    t_2 = cos(phi2) * t_1
    t_3 = (cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))
    if (phi2 <= (-9.5d+197)) then
        tmp = atan2(t_2, (sin(phi2) - sqrt((t_0 ** 2.0d0))))
    else if (phi2 <= (-7d-25)) then
        tmp = atan2((sin(lambda1) * cos(phi2)), t_3)
    else if (phi2 <= (-1.46d-47)) then
        tmp = atan2((cos(phi2) * ((lambda1 * (cos(lambda2) + (lambda1 * (sin(lambda2) * 0.5d0)))) - sin(lambda2))), (sin(phi2) - t_0))
    else if (phi2 <= 3200000000.0d0) then
        tmp = atan2(t_1, t_3)
    else
        tmp = atan2(t_2, (sin(phi2) - ((cos(phi2) * (sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2))))) / 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * Math.cos((lambda2 - lambda1));
	double t_1 = Math.sin((lambda1 - lambda2));
	double t_2 = Math.cos(phi2) * t_1;
	double t_3 = (Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos(phi2) * Math.sin(phi1)) * Math.cos((lambda1 - lambda2)));
	double tmp;
	if (phi2 <= -9.5e+197) {
		tmp = Math.atan2(t_2, (Math.sin(phi2) - Math.sqrt(Math.pow(t_0, 2.0))));
	} else if (phi2 <= -7e-25) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), t_3);
	} else if (phi2 <= -1.46e-47) {
		tmp = Math.atan2((Math.cos(phi2) * ((lambda1 * (Math.cos(lambda2) + (lambda1 * (Math.sin(lambda2) * 0.5)))) - Math.sin(lambda2))), (Math.sin(phi2) - t_0));
	} else if (phi2 <= 3200000000.0) {
		tmp = Math.atan2(t_1, t_3);
	} else {
		tmp = Math.atan2(t_2, (Math.sin(phi2) - ((Math.cos(phi2) * (Math.sin((lambda2 + (phi1 - lambda1))) + Math.sin((phi1 + (lambda1 - lambda2))))) / 2.0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = phi1 * math.cos((lambda2 - lambda1))
	t_1 = math.sin((lambda1 - lambda2))
	t_2 = math.cos(phi2) * t_1
	t_3 = (math.cos(phi1) * math.sin(phi2)) - ((math.cos(phi2) * math.sin(phi1)) * math.cos((lambda1 - lambda2)))
	tmp = 0
	if phi2 <= -9.5e+197:
		tmp = math.atan2(t_2, (math.sin(phi2) - math.sqrt(math.pow(t_0, 2.0))))
	elif phi2 <= -7e-25:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), t_3)
	elif phi2 <= -1.46e-47:
		tmp = math.atan2((math.cos(phi2) * ((lambda1 * (math.cos(lambda2) + (lambda1 * (math.sin(lambda2) * 0.5)))) - math.sin(lambda2))), (math.sin(phi2) - t_0))
	elif phi2 <= 3200000000.0:
		tmp = math.atan2(t_1, t_3)
	else:
		tmp = math.atan2(t_2, (math.sin(phi2) - ((math.cos(phi2) * (math.sin((lambda2 + (phi1 - lambda1))) + math.sin((phi1 + (lambda1 - lambda2))))) / 2.0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(phi1 * cos(Float64(lambda2 - lambda1)))
