Bearing on a great circle

Percentage Accurate: 78.8% → 99.7%

Time: 35.1s

Alternatives: 27

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: 78.8% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-*l* 82.7%

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

3. Simplified 82.7%

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

   1. cos-diff 82.9%

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

   2. +-commutative 82.9%

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

   3. *-commutative 82.9%

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

6. Applied egg-rr 82.9%

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

   1. sin-diff 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 93.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := \left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -8300000:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0}\\ \mathbf{elif}\;\phi_2 \leq 65000:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{\sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (-
          (* (cos phi1) (sin phi2))
          (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
        (t_1
         (*
          (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
          (cos phi2))))
   (if (<= phi2 -8300000.0)
     (atan2 t_1 t_0)
     (if (<= phi2 65000.0)
       (atan2
        t_1
        (-
         (sin phi2)
         (*
          (cos phi2)
          (*
           (sin phi1)
           (+
            (* (sin lambda1) (sin lambda2))
            (* (cos lambda2) (cos lambda1)))))))
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
        t_0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))));
	double t_1 = ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2);
	double tmp;
	if (phi2 <= -8300000.0) {
		tmp = atan2(t_1, t_0);
	} else if (phi2 <= 65000.0) {
		tmp = atan2(t_1, (sin(phi2) - (cos(phi2) * (sin(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))))));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), t_0);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))))
	t_1 = Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2))
	tmp = 0.0
	if (phi2 <= -8300000.0)
		tmp = atan(t_1, t_0);
	elseif (phi2 <= 65000.0)
		tmp = atan(t_1, Float64(sin(phi2) - Float64(cos(phi2) * Float64(sin(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1)))))));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), t_0);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -8300000.0], N[ArcTan[t$95$1 / t$95$0], $MachinePrecision], If[LessEqual[phi2, 65000.0], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * 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]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -8.3e6

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

      2. associate-*l* 86.6%

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

   3. Simplified 86.6%

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

      1. sin-diff 99.7%

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

   6. Applied egg-rr 92.0%

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

   if -8.3e6 < phi2 < 65000

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-*l* 85.4%

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

   3. Simplified 85.4%

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

      1. cos-diff 85.6%

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

      2. +-commutative 85.6%

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

      3. *-commutative 85.6%

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

   6. Applied egg-rr 85.6%

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

      1. sin-diff 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in phi1 around 0 99.4%

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

   if 65000 < phi2

   1. Initial program 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l* 74.2%

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

   3. Simplified 74.2%

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

      1. sin-diff 86.6%

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

      2. cancel-sign-sub-inv 86.6%

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

      3. fma-def 86.6%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8300000:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_2 \leq 65000:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-*l* 82.7%

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

3. Simplified 82.7%

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

   1. sin-diff 89.8%

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

   2. cancel-sign-sub-inv 89.8%

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

   3. fma-def 89.8%

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

6. Applied egg-rr 89.8%

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

7. Final simplification 89.8%

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

Alternative 4: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -2.8 \cdot 10^{-5} \lor \neg \left(\lambda_2 \leq 2 \cdot 10^{-33}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - \lambda_2 \cdot \cos \lambda_1\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda2 -2.8e-5) (not (<= lambda2 2e-33)))
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
       (cos phi2))
      (- t_0 (* (cos lambda2) (* (cos phi2) (sin phi1)))))
     (atan2
      (* (cos phi2) (- (sin lambda1) (* lambda2 (cos lambda1))))
      (- t_0 (* (cos phi2) (* (cos lambda1) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda2 <= -2.8e-5) || !(lambda2 <= 2e-33)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * (sin(lambda1) - (lambda2 * cos(lambda1)))), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda2 <= (-2.8d-5)) .or. (.not. (lambda2 <= 2d-33))) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos(lambda2) * (cos(phi2) * sin(phi1)))))
    else
        tmp = atan2((cos(phi2) * (sin(lambda1) - (lambda2 * cos(lambda1)))), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda2 <= -2.8e-5) || !(lambda2 <= 2e-33)) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (t_0 - (Math.cos(lambda2) * (Math.cos(phi2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * (Math.sin(lambda1) - (lambda2 * Math.cos(lambda1)))), (t_0 - (Math.cos(phi2) * (Math.cos(lambda1) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda2 <= -2.8e-5) or not (lambda2 <= 2e-33):
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (t_0 - (math.cos(lambda2) * (math.cos(phi2) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * (math.sin(lambda1) - (lambda2 * math.cos(lambda1)))), (t_0 - (math.cos(phi2) * (math.cos(lambda1) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda2 <= -2.8e-5) || !(lambda2 <= 2e-33))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(cos(lambda2) * Float64(cos(phi2) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * Float64(sin(lambda1) - Float64(lambda2 * cos(lambda1)))), Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda1) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda2 <= -2.8e-5) || ~((lambda2 <= 2e-33)))
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	else
		tmp = atan2((cos(phi2) * (sin(lambda1) - (lambda2 * cos(lambda1)))), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda2, -2.8e-5], N[Not[LessEqual[lambda2, 2e-33]], $MachinePrecision]], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] - N[(lambda2 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -2.79999999999999996e-5 or 2.0000000000000001e-33 < lambda2

   1. Initial program 63.5%

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

      1. *-commutative 63.5%

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

      2. associate-*l* 63.5%

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

   3. Simplified 63.5%

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

      1. cos-diff 63.9%

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

      2. +-commutative 63.9%

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

      3. *-commutative 63.9%

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

   6. Applied egg-rr 63.9%

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

      1. sin-diff 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in lambda1 around 0 78.4%

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

   if -2.79999999999999996e-5 < lambda2 < 2.0000000000000001e-33

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

      1. sin-diff 99.6%

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

      2. cancel-sign-sub-inv 99.6%

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

      3. fma-def 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in lambda2 around 0 99.6%