	t_1 = sin(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * t_1)
	t_3 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if (phi2 <= -9.5e+197)
		tmp = atan(t_2, Float64(sin(phi2) - sqrt((t_0 ^ 2.0))));
	elseif (phi2 <= -7e-25)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), t_3);
	elseif (phi2 <= -1.46e-47)
		tmp = atan(Float64(cos(phi2) * Float64(Float64(lambda1 * Float64(cos(lambda2) + Float64(lambda1 * Float64(sin(lambda2) * 0.5)))) - sin(lambda2))), Float64(sin(phi2) - t_0));
	elseif (phi2 <= 3200000000.0)
		tmp = atan(t_1, t_3);
	else
		tmp = atan(t_2, Float64(sin(phi2) - Float64(Float64(cos(phi2) * Float64(sin(Float64(lambda2 + Float64(phi1 - lambda1))) + sin(Float64(phi1 + Float64(lambda1 - lambda2))))) / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = phi1 * cos((lambda2 - lambda1));
	t_1 = sin((lambda1 - lambda2));
	t_2 = cos(phi2) * t_1;
	t_3 = (cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)));
	tmp = 0.0;
	if (phi2 <= -9.5e+197)
		tmp = atan2(t_2, (sin(phi2) - sqrt((t_0 ^ 2.0))));
	elseif (phi2 <= -7e-25)
		tmp = atan2((sin(lambda1) * cos(phi2)), t_3);
	elseif (phi2 <= -1.46e-47)
		tmp = atan2((cos(phi2) * ((lambda1 * (cos(lambda2) + (lambda1 * (sin(lambda2) * 0.5)))) - sin(lambda2))), (sin(phi2) - t_0));
	elseif (phi2 <= 3200000000.0)
		tmp = atan2(t_1, t_3);
	else
		tmp = atan2(t_2, (sin(phi2) - ((cos(phi2) * (sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2))))) / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = 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]}, If[LessEqual[phi2, -9.5e+197], N[ArcTan[t$95$2 / N[(N[Sin[phi2], $MachinePrecision] - N[Sqrt[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, -7e-25], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$3], $MachinePrecision], If[LessEqual[phi2, -1.46e-47], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(lambda1 * N[(N[Cos[lambda2], $MachinePrecision] + N[(lambda1 * N[(N[Sin[lambda2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 3200000000.0], N[ArcTan[t$95$1 / t$95$3], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[N[(lambda2 + N[(phi1 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(phi1 + N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if phi2 < -9.4999999999999997e197

   1. Initial program 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-*l* 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in phi1 around 0 59.3%

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

      1. sub-neg 59.3%

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

      2. remove-double-neg 59.3%

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

      3. mul-1-neg 59.3%

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

      4. distribute-neg-in 59.3%

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

      5. +-commutative 59.3%

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

      6. *-commutative 59.3%

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

      7. cos-neg 59.3%

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

      8. mul-1-neg 59.3%

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

      9. unsub-neg 59.3%

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

   7. Simplified 59.3%

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

   8. Taylor expanded in phi1 around 0 59.3%

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

   9. Taylor expanded in phi2 around 0 54.0%

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

       1. add-sqr-sqrt 31.3%

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

       2. sqrt-unprod 60.9%

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

       3. pow2 60.9%

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

   11. Applied egg-rr 60.9%

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

   if -9.4999999999999997e197 < phi2 < -7.0000000000000004e-25

   1. Initial program 78.5%

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

   3. Taylor expanded in lambda2 around 0 51.7%

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

   if -7.0000000000000004e-25 < phi2 < -1.46e-47

   1. Initial program 57.8%

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

      1. *-commutative 57.8%

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

      2. associate-*l* 57.8%

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

   3. Simplified 57.8%

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

   5. Taylor expanded in phi1 around 0 57.8%

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

      1. sub-neg 57.8%

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

      2. remove-double-neg 57.8%

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

      3. mul-1-neg 57.8%

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

      4. distribute-neg-in 57.8%

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

      5. +-commutative 57.8%

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

      6. *-commutative 57.8%

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

      7. cos-neg 57.8%

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

      8. mul-1-neg 57.8%

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

      9. unsub-neg 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in phi1 around 0 57.8%

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

   9. Taylor expanded in phi2 around 0 57.8%

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

   10. Taylor expanded in lambda1 around 0 88.6%

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

       1. +-commutative 88.6%

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

       2. sin-neg 88.6%

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

       3. unsub-neg 88.6%

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

       4. *-commutative 88.6%

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

       5. associate-*r* 88.6%

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

       6. *-commutative 88.6%

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

       7. cos-neg 88.6%

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

       8. *-commutative 88.6%

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

       9. sin-neg 88.6%

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

       10. distribute-lft-neg-out 88.6%

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

       11. distribute-rgt-neg-in 88.6%

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

       12. metadata-eval 88.6%

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

   12. Simplified 88.6%

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

   if -1.46e-47 < phi2 < 3.2e9

   1. Initial program 85.8%

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

      1. add-sqr-sqrt 44.6%

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

      2. pow2 44.6%

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

   4. Applied egg-rr 44.6%

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

   5. Taylor expanded in phi2 around 0 84.9%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + -0.5 \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 84.9%