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

   8. Taylor expanded in lambda2 around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. neg-mul-1 99.6%

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

   10. Simplified 99.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.8 \cdot 10^{-5} \lor \neg \left(\lambda_2 \leq 2 \cdot 10^{-33}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - \lambda_2 \cdot \cos \lambda_1\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1.75 \cdot 10^{-5} \lor \neg \left(\lambda_1 \leq 0.00039\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda1 -1.75e-5) (not (<= lambda1 0.00039)))
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
       (cos phi2))
      (- t_0 (* (sin phi1) (* (cos lambda1) (cos phi2)))))
     (atan2
      (* (cos phi2) (- (* lambda1 (cos lambda2)) (sin lambda2)))
      (-
       t_0
       (*
        (cos phi2)
        (* (sin phi1) (+ (cos lambda2) (* lambda1 (sin lambda2))))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda1 <= -1.75e-5) || !(lambda1 <= 0.00039)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * (cos(lambda1) * cos(phi2)))));
	} else {
		tmp = atan2((cos(phi2) * ((lambda1 * cos(lambda2)) - sin(lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * (cos(lambda2) + (lambda1 * sin(lambda2)))))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda1 <= (-1.75d-5)) .or. (.not. (lambda1 <= 0.00039d0))) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * (cos(lambda1) * cos(phi2)))))
    else
        tmp = atan2((cos(phi2) * ((lambda1 * cos(lambda2)) - sin(lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * (cos(lambda2) + (lambda1 * sin(lambda2)))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda1 <= -1.75e-5) || !(lambda1 <= 0.00039)) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (t_0 - (Math.sin(phi1) * (Math.cos(lambda1) * Math.cos(phi2)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * ((lambda1 * Math.cos(lambda2)) - Math.sin(lambda2))), (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * (Math.cos(lambda2) + (lambda1 * Math.sin(lambda2)))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda1 <= -1.75e-5) or not (lambda1 <= 0.00039):
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (t_0 - (math.sin(phi1) * (math.cos(lambda1) * math.cos(phi2)))))
	else:
		tmp = math.atan2((math.cos(phi2) * ((lambda1 * math.cos(lambda2)) - math.sin(lambda2))), (t_0 - (math.cos(phi2) * (math.sin(phi1) * (math.cos(lambda2) + (lambda1 * math.sin(lambda2)))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda1 <= -1.75e-5) || !(lambda1 <= 0.00039))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * Float64(cos(lambda1) * cos(phi2)))));
	else
		tmp = atan(Float64(cos(phi2) * Float64(Float64(lambda1 * cos(lambda2)) - sin(lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * Float64(cos(lambda2) + Float64(lambda1 * sin(lambda2)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda1 <= -1.75e-5) || ~((lambda1 <= 0.00039)))
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (sin(phi1) * (cos(lambda1) * cos(phi2)))));
	else
		tmp = atan2((cos(phi2) * ((lambda1 * cos(lambda2)) - sin(lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * (cos(lambda2) + (lambda1 * sin(lambda2)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -1.75e-5], N[Not[LessEqual[lambda1, 0.00039]], $MachinePrecision]], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(lambda1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] + N[(lambda1 * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.7499999999999998e-5 or 3.89999999999999993e-4 < lambda1

   1. Initial program 67.7%

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

      1. *-commutative 67.7%

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

      2. associate-*l* 67.7%

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

   3. Simplified 67.7%

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

      1. cos-diff 68.0%

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

      2. +-commutative 68.0%

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

      3. *-commutative 68.0%

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

   6. Applied egg-rr 68.0%

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

      1. sin-diff 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in lambda2 around 0 81.1%

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

       1. associate-*r* 81.1%

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

       2. *-commutative 81.1%

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

   11. Simplified 81.1%

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

   if -1.7499999999999998e-5 < lambda1 < 3.89999999999999993e-4

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda1 around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. cos-neg 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. sin-neg 99.6%

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

      5. remove-double-neg 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in lambda1 around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. sin-neg 99.8%

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

      3. unsub-neg 99.8%

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

      4. *-commutative 99.8%

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

      5. cos-neg 99.8%

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

   10. Simplified 99.8%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.75 \cdot 10^{-5} \lor \neg \left(\lambda_1 \leq 0.00039\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.45 \cdot 10^{+42} \lor \neg \left(\phi_1 \leq 1.46 \cdot 10^{-13}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, t_0\right)}{t_1 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot t_0\right)}{t_1 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- (sin lambda2))) (t_1 (* (cos phi1) (sin phi2))))
   (if (or (<= phi1 -1.45e+42) (not (<= phi1 1.46e-13)))
     (atan2
      (* (cos phi2) (fma (sin lambda1) (cos lambda2) t_0))
      (- t_1 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (atan2
      (* (cos phi2) (fma (sin lambda1) (cos lambda2) (* (cos lambda1) t_0)))
      (- t_1 (* (cos lambda1) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = -sin(lambda2);
	double t_1 = cos(phi1) * sin(phi2);
	double tmp;
	if ((phi1 <= -1.45e+42) || !(phi1 <= 1.46e-13)) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), t_0)), (t_1 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * t_0))), (t_1 - (cos(lambda1) * sin(phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(-sin(lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((phi1 <= -1.45e+42) || !(phi1 <= 1.46e-13))
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), t_0)), Float64(t_1 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * t_0))), Float64(t_1 - Float64(cos(lambda1) * sin(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = (-N[Sin[lambda2], $MachinePrecision])}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -1.45e+42], N[Not[LessEqual[phi1, 1.46e-13]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.4499999999999999e42 or 1.46000000000000009e-13 < phi1

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-*l* 80.3%

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

   3. Simplified 80.3%

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

      1. sin-diff 83.9%

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

      2. cancel-sign-sub-inv 83.9%

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

      3. fma-def 83.9%

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

   6. Applied egg-rr 83.9%

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

   7. Taylor expanded in lambda1 around 0 81.0%

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

      1. mul-1-neg 81.0%

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

   9. Simplified 81.0%

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

   if -1.4499999999999999e42 < phi1 < 1.46000000000000009e-13

   1. Initial program 85.5%

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

      1. *-commutative 85.5%

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

      2. associate-*l* 85.5%

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

   3. Simplified 85.5%

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

      1. sin-diff 96.4%

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

      2. cancel-sign-sub-inv 96.4%

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

      3. fma-def 96.4%

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

   6. Applied egg-rr 96.4%

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

   7. Taylor expanded in lambda2 around 0 96.5%

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

   8. Taylor expanded in phi2 around 0 96.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.45 \cdot 10^{+42} \lor \neg \left(\phi_1 \leq 1.46 \cdot 10^{-13}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, -\sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+34} \lor \neg \left(\phi_1 \leq 1.45 \cdot 10^{-13}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{t_0 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= phi1 -8.5e+34) (not (<= phi1 1.45e-13)))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (atan2
      (*
       (cos phi2)
       (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
      (- t_0 (* (cos lambda1) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((phi1 <= -8.5e+34) || !(phi1 <= 1.45e-13)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), (t_0 - (cos(lambda1) * sin(phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((phi1 <= -8.5e+34) || !(phi1 <= 1.45e-13))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), Float64(t_0 - Float64(cos(lambda1) * sin(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -8.5e+34], N[Not[LessEqual[phi1, 1.45e-13]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.5000000000000003e34 or 1.4499999999999999e-13 < phi1