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

      2. distribute-rgt1-in 84.9%

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

   7. Simplified 84.9%

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

   8. Taylor expanded in phi2 around 0 85.1%

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

   if 3.2e9 < phi2

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-*l* 77.1%

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

   3. Simplified 77.1%

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

      1. *-commutative 77.1%

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

      2. sin-cos-mult 57.3%

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

      3. associate-*l/ 57.3%

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

      4. associate--r- 57.3%

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

      5. +-commutative 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in phi1 around 0 54.9%

      \[\leadsto \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}{\color{blue}{\sin \phi_2} - \frac{\left(\sin \left(\left(\phi_1 - \lambda_1\right) + \lambda_2\right) + \sin \left(\left(\lambda_1 - \lambda_2\right) + \phi_1\right)\right) \cdot \cos \phi_2}{2}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.5 \cdot 10^{+197}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sqrt{{\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\\ \mathbf{elif}\;\phi_2 \leq -7 \cdot 10^{-25}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq -1.46 \cdot 10^{-47}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \left(\sin \lambda_2 \cdot 0.5\right)\right) - \sin \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\phi_2 \leq 3200000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)\right)}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. *-commutative 81.3%

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

   2. associate-*l* 81.3%

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

3. Simplified 81.3%

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

5. Final simplification 81.3%

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

Alternative 17: 80.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

3. Final simplification 81.3%

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

Alternative 18: 66.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t\_0\\ \mathbf{if}\;\phi_2 \leq -8100:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - \sqrt{{\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\\ \mathbf{elif}\;\phi_2 \leq 3200000000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)\right)}{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (* (cos phi2) t_0)))
   (if (<= phi2 -8100.0)
     (atan2
      t_1
      (- (sin phi2) (sqrt (pow (* phi1 (cos (- lambda2 lambda1))) 2.0))))
     (if (<= phi2 3200000000.0)
       (atan2
        t_0
        (-
         (* (cos phi1) (sin phi2))
         (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
       (atan2
        t_1
        (-
         (sin phi2)
         (/
          (*
           (cos phi2)
           (+
            (sin (+ lambda2 (- phi1 lambda1)))
            (sin (+ phi1 (- lambda1 lambda2)))))
          2.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos(phi2) * t_0;
	double tmp;
	if (phi2 <= -8100.0) {
		tmp = atan2(t_1, (sin(phi2) - sqrt(pow((phi1 * cos((lambda2 - lambda1))), 2.0))));
	} else if (phi2 <= 3200000000.0) {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2(t_1, (sin(phi2) - ((cos(phi2) * (sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2))))) / 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = cos(phi2) * t_0
    if (phi2 <= (-8100.0d0)) then
        tmp = atan2(t_1, (sin(phi2) - sqrt(((phi1 * cos((lambda2 - lambda1))) ** 2.0d0))))
    else if (phi2 <= 3200000000.0d0) then
        tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))))
    else
        tmp = atan2(t_1, (sin(phi2) - ((cos(phi2) * (sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2))))) / 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * t_0;
	double tmp;
	if (phi2 <= -8100.0) {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - Math.sqrt(Math.pow((phi1 * Math.cos((lambda2 - lambda1))), 2.0))));
	} else if (phi2 <= 3200000000.0) {
		tmp = Math.atan2(t_0, ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos(phi2) * Math.sin(phi1)) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - ((Math.cos(phi2) * (Math.sin((lambda2 + (phi1 - lambda1))) + Math.sin((phi1 + (lambda1 - lambda2))))) / 2.0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.cos(phi2) * t_0
	tmp = 0
	if phi2 <= -8100.0:
		tmp = math.atan2(t_1, (math.sin(phi2) - math.sqrt(math.pow((phi1 * math.cos((lambda2 - lambda1))), 2.0))))
	elif phi2 <= 3200000000.0:
		tmp = math.atan2(t_0, ((math.cos(phi1) * math.sin(phi2)) - ((math.cos(phi2) * math.sin(phi1)) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2(t_1, (math.sin(phi2) - ((math.cos(phi2) * (math.sin((lambda2 + (phi1 - lambda1))) + math.sin((phi1 + (lambda1 - lambda2))))) / 2.0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * t_0)
	tmp = 0.0
	if (phi2 <= -8100.0)
		tmp = atan(t_1, Float64(sin(phi2) - sqrt((Float64(phi1 * cos(Float64(lambda2 - lambda1))) ^ 2.0))));
	elseif (phi2 <= 3200000000.0)
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(t_1, Float64(sin(phi2) - Float64(Float64(cos(phi2) * Float64(sin(Float64(lambda2 + Float64(phi1 - lambda1))) + sin(Float64(phi1 + Float64(lambda1 - lambda2))))) / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = cos(phi2) * t_0;
	tmp = 0.0;
	if (phi2 <= -8100.0)
		tmp = atan2(t_1, (sin(phi2) - sqrt(((phi1 * cos((lambda2 - lambda1))) ^ 2.0))));
	elseif (phi2 <= 3200000000.0)
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	else
		tmp = atan2(t_1, (sin(phi2) - ((cos(phi2) * (sin((lambda2 + (phi1 - lambda1))) + sin((phi1 + (lambda1 - lambda2))))) / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[phi2, -8100.0], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[Sqrt[N[Power[N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 3200000000.0], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[N[(lambda2 + N[(phi1 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(phi1 + N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -8100

   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. Step-by-step derivation

      1. *-commutative 82.0%

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

      2. associate-*l* 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in phi1 around 0 47.2%

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

      1. sub-neg 47.2%

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

      2. remove-double-neg 47.2%

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

      3. mul-1-neg 47.2%

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

      4. distribute-neg-in 47.2%

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

      5. +-commutative 47.2%

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

      6. *-commutative 47.2%

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

      7. cos-neg 47.2%

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

      8. mul-1-neg 47.2%

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

      9. unsub-neg 47.2%

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

   7. Simplified 47.2%

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

   8. Taylor expanded in phi1 around 0 47.0%

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

   9. Taylor expanded in phi2 around 0 41.8%

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

       1. add-sqr-sqrt 25.2%

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

       2. sqrt-unprod 50.3%

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

       3. pow2 50.3%

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

   11. Applied egg-rr 50.3%

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

   if -8100 < phi2 < 3.2e9

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 43.5%

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

      2. pow2 43.5%

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

   4. Applied egg-rr 43.5%

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

   5. Taylor expanded in phi2 around 0 81.7%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + -0.5 \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 81.7%

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

      2. distribute-rgt1-in 81.7%

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

   7. Simplified 81.7%

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

   8. Taylor expanded in phi2 around 0 81.6%

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

   if 3.2e9 < phi2

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

      2. associate-*l* 77.1%

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

   3. Simplified 77.1%

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

      1. *-commutative 77.1%

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

      2. sin-cos-mult 57.3%

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

      3. associate-*l/ 57.3%

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

      4. associate--r- 57.3%

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

      5. +-commutative 57.3%

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

   6. Applied egg-rr 57.3%

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

   7. Taylor expanded in phi1 around 0 54.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8100:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sqrt{{\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\\ \mathbf{elif}\;\phi_2 \leq 3200000000:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \frac{\cos \phi_2 \cdot \left(\sin \left(\lambda_2 + \left(\phi_1 - \lambda_1\right)\right) + \sin \left(\phi_1 + \left(\lambda_1 - \lambda_2\right)\right)\right)}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 66.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -8100 \lor \neg \left(\phi_2 \leq 3200000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_2 - \sqrt{{\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (or (<= phi2 -8100.0) (not (<= phi2 3200000000.0)))
     (atan2
      (* (cos phi2) t_0)
      (- (sin phi2) (sqrt (pow (* phi1 (cos (- lambda2 lambda1))) 2.0))))
     (atan2
      t_0
      (-
       (* (cos phi1) (sin phi2))
       (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -8100.0) || !(phi2 <= 3200000000.0)) {
		tmp = atan2((cos(phi2) * t_0), (sin(phi2) - sqrt(pow((phi1 * cos((lambda2 - lambda1))), 2.0))));
	} else {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if ((phi2 <= (-8100.0d0)) .or. (.not. (phi2 <= 3200000000.0d0))) then
        tmp = atan2((cos(phi2) * t_0), (sin(phi2) - sqrt(((phi1 * cos((lambda2 - lambda1))) ** 2.0d0))))
    else
        tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -8100.0) || !(phi2 <= 3200000000.0)) {
		tmp = Math.atan2((Math.cos(phi2) * t_0), (Math.sin(phi2) - Math.sqrt(Math.pow((phi1 * Math.cos((lambda2 - lambda1))), 2.0))));
	} else {
		tmp = Math.atan2(t_0, ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos(phi2) * Math.sin(phi1)) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -8100.0) or not (phi2 <= 3200000000.0):
		tmp = math.atan2((math.cos(phi2) * t_0), (math.sin(phi2) - math.sqrt(math.pow((phi1 * math.cos((lambda2 - lambda1))), 2.0))))
	else:
		tmp = math.atan2(t_0, ((math.cos(phi1) * math.sin(phi2)) - ((math.cos(phi2) * math.sin(phi1)) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi2 <= -8100.0) || !(phi2 <= 3200000000.0))
		tmp = atan(Float64(cos(phi2) * t_0), Float64(sin(phi2) - sqrt((Float64(phi1 * cos(Float64(lambda2 - lambda1))) ^ 2.0))));
	else
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -8100.0) || ~((phi2 <= 3200000000.0)))
		tmp = atan2((cos(phi2) * t_0), (sin(phi2) - sqrt(((phi1 * cos((lambda2 - lambda1))) ^ 2.0))));
	else
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -8100.0], N[Not[LessEqual[phi2, 3200000000.0]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[Sqrt[N[Power[N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -8100 or 3.2e9 < phi2

   1. Initial program 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*l* 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in phi1 around 0 46.8%

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

      1. sub-neg 46.8%

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

      2. remove-double-neg 46.8%

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

      3. mul-1-neg 46.8%

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

      4. distribute-neg-in 46.8%

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

      5. +-commutative 46.8%

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

      6. *-commutative 46.8%

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

      7. cos-neg 46.8%

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

      8. mul-1-neg 46.8%

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

      9. unsub-neg 46.8%

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

   7. Simplified 46.8%

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

   8. Taylor expanded in phi1 around 0 46.8%

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

   9. Taylor expanded in phi2 around 0 45.5%

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

       1. add-sqr-sqrt 28.5%

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

       2. sqrt-unprod 52.6%

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

       3. pow2 52.6%

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

   11. Applied egg-rr 52.6%

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

   if -8100 < phi2 < 3.2e9

   1. Initial program 83.1%

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

      1. add-sqr-sqrt 43.5%

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

      2. pow2 43.5%

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

   4. Applied egg-rr 43.5%

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

   5. Taylor expanded in phi2 around 0 81.7%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(\lambda_1 - \lambda_2\right) + -0.5 \cdot \left({\phi_2}^{2} \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 81.7%