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*l* 79.4%

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

   3. Simplified 79.4%

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

   if -8.5000000000000003e34 < phi1 < 1.4499999999999999e-13

   1. Initial program 86.6%

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

      1. *-commutative 86.6%

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

      2. associate-*l* 86.6%

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

   3. Simplified 86.6%

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

      1. sin-diff 97.7%

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

      2. cancel-sign-sub-inv 97.7%

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

      3. fma-def 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in lambda2 around 0 97.7%

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

   8. Taylor expanded in phi2 around 0 97.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{+34} \lor \neg \left(\phi_1 \leq 1.45 \cdot 10^{-13}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \sin \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.25 \cdot 10^{+26} \lor \neg \left(\phi_1 \leq 9.2 \cdot 10^{-14}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= phi1 -1.25e+26) (not (<= phi1 9.2e-14)))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (atan2
      (*
       (cos phi2)
       (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
      (- t_0 (* (cos phi2) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((phi1 <= -1.25e+26) || !(phi1 <= 9.2e-14)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), (t_0 - (cos(phi2) * sin(phi1))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((phi1 <= -1.25e+26) || !(phi1 <= 9.2e-14))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), Float64(t_0 - Float64(cos(phi2) * sin(phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -1.25e+26], N[Not[LessEqual[phi1, 9.2e-14]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.25e26 or 9.19999999999999993e-14 < phi1

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. associate-*l* 79.7%

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

   3. Simplified 79.7%

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

   if -1.25e26 < phi1 < 9.19999999999999993e-14

   1. Initial program 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-*l* 86.4%

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

   3. Simplified 86.4%

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

      1. sin-diff 97.7%

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

      2. cancel-sign-sub-inv 97.7%

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

      3. fma-def 97.7%

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

   6. Applied egg-rr 97.7%

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

   7. Taylor expanded in lambda2 around 0 97.7%

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

   8. Taylor expanded in lambda1 around 0 97.8%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.25 \cdot 10^{+26} \lor \neg \left(\phi_1 \leq 9.2 \cdot 10^{-14}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \sin \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-*l* 82.7%

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

3. Simplified 82.7%

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

   1. sin-diff 99.7%

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

6. Applied egg-rr 89.8%

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

7. Final simplification 89.8%

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

Alternative 10: 87.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.6 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 10^{-13}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -6.6e-7) (not (<= phi1 1e-13)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
   (atan2
    (*
     (- (* (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 <= -6.6e-7) || !(phi1 <= 1e-13)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (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 <= (-6.6d-7)) .or. (.not. (phi1 <= 1d-13))) then
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
    else
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (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 <= -6.6e-7) || !(phi1 <= 1e-13)) {
		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.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (Math.sin(phi2) - (phi1 * Math.cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -6.6e-7) or not (phi1 <= 1e-13):
		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.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (math.sin(phi2) - (phi1 * math.cos((lambda2 - lambda1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -6.6e-7) || !(phi1 <= 1e-13))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), 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 <= -6.6e-7) || ~((phi1 <= 1e-13)))
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	else
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (sin(phi2) - (phi1 * cos((lambda2 - lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -6.6e-7], N[Not[LessEqual[phi1, 1e-13]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[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 < -6.6000000000000003e-7 or 1e-13 < phi1

   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. Step-by-step derivation

      1. *-commutative 78.5%

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

      2. associate-*l* 78.6%

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

   3. Simplified 78.6%

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

   if -6.6000000000000003e-7 < phi1 < 1e-13

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l* 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in phi1 around 0 88.0%

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

      1. sub-neg 88.0%

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

      2. remove-double-neg 88.0%

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

      3. mul-1-neg 88.0%

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

      4. distribute-neg-in 88.0%

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

      5. +-commutative 88.0%

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

      6. cos-neg 88.0%

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

      7. mul-1-neg 88.0%

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

      8. unsub-neg 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in phi1 around 0 88.0%

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

   9. Taylor expanded in phi2 around 0 88.0%

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

       1. sin-diff 99.9%

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

   11. Applied egg-rr 99.7%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.6 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 10^{-13}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -0.0155 \lor \neg \left(\lambda_2 \leq 5.7 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda2 -0.0155) (not (<= lambda2 5.7e-5)))
     (atan2
      (* (cos phi2) (sin (- lambda2)))
      (- t_0 (* (cos lambda2) (* (cos phi2) (sin phi1)))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* (cos phi2) (* (cos lambda1) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda2 <= -0.0155) || !(lambda2 <= 5.7e-5)) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda2 <= (-0.0155d0)) .or. (.not. (lambda2 <= 5.7d-5))) then
        tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(lambda2) * (cos(phi2) * sin(phi1)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda2 <= -0.0155) || !(lambda2 <= 5.7e-5)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (t_0 - (Math.cos(lambda2) * (Math.cos(phi2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (Math.cos(phi2) * (Math.cos(lambda1) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda2 <= -0.0155) or not (lambda2 <= 5.7e-5):
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), (t_0 - (math.cos(lambda2) * (math.cos(phi2) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (math.cos(phi2) * (math.cos(lambda1) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda2 <= -0.0155) || !(lambda2 <= 5.7e-5))
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_0 - Float64(cos(lambda2) * Float64(cos(phi2) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda1) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda2 <= -0.0155) || ~((lambda2 <= 5.7e-5)))
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda2, -0.0155], N[Not[LessEqual[lambda2, 5.7e-5]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -0.0155 or 5.7000000000000003e-5 < lambda2

   1. Initial program 62.6%

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

      1. *-commutative 62.6%

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

      2. associate-*l* 62.6%

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

   3. Simplified 62.6%

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

      1. cos-diff 63.0%

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

      2. +-commutative 63.0%

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

      3. *-commutative 63.0%

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

   6. Applied egg-rr 63.0%

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

   7. Taylor expanded in lambda1 around 0 59.7%

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

   8. Taylor expanded in lambda1 around 0 59.4%

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

   if -0.0155 < lambda2 < 5.7000000000000003e-5

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in lambda2 around 0 99.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.0155 \lor \neg \left(\lambda_2 \leq 5.7 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)\\ \mathbf{if}\;\lambda_1 \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2}\\ \mathbf{elif}\;\lambda_1 \leq 0.0068:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (- t_0 (* (cos phi2) (* (cos lambda1) (sin phi1))))))
   (if (<= lambda1 -5.8e-6)
     (atan2 t_1 t_2)
     (if (<= lambda1 0.0068)
       (atan2 t_1 (- t_0 (* (cos phi2) (* (cos lambda2) (sin phi1)))))
       (atan2 (* (sin lambda1) (cos phi2)) t_2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double t_2 = t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)));
	double tmp;
	if (lambda1 <= -5.8e-6) {
		tmp = atan2(t_1, t_2);
	} else if (lambda1 <= 0.0068) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	} else {
		tmp = atan2((sin(lambda1) * cos(phi2)), t_2);
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    t_2 = t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))
    if (lambda1 <= (-5.8d-6)) then
        tmp = atan2(t_1, t_2)
    else if (lambda1 <= 0.0068d0) then
        tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))))
    else
        tmp = atan2((sin(lambda1) * cos(phi2)), t_2)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double t_2 = t_0 - (Math.cos(phi2) * (Math.cos(lambda1) * Math.sin(phi1)));
	double tmp;
	if (lambda1 <= -5.8e-6) {
		tmp = Math.atan2(t_1, t_2);
	} else if (lambda1 <= 0.0068) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), t_2);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	t_2 = t_0 - (math.cos(phi2) * (math.cos(lambda1) * math.sin(phi1)))
	tmp = 0
	if lambda1 <= -5.8e-6:
		tmp = math.atan2(t_1, t_2)
	elif lambda1 <= 0.0068:
		tmp = math.atan2(t_1, (t_0 - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), t_2)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_2 = Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda1) * sin(phi1))))
	tmp = 0.0
	if (lambda1 <= -5.8e-6)
		tmp = atan(t_1, t_2);
	elseif (lambda1 <= 0.0068)
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	else
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	t_2 = t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)));
	tmp = 0.0;
	if (lambda1 <= -5.8e-6)
		tmp = atan2(t_1, t_2);
	elseif (lambda1 <= 0.0068)
		tmp = atan2(t_1, (t_0 - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	else
		tmp = atan2((sin(lambda1) * cos(phi2)), t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -5.8e-6], N[ArcTan[t$95$1 / t$95$2], $MachinePrecision], If[LessEqual[lambda1, 0.0068], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / t$95$2], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -5.8000000000000004e-6

   1. Initial program 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*l* 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in lambda2 around 0 72.4%

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

   if -5.8000000000000004e-6 < lambda1 < 0.00679999999999999962

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda1 around 0 99.5%

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

      1. cos-neg 99.5%

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

   7. Simplified 99.5%

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

   if 0.00679999999999999962 < lambda1

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l* 62.9%

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

   3. Simplified 62.9%

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

      1. sin-diff 78.7%

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

      2. cancel-sign-sub-inv 78.7%

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

      3. fma-def 78.7%

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

   6. Applied egg-rr 78.7%

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

   7. Taylor expanded in lambda2 around 0 78.7%

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

   8. Taylor expanded in lambda2 around 0 66.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{elif}\;\lambda_1 \leq 0.0068:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -6.2 \cdot 10^{-16} \lor \neg \left(\lambda_1 \leq 0.00155\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_0 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda1 -6.2e-16) (not (<= lambda1 0.00155)))
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* (cos phi2) (* (cos lambda1) (sin phi1)))))
     (atan2
      (* (cos phi2) (sin (- lambda2)))
      (- t_0 (* (cos lambda2) (* (cos phi2) (sin phi1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda1 <= -6.2e-16) || !(lambda1 <= 0.00155)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda1 <= (-6.2d-16)) .or. (.not. (lambda1 <= 0.00155d0))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))))
    else
        tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(lambda2) * (cos(phi2) * sin(phi1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda1 <= -6.2e-16) || !(lambda1 <= 0.00155)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - (Math.cos(phi2) * (Math.cos(lambda1) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (t_0 - (Math.cos(lambda2) * (Math.cos(phi2) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda1 <= -6.2e-16) or not (lambda1 <= 0.00155):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - (math.cos(phi2) * (math.cos(lambda1) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), (t_0 - (math.cos(lambda2) * (math.cos(phi2) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda1 <= -6.2e-16) || !(lambda1 <= 0.00155))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda1) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_0 - Float64(cos(lambda2) * Float64(cos(phi2) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda1 <= -6.2e-16) || ~((lambda1 <= 0.00155)))
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	else
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_0 - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -6.2e-16], N[Not[LessEqual[lambda1, 0.00155]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.2000000000000002e-16 or 0.00154999999999999995 < lambda1

   1. Initial program 68.4%

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

      1. *-commutative 68.4%

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

      2. associate-*l* 68.5%

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

   3. Simplified 68.5%

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

      1. sin-diff 81.5%

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

      2. cancel-sign-sub-inv 81.5%

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

      3. fma-def 81.5%

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

   6. Applied egg-rr 81.5%

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

   7. Taylor expanded in lambda2 around 0 81.0%

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

   8. Taylor expanded in lambda2 around 0 68.0%

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

   if -6.2000000000000002e-16 < lambda1 < 0.00154999999999999995

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. cos-diff 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in lambda1 around 0 87.5%