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

      2. distribute-rgt1-in 81.7%

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

   7. Simplified 81.7%

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

   8. Taylor expanded in phi2 around 0 81.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8100 \lor \neg \left(\phi_2 \leq 3200000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sqrt{{\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 52.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{+24} \lor \neg \left(\phi_1 \leq 4.3 \cdot 10^{+134}\right):\\ \;\;\;\;\tan^{-1}_* \frac{{\left(\sqrt{t\_0}\right)}^{2}}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - \cos \lambda_2 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi1 -1e+24) (not (<= phi1 4.3e+134)))
     (atan2 (pow (sqrt t_0) 2.0) (* (sin phi1) (- (cos (- lambda1 lambda2)))))
     (atan2 t_0 (- (sin phi2) (* (cos lambda2) phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -1e+24) || !(phi1 <= 4.3e+134)) {
		tmp = atan2(pow(sqrt(t_0), 2.0), (sin(phi1) * -cos((lambda1 - lambda2))));
	} else {
		tmp = atan2(t_0, (sin(phi2) - (cos(lambda2) * phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if ((phi1 <= (-1d+24)) .or. (.not. (phi1 <= 4.3d+134))) then
        tmp = atan2((sqrt(t_0) ** 2.0d0), (sin(phi1) * -cos((lambda1 - lambda2))))
    else
        tmp = atan2(t_0, (sin(phi2) - (cos(lambda2) * 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 <= -1e+24) || !(phi1 <= 4.3e+134)) {
		tmp = Math.atan2(Math.pow(Math.sqrt(t_0), 2.0), (Math.sin(phi1) * -Math.cos((lambda1 - lambda2))));
	} else {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - (Math.cos(lambda2) * phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi1 <= -1e+24) or not (phi1 <= 4.3e+134):
		tmp = math.atan2(math.pow(math.sqrt(t_0), 2.0), (math.sin(phi1) * -math.cos((lambda1 - lambda2))))
	else:
		tmp = math.atan2(t_0, (math.sin(phi2) - (math.cos(lambda2) * phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if ((phi1 <= -1e+24) || !(phi1 <= 4.3e+134))
		tmp = atan((sqrt(t_0) ^ 2.0), Float64(sin(phi1) * Float64(-cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(t_0, Float64(sin(phi2) - Float64(cos(lambda2) * 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 <= -1e+24) || ~((phi1 <= 4.3e+134)))
		tmp = atan2((sqrt(t_0) ^ 2.0), (sin(phi1) * -cos((lambda1 - lambda2))));
	else
		tmp = atan2(t_0, (sin(phi2) - (cos(lambda2) * phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -1e+24], N[Not[LessEqual[phi1, 4.3e+134]], $MachinePrecision]], N[ArcTan[N[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda2], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.9999999999999998e23 or 4.3e134 < phi1

   1. Initial program 78.6%

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

      1. add-sqr-sqrt 43.2%

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

      2. pow2 43.2%

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

   4. Applied egg-rr 43.2%

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

   5. Taylor expanded in phi2 around 0 28.5%

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

      1. associate-*r* 28.5%

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

      2. neg-mul-1 28.5%

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

   7. Simplified 28.5%

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

   if -9.9999999999999998e23 < phi1 < 4.3e134

   1. Initial program 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l* 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in phi1 around 0 67.5%

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

      1. sub-neg 67.5%

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

      2. remove-double-neg 67.5%

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

      3. mul-1-neg 67.5%

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

      4. distribute-neg-in 67.5%

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

      5. +-commutative 67.5%

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

      6. *-commutative 67.5%

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

      7. cos-neg 67.5%

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

      8. mul-1-neg 67.5%

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

      9. unsub-neg 67.5%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in phi1 around 0 67.5%