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

   8. Taylor expanded in lambda1 around 0 87.5%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.2 \cdot 10^{-16} \lor \neg \left(\lambda_1 \leq 0.00155\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \phi_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -5.2 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\lambda_1 \leq 0.00155:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_1 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_0}{t_1 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda1) (cos phi2))) (t_1 (* (cos phi1) (sin phi2))))
   (if (<= lambda1 -5.2e-16)
     (atan2
      t_0
      (- t_1 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
     (if (<= lambda1 0.00155)
       (atan2
        (* (cos phi2) (sin (- lambda2)))
        (- t_1 (* (cos lambda2) (* (cos phi2) (sin phi1)))))
       (atan2 t_0 (- t_1 (* (cos phi2) (* (cos lambda1) (sin phi1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(phi2);
	double t_1 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -5.2e-16) {
		tmp = atan2(t_0, (t_1 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else if (lambda1 <= 0.00155) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_1 - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	} else {
		tmp = atan2(t_0, (t_1 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(lambda1) * cos(phi2)
    t_1 = cos(phi1) * sin(phi2)
    if (lambda1 <= (-5.2d-16)) then
        tmp = atan2(t_0, (t_1 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
    else if (lambda1 <= 0.00155d0) then
        tmp = atan2((cos(phi2) * sin(-lambda2)), (t_1 - (cos(lambda2) * (cos(phi2) * sin(phi1)))))
    else
        tmp = atan2(t_0, (t_1 - (cos(phi2) * (cos(lambda1) * sin(phi1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(lambda1) * Math.cos(phi2);
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -5.2e-16) {
		tmp = Math.atan2(t_0, (t_1 - (Math.cos(phi2) * (Math.sin(phi1) * Math.cos((lambda1 - lambda2))))));
	} else if (lambda1 <= 0.00155) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (t_1 - (Math.cos(lambda2) * (Math.cos(phi2) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2(t_0, (t_1 - (Math.cos(phi2) * (Math.cos(lambda1) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(lambda1) * math.cos(phi2)
	t_1 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -5.2e-16:
		tmp = math.atan2(t_0, (t_1 - (math.cos(phi2) * (math.sin(phi1) * math.cos((lambda1 - lambda2))))))
	elif lambda1 <= 0.00155:
		tmp = math.atan2((math.cos(phi2) * math.sin(-lambda2)), (t_1 - (math.cos(lambda2) * (math.cos(phi2) * math.sin(phi1)))))
	else:
		tmp = math.atan2(t_0, (t_1 - (math.cos(phi2) * (math.cos(lambda1) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(phi2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -5.2e-16)
		tmp = atan(t_0, Float64(t_1 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	elseif (lambda1 <= 0.00155)
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_1 - Float64(cos(lambda2) * Float64(cos(phi2) * sin(phi1)))));
	else
		tmp = atan(t_0, Float64(t_1 - Float64(cos(phi2) * Float64(cos(lambda1) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(lambda1) * cos(phi2);
	t_1 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -5.2e-16)
		tmp = atan2(t_0, (t_1 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	elseif (lambda1 <= 0.00155)
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_1 - (cos(lambda2) * (cos(phi2) * sin(phi1)))));
	else
		tmp = atan2(t_0, (t_1 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -5.2e-16], N[ArcTan[t$95$0 / N[(t$95$1 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda1, 0.00155], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[Cos[lambda2], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(t$95$1 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -5.1999999999999997e-16

   1. Initial program 73.9%

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

      1. *-commutative 73.9%

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

      2. associate-*l* 73.9%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in lambda2 around 0 69.1%

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

   if -5.1999999999999997e-16 < lambda1 < 0.00154999999999999995

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

   3. Simplified 99.7%

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

      1. cos-diff 99.7%

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

      2. +-commutative 99.7%

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

      3. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in lambda1 around 0 87.5%

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

   8. Taylor expanded in lambda1 around 0 87.5%

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

   if 0.00154999999999999995 < lambda1

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

      2. associate-*l* 62.9%

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

   3. Simplified 62.9%

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

      1. sin-diff 78.7%

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

      2. cancel-sign-sub-inv 78.7%

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

      3. fma-def 78.7%

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

   6. Applied egg-rr 78.7%

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

   7. Taylor expanded in lambda2 around 0 78.7%

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

   8. Taylor expanded in lambda2 around 0 66.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5.2 \cdot 10^{-16}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\lambda_1 \leq 0.00155:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -4.1 \cdot 10^{-5} \lor \neg \left(\lambda_1 \leq 0.056\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda1 -4.1e-5) (not (<= lambda1 0.056)))
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* (cos phi2) (* (cos lambda1) (sin phi1)))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* (cos phi2) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda1 <= -4.1e-5) || !(lambda1 <= 0.056)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * sin(phi1))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda1 <= (-4.1d-5)) .or. (.not. (lambda1 <= 0.056d0))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * sin(phi1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda1 <= -4.1e-5) || !(lambda1 <= 0.056)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - (Math.cos(phi2) * (Math.cos(lambda1) * Math.sin(phi1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (Math.cos(phi2) * Math.sin(phi1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda1 <= -4.1e-5) or not (lambda1 <= 0.056):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - (math.cos(phi2) * (math.cos(lambda1) * math.sin(phi1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (math.cos(phi2) * math.sin(phi1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda1 <= -4.1e-5) || !(lambda1 <= 0.056))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(cos(phi2) * Float64(cos(lambda1) * sin(phi1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * sin(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda1 <= -4.1e-5) || ~((lambda1 <= 0.056)))
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (cos(phi2) * (cos(lambda1) * sin(phi1)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * sin(phi1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -4.1e-5], N[Not[LessEqual[lambda1, 0.056]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.10000000000000005e-5 or 0.0560000000000000012 < lambda1

   1. Initial program 67.5%

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

      1. *-commutative 67.5%

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

      2. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

      1. sin-diff 80.9%

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

      2. cancel-sign-sub-inv 80.9%

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

      3. fma-def 80.9%

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

   6. Applied egg-rr 80.9%

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

   7. Taylor expanded in lambda2 around 0 80.9%

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

   8. Taylor expanded in lambda2 around 0 68.1%

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

   if -4.10000000000000005e-5 < lambda1 < 0.0560000000000000012

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in lambda1 around 0 99.6%

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

      1. mul-1-neg 99.6%

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

      2. cos-neg 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. sin-neg 99.6%

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

      5. remove-double-neg 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in lambda2 around 0 80.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.1 \cdot 10^{-5} \lor \neg \left(\lambda_1 \leq 0.056\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \sin \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 78.8% 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 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-*l* 82.7%

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

3. Simplified 82.7%

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

5. Final simplification 82.7%

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

Alternative 17: 68.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -0.47 \lor \neg \left(\phi_2 \leq 65000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\frac{t_1 \cdot 2}{2}}{t_0 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (sin (- lambda1 lambda2))))
   (if (or (<= phi2 -0.47) (not (<= phi2 65000.0)))
     (atan2 (* (cos phi2) t_1) (- t_0 (* (cos phi2) (sin phi1))))
     (atan2
      (/ (* t_1 2.0) 2.0)
      (- t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -0.47) || !(phi2 <= 65000.0)) {
		tmp = atan2((cos(phi2) * t_1), (t_0 - (cos(phi2) * sin(phi1))));
	} else {
		tmp = atan2(((t_1 * 2.0) / 2.0), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin((lambda1 - lambda2))
    if ((phi2 <= (-0.47d0)) .or. (.not. (phi2 <= 65000.0d0))) then
        tmp = atan2((cos(phi2) * t_1), (t_0 - (cos(phi2) * sin(phi1))))
    else
        tmp = atan2(((t_1 * 2.0d0) / 2.0d0), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -0.47) || !(phi2 <= 65000.0)) {
		tmp = Math.atan2((Math.cos(phi2) * t_1), (t_0 - (Math.cos(phi2) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(((t_1 * 2.0) / 2.0), (t_0 - (Math.cos(phi2) * (Math.sin(phi1) * Math.cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -0.47) or not (phi2 <= 65000.0):
		tmp = math.atan2((math.cos(phi2) * t_1), (t_0 - (math.cos(phi2) * math.sin(phi1))))
	else:
		tmp = math.atan2(((t_1 * 2.0) / 2.0), (t_0 - (math.cos(phi2) * (math.sin(phi1) * math.cos((lambda1 - lambda2))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi2 <= -0.47) || !(phi2 <= 65000.0))
		tmp = atan(Float64(cos(phi2) * t_1), Float64(t_0 - Float64(cos(phi2) * sin(phi1))));
	else
		tmp = atan(Float64(Float64(t_1 * 2.0) / 2.0), Float64(t_0 - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -0.47) || ~((phi2 <= 65000.0)))
		tmp = atan2((cos(phi2) * t_1), (t_0 - (cos(phi2) * sin(phi1))));
	else
		tmp = atan2(((t_1 * 2.0) / 2.0), (t_0 - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -0.47], N[Not[LessEqual[phi2, 65000.0]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(t$95$1 * 2.0), $MachinePrecision] / 2.0), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.46999999999999997 or 65000 < phi2

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*l* 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in lambda1 around 0 63.5%

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

      1. mul-1-neg 63.5%

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

      2. cos-neg 63.5%

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

      3. distribute-rgt-neg-in 63.5%

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

      4. sin-neg 63.5%

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

      5. remove-double-neg 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in lambda2 around 0 55.7%

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

   if -0.46999999999999997 < phi2 < 65000

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*l* 85.2%

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

   3. Simplified 85.2%

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

      1. sin-cos-mult 81.5%

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

      2. associate--l- 81.5%

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

      3. associate-+l- 81.5%

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

   6. Applied egg-rr 81.5%

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

   7. Taylor expanded in phi2 around 0 85.3%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.47 \lor \neg \left(\phi_2 \leq 65000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot 2}{2}}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 54.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-35} \lor \neg \left(\phi_1 \leq 4.6 \cdot 10^{+111}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -3.5e-35) (not (<= phi1 4.6e+111)))
   (atan2
    (* (sin lambda1) (cos phi2))
    (- (* (cos phi1) (sin phi2)) (* (sin phi1) (cos (- lambda2 lambda1)))))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (- (sin phi2) (* (cos lambda1) phi1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -3.5e-35) || !(phi1 <= 4.6e+111)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda1) * phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi1 <= (-3.5d-35)) .or. (.not. (phi1 <= 4.6d+111))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda2 - lambda1)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda1) * phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -3.5e-35) || !(phi1 <= 4.6e+111)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), ((Math.cos(phi1) * Math.sin(phi2)) - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.sin(phi2) - (Math.cos(lambda1) * phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -3.5e-35) or not (phi1 <= 4.6e+111):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), ((math.cos(phi1) * math.sin(phi2)) - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.sin(phi2) - (math.cos(lambda1) * phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -3.5e-35) || !(phi1 <= 4.6e+111))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(sin(phi2) - Float64(cos(lambda1) * phi1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -3.5e-35) || ~((phi1 <= 4.6e+111)))
		tmp = atan2((sin(lambda1) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda1) * phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -3.5e-35], N[Not[LessEqual[phi1, 4.6e+111]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.49999999999999996e-35 or 4.60000000000000004e111 < phi1

   1. Initial program 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-*l* 75.0%

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

   3. Simplified 75.0%

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

   5. Taylor expanded in lambda2 around 0 48.0%

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

   6. Taylor expanded in phi2 around 0 34.9%

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

      1. sub-neg 34.9%

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

      2. remove-double-neg 34.9%

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

      3. distribute-neg-in 34.9%

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

      4. +-commutative 34.9%

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

      5. neg-mul-1 34.9%

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

      6. cos-neg 34.9%

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

      7. neg-mul-1 34.9%

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

      8. sub-neg 34.9%

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

   8. Simplified 34.9%

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

   if -3.49999999999999996e-35 < phi1 < 4.60000000000000004e111

   1. Initial program 89.6%

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

      1. *-commutative 89.6%

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

      2. associate-*l* 89.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in phi1 around 0 79.8%

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

      1. sub-neg 79.8%

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

      2. remove-double-neg 79.8%

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

      3. mul-1-neg 79.8%

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

      4. distribute-neg-in 79.8%

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

      5. +-commutative 79.8%

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

      6. cos-neg 79.8%

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

      7. mul-1-neg 79.8%

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

      8. unsub-neg 79.8%

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

   7. Simplified 79.8%

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

   8. Taylor expanded in phi1 around 0 79.8%

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

   9. Taylor expanded in phi2 around 0 79.8%

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

   10. Taylor expanded in lambda2 around 0 79.9%

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

       1. cos-neg 79.9%

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

   12. Simplified 79.9%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-35} \lor \neg \left(\phi_1 \leq 4.6 \cdot 10^{+111}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -1.15 \cdot 10^{+160} \lor \neg \left(\lambda_1 \leq 640000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= lambda1 -1.15e+160) (not (<= lambda1 640000000.0)))
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* (sin phi1) (cos (- lambda2 lambda1)))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- t_0 (* (cos phi2) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((lambda1 <= -1.15e+160) || !(lambda1 <= 640000000.0)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * sin(phi1))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((lambda1 <= (-1.15d+160)) .or. (.not. (lambda1 <= 640000000.0d0))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * sin(phi1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((lambda1 <= -1.15e+160) || !(lambda1 <= 640000000.0)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 - (Math.cos(phi2) * Math.sin(phi1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (lambda1 <= -1.15e+160) or not (lambda1 <= 640000000.0):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 - (math.cos(phi2) * math.sin(phi1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((lambda1 <= -1.15e+160) || !(lambda1 <= 640000000.0))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(cos(phi2) * sin(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((lambda1 <= -1.15e+160) || ~((lambda1 <= 640000000.0)))
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (cos(phi2) * sin(phi1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -1.15e+160], N[Not[LessEqual[lambda1, 640000000.0]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.14999999999999994e160 or 6.4e8 < lambda1