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

   9. Taylor expanded in phi2 around 0 68.0%

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

   10. Taylor expanded in lambda1 around 0 68.2%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{+24} \lor \neg \left(\phi_1 \leq 4.3 \cdot 10^{+134}\right):\\ \;\;\;\;\tan^{-1}_* \frac{{\left(\sqrt{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}\right)}^{2}}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_2 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 52.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 4.6 \cdot 10^{+134}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2 - \sqrt{{\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{{\left(\sqrt{t\_0}\right)}^{2}}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 4.6e+134)
     (atan2
      t_0
      (- (sin phi2) (sqrt (pow (* phi1 (cos (- lambda2 lambda1))) 2.0))))
     (atan2
      (pow (sqrt t_0) 2.0)
      (* (sin phi1) (- (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 4.6e+134) {
		tmp = atan2(t_0, (sin(phi2) - sqrt(pow((phi1 * cos((lambda2 - lambda1))), 2.0))));
	} else {
		tmp = atan2(pow(sqrt(t_0), 2.0), (sin(phi1) * -cos((lambda1 - lambda2))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= 4.6d+134) then
        tmp = atan2(t_0, (sin(phi2) - sqrt(((phi1 * cos((lambda2 - lambda1))) ** 2.0d0))))
    else
        tmp = atan2((sqrt(t_0) ** 2.0d0), (sin(phi1) * -cos((lambda1 - lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 4.6e+134) {
		tmp = Math.atan2(t_0, (Math.sin(phi2) - Math.sqrt(Math.pow((phi1 * Math.cos((lambda2 - lambda1))), 2.0))));
	} else {
		tmp = Math.atan2(Math.pow(Math.sqrt(t_0), 2.0), (Math.sin(phi1) * -Math.cos((lambda1 - lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= 4.6e+134:
		tmp = math.atan2(t_0, (math.sin(phi2) - math.sqrt(math.pow((phi1 * math.cos((lambda2 - lambda1))), 2.0))))
	else:
		tmp = math.atan2(math.pow(math.sqrt(t_0), 2.0), (math.sin(phi1) * -math.cos((lambda1 - lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= 4.6e+134)
		tmp = atan(t_0, Float64(sin(phi2) - sqrt((Float64(phi1 * cos(Float64(lambda2 - lambda1))) ^ 2.0))));
	else
		tmp = atan((sqrt(t_0) ^ 2.0), Float64(sin(phi1) * Float64(-cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= 4.6e+134)
		tmp = atan2(t_0, (sin(phi2) - sqrt(((phi1 * cos((lambda2 - lambda1))) ^ 2.0))));
	else
		tmp = atan2((sqrt(t_0) ^ 2.0), (sin(phi1) * -cos((lambda1 - lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 4.6e+134], N[ArcTan[t$95$0 / N[(N[Sin[phi2], $MachinePrecision] - N[Sqrt[N[Power[N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[Power[N[Sqrt[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 4.5999999999999996e134

   1. Initial program 82.1%

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

      1. *-commutative 82.1%

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

      2. associate-*l* 82.1%

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

   3. Simplified 82.1%

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

   5. Taylor expanded in phi1 around 0 54.3%

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

      1. sub-neg 54.3%

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

      2. remove-double-neg 54.3%

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

      3. mul-1-neg 54.3%

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

      4. distribute-neg-in 54.3%

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

      5. +-commutative 54.3%

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

      6. *-commutative 54.3%

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

      7. cos-neg 54.3%

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

      8. mul-1-neg 54.3%

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

      9. unsub-neg 54.3%

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

   7. Simplified 54.3%

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

   8. Taylor expanded in phi1 around 0 54.3%

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

   9. Taylor expanded in phi2 around 0 53.4%

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

       1. add-sqr-sqrt 32.5%

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

       2. sqrt-unprod 57.3%

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

       3. pow2 57.3%

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

   11. Applied egg-rr 57.3%

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

   if 4.5999999999999996e134 < phi1

   1. Initial program 77.2%

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

      1. add-sqr-sqrt 46.4%

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

      2. pow2 46.4%

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

   4. Applied egg-rr 46.4%

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

   5. Taylor expanded in phi2 around 0 31.4%

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

      1. associate-*r* 31.4%

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

      2. neg-mul-1 31.4%

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

   7. Simplified 31.4%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 4.6 \cdot 10^{+134}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \sqrt{{\left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{{\left(\sqrt{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}\right)}^{2}}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 48.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 81.3%