   1. Initial program 67.6%

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

      1. *-commutative 67.6%

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

      2. associate-*l* 67.6%

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

   3. Simplified 67.6%

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

   5. Taylor expanded in lambda2 around 0 70.2%

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

   6. Taylor expanded in phi2 around 0 59.5%

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

      1. sub-neg 59.5%

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

      2. remove-double-neg 59.5%

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

      3. distribute-neg-in 59.5%

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

      4. +-commutative 59.5%

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

      5. neg-mul-1 59.5%

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

      6. cos-neg 59.5%

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

      7. neg-mul-1 59.5%

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

      8. sub-neg 59.5%

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

   8. Simplified 59.5%

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

   if -1.14999999999999994e160 < lambda1 < 6.4e8

   1. Initial program 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-*l* 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in lambda1 around 0 87.4%

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

      1. mul-1-neg 87.4%

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

      2. cos-neg 87.4%

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

      3. distribute-rgt-neg-in 87.4%

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

      4. sin-neg 87.4%

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

      5. remove-double-neg 87.4%

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

   7. Simplified 87.4%

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

   8. Taylor expanded in lambda2 around 0 73.2%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.15 \cdot 10^{+160} \lor \neg \left(\lambda_1 \leq 640000000\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \sin \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 54.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -0.025:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_1 \cdot \sin \phi_2 - t_0}\\ \mathbf{elif}\;\phi_1 \leq 3.8 \cdot 10^{+111}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
   (if (<= phi1 -0.025)
     (atan2 (sin lambda1) (- (* (cos phi1) (sin phi2)) t_0))
     (if (<= phi1 3.8e+111)
       (atan2
        (* (cos phi2) (sin (- lambda1 lambda2)))
        (- (sin phi2) (* (cos lambda1) phi1)))
       (atan2 (* (sin lambda1) (cos phi2)) (- (sin phi2) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2)));
	double tmp;
	if (phi1 <= -0.025) {
		tmp = atan2(sin(lambda1), ((cos(phi1) * sin(phi2)) - t_0));
	} else if (phi1 <= 3.8e+111) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda1) * phi1)));
	} else {
		tmp = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2)))
    if (phi1 <= (-0.025d0)) then
        tmp = atan2(sin(lambda1), ((cos(phi1) * sin(phi2)) - t_0))
    else if (phi1 <= 3.8d+111) then
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda1) * phi1)))
    else
        tmp = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * (Math.sin(phi1) * Math.cos((lambda1 - lambda2)));
	double tmp;
	if (phi1 <= -0.025) {
		tmp = Math.atan2(Math.sin(lambda1), ((Math.cos(phi1) * Math.sin(phi2)) - t_0));
	} else if (phi1 <= 3.8e+111) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.sin(phi2) - (Math.cos(lambda1) * phi1)));
	} else {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (Math.sin(phi2) - t_0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * (math.sin(phi1) * math.cos((lambda1 - lambda2)))
	tmp = 0
	if phi1 <= -0.025:
		tmp = math.atan2(math.sin(lambda1), ((math.cos(phi1) * math.sin(phi2)) - t_0))
	elif phi1 <= 3.8e+111:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.sin(phi2) - (math.cos(lambda1) * phi1)))
	else:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (math.sin(phi2) - t_0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if (phi1 <= -0.025)
		tmp = atan(sin(lambda1), Float64(Float64(cos(phi1) * sin(phi2)) - t_0));
	elseif (phi1 <= 3.8e+111)
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(sin(phi2) - Float64(cos(lambda1) * phi1)));
	else
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(sin(phi2) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2)));
	tmp = 0.0;
	if (phi1 <= -0.025)
		tmp = atan2(sin(lambda1), ((cos(phi1) * sin(phi2)) - t_0));
	elseif (phi1 <= 3.8e+111)
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (cos(lambda1) * phi1)));
	else
		tmp = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.025], N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.8e+111], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.025000000000000001

   1. Initial program 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-*l* 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in lambda2 around 0 52.0%

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

   6. Taylor expanded in phi2 around 0 32.4%

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

   if -0.025000000000000001 < phi1 < 3.79999999999999976e111

   1. Initial program 88.1%

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

      1. *-commutative 88.1%

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

      2. associate-*l* 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in phi1 around 0 78.6%

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

      1. sub-neg 78.6%

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

      2. remove-double-neg 78.6%

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

      3. mul-1-neg 78.6%

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

      4. distribute-neg-in 78.6%

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

      5. +-commutative 78.6%

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

      6. cos-neg 78.6%

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

      7. mul-1-neg 78.6%

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

      8. unsub-neg 78.6%

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

   7. Simplified 78.6%

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

   8. Taylor expanded in phi1 around 0 78.6%

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

   9. Taylor expanded in phi2 around 0 78.6%

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

   10. Taylor expanded in lambda2 around 0 78.6%

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

       1. cos-neg 78.6%

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

   12. Simplified 78.6%

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

   if 3.79999999999999976e111 < phi1

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. associate-*l* 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in lambda2 around 0 42.6%

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

   6. Taylor expanded in phi1 around 0 33.9%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.025:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.8 \cdot 10^{+111}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \cos \lambda_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 53.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.032:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \left|\cos \left(\lambda_2 - \lambda_1\right)\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.032)
   (atan2
    (sin lambda1)
    (-
     (* (cos phi1) (sin phi2))
     (* (cos phi2) (* (sin phi1) (cos (- lambda1 lambda2))))))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (- (sin phi2) (* phi1 (fabs (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.032) {
		tmp = atan2(sin(lambda1), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (phi1 * fabs(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 <= (-0.032d0)) then
        tmp = atan2(sin(lambda1), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (phi1 * abs(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 <= -0.032) {
		tmp = Math.atan2(Math.sin(lambda1), ((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 - lambda2))), (Math.sin(phi2) - (phi1 * Math.abs(Math.cos((lambda2 - lambda1))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.032:
		tmp = math.atan2(math.sin(lambda1), ((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 - lambda2))), (math.sin(phi2) - (phi1 * math.fabs(math.cos((lambda2 - lambda1))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.032)
		tmp = atan(sin(lambda1), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(sin(phi1) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(sin(phi2) - Float64(phi1 * abs(cos(Float64(lambda2 - lambda1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.032)
		tmp = atan2(sin(lambda1), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (sin(phi1) * cos((lambda1 - lambda2))))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (phi1 * abs(cos((lambda2 - lambda1))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.032], N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(phi1 * N[Abs[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.032000000000000001