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

   1. *-commutative 81.3%

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

   2. associate-*l* 81.3%

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

3. Simplified 81.3%

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

5. Taylor expanded in phi1 around 0 47.1%

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

   1. sub-neg 47.1%

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

   2. remove-double-neg 47.1%

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

   3. mul-1-neg 47.1%

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

   4. distribute-neg-in 47.1%

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

   5. +-commutative 47.1%

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

   6. *-commutative 47.1%

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

   7. cos-neg 47.1%

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

   8. mul-1-neg 47.1%

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

   9. unsub-neg 47.1%

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

7. Simplified 47.1%

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

8. Taylor expanded in phi1 around 0 47.1%

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

9. Taylor expanded in lambda2 around 0 46.7%

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

    1. cos-neg 46.7%

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

11. Simplified 46.7%

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

12. Final simplification 46.7%

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

Alternative 23: 48.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(phi2) * (cos(lambda2) * phi1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. *-commutative 81.3%

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

   2. associate-*l* 81.3%

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

3. Simplified 81.3%

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

5. Taylor expanded in phi1 around 0 47.1%

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

   1. sub-neg 47.1%

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

   2. remove-double-neg 47.1%

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

   3. mul-1-neg 47.1%

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

   4. distribute-neg-in 47.1%

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

   5. +-commutative 47.1%

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

   6. *-commutative 47.1%

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

   7. cos-neg 47.1%

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

   8. mul-1-neg 47.1%

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

   9. unsub-neg 47.1%

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

7. Simplified 47.1%

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

8. Taylor expanded in phi1 around 0 47.1%

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

9. Taylor expanded in lambda1 around 0 47.8%

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

    1. associate-*r* 47.8%

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

11. Simplified 47.8%

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

12. Final simplification 47.8%

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

Alternative 24: 48.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (phi1 * cos((lambda2 - lambda1)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. *-commutative 81.3%

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

   2. associate-*l* 81.3%

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

3. Simplified 81.3%

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

5. Taylor expanded in phi1 around 0 47.1%

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

   1. sub-neg 47.1%

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

   2. remove-double-neg 47.1%

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

   3. mul-1-neg 47.1%

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

   4. distribute-neg-in 47.1%

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

   5. +-commutative 47.1%

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

   6. *-commutative 47.1%

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

   7. cos-neg 47.1%

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

   8. mul-1-neg 47.1%

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

   9. unsub-neg 47.1%

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

7. Simplified 47.1%

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

8. Taylor expanded in phi1 around 0 47.1%

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

9. Taylor expanded in phi2 around 0 46.5%

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

10. Final simplification 46.5%

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

Alternative 25: 31.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - (phi1 * cos((lambda2 - lambda1)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. *-commutative 81.3%

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

   2. associate-*l* 81.3%

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

3. Simplified 81.3%

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

5. Taylor expanded in phi1 around 0 47.1%

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

   1. sub-neg 47.1%

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

   2. remove-double-neg 47.1%

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

   3. mul-1-neg 47.1%

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

   4. distribute-neg-in 47.1%

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

   5. +-commutative 47.1%

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

   6. *-commutative 47.1%

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

   7. cos-neg 47.1%

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

   8. mul-1-neg 47.1%

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

   9. unsub-neg 47.1%

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

7. Simplified 47.1%

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

8. Taylor expanded in phi1 around 0 47.1%

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

9. Taylor expanded in phi2 around 0 46.5%

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

10. Taylor expanded in lambda2 around 0 30.3%

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

11. Final simplification 30.3%

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

Alternative 26: 48.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda2) * phi1)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.3%

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

   1. *-commutative 81.3%

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

   2. associate-*l* 81.3%

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

3. Simplified 81.3%

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

5. Taylor expanded in phi1 around 0 47.1%

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

   1. sub-neg 47.1%

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

   2. remove-double-neg 47.1%

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

   3. mul-1-neg 47.1%

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

   4. distribute-neg-in 47.1%

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

   5. +-commutative 47.1%

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

   6. *-commutative 47.1%

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

   7. cos-neg 47.1%

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

   8. mul-1-neg 47.1%

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

   9. unsub-neg 47.1%

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

7. Simplified 47.1%

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

8. Taylor expanded in phi1 around 0 47.1%

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

9. Taylor expanded in phi2 around 0 46.5%

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

10. Taylor expanded in lambda1 around 0 46.5%

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

11. Final simplification 46.5%

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

Reproduce

herbie shell --seed 2024054 
(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))))))