   1. Initial program 75.3%

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

      1. *-commutative 75.3%

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

      2. associate-*l* 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in lambda2 around 0 52.0%

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

   6. Taylor expanded in phi2 around 0 32.4%

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

   if -0.032000000000000001 < phi1

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. associate-*l* 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in phi1 around 0 63.8%

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

      1. sub-neg 63.8%

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

      2. remove-double-neg 63.8%

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

      3. mul-1-neg 63.8%

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

      4. distribute-neg-in 63.8%

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

      5. +-commutative 63.8%

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

      6. cos-neg 63.8%

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

      7. mul-1-neg 63.8%

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

      8. unsub-neg 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in phi1 around 0 63.8%

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

   9. Taylor expanded in phi2 around 0 64.0%

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

       1. cos-diff 64.1%

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

       2. *-commutative 64.1%

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

       3. *-commutative 64.1%

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

       4. cos-diff 64.0%

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

       5. add-sqr-sqrt 47.6%

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

       6. sqrt-prod 65.3%

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

       7. rem-sqrt-square 65.3%

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

       8. cos-diff 65.3%

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

       9. *-commutative 65.3%

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

       10. *-commutative 65.3%

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

       11. cos-diff 65.3%

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

   11. Applied egg-rr 65.3%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.032:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \left|\cos \left(\lambda_2 - \lambda_1\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 48.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{+63}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - \cos \phi_2 \cdot \left(\phi_1 \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \left|t_0\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -5e+63)
     (atan2
      (* (sin lambda1) (cos phi2))
      (- (sin phi2) (* (cos phi2) (* phi1 t_0))))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (- (sin phi2) (* phi1 (fabs t_0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -5e+63) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - (cos(phi2) * (phi1 * t_0))));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (phi1 * fabs(t_0))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi1 <= (-5d+63)) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - (cos(phi2) * (phi1 * t_0))))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (phi1 * abs(t_0))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -5e+63) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (Math.sin(phi2) - (Math.cos(phi2) * (phi1 * t_0))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.sin(phi2) - (phi1 * Math.abs(t_0))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -5e+63:
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (math.sin(phi2) - (math.cos(phi2) * (phi1 * t_0))))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.sin(phi2) - (phi1 * math.fabs(t_0))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -5e+63)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(sin(phi2) - Float64(cos(phi2) * Float64(phi1 * t_0))));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(sin(phi2) - Float64(phi1 * abs(t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -5e+63)
		tmp = atan2((sin(lambda1) * cos(phi2)), (sin(phi2) - (cos(phi2) * (phi1 * t_0))));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - (phi1 * abs(t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -5e+63], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(phi1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(phi1 * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.00000000000000011e63

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-*l* 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in phi1 around 0 15.9%

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

      1. sub-neg 15.9%

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

      2. remove-double-neg 15.9%

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

      3. mul-1-neg 15.9%

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

      4. distribute-neg-in 15.9%

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

      5. +-commutative 15.9%

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

      6. cos-neg 15.9%

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

      7. mul-1-neg 15.9%

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

      8. unsub-neg 15.9%

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

   7. Simplified 15.9%

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

   8. Taylor expanded in phi1 around 0 15.9%

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

   9. Taylor expanded in lambda2 around 0 12.9%

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

   if -5.00000000000000011e63 < phi1

   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. Step-by-step derivation

      1. *-commutative 83.9%

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

      2. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in phi1 around 0 60.5%

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

      1. sub-neg 60.5%

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

      2. remove-double-neg 60.5%

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

      3. mul-1-neg 60.5%

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

      4. distribute-neg-in 60.5%

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

      5. +-commutative 60.5%

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

      6. cos-neg 60.5%

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

      7. mul-1-neg 60.5%

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

      8. unsub-neg 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in phi1 around 0 60.5%

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

   9. Taylor expanded in phi2 around 0 60.7%

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

       1. cos-diff 60.7%

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

       2. *-commutative 60.7%

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

       3. *-commutative 60.7%

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

       4. cos-diff 60.7%

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

       5. add-sqr-sqrt 45.0%

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

       6. sqrt-prod 61.8%

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

       7. rem-sqrt-square 61.8%

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

       8. cos-diff 61.8%

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

       9. *-commutative 61.8%

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

       10. *-commutative 61.8%

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

       11. cos-diff 61.8%

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

   11. Applied egg-rr 61.8%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{+63}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\sin \phi_2 - \cos \phi_2 \cdot \left(\phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \phi_1 \cdot \left|\cos \left(\lambda_2 - \lambda_1\right)\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 47.9% accurate, 1.3× speedup

Math

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

FPCore

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

Fortran

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

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.sin(phi2) - (Math.cos(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) - (math.cos(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(cos(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) - (cos(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[(N[Cos[phi2], $MachinePrecision] * N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-*l* 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in phi1 around 0 50.0%

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

   1. sub-neg 50.0%

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

   2. remove-double-neg 50.0%

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

   3. mul-1-neg 50.0%

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

   4. distribute-neg-in 50.0%

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

   5. +-commutative 50.0%

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

   6. cos-neg 50.0%

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

   7. mul-1-neg 50.0%

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

   8. unsub-neg 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in phi1 around 0 50.0%

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

9. Final simplification 50.0%

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

Alternative 24: 47.8% accurate, 1.3× speedup

Math

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

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 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-*l* 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in phi1 around 0 50.0%

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

   1. sub-neg 50.0%

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

   2. remove-double-neg 50.0%

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

   3. mul-1-neg 50.0%

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

   4. distribute-neg-in 50.0%

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

   5. +-commutative 50.0%

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

   6. cos-neg 50.0%

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

   7. mul-1-neg 50.0%

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

   8. unsub-neg 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in phi1 around 0 50.0%

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

9. Taylor expanded in lambda1 around 0 49.6%

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

10. Final simplification 49.6%

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

Alternative 25: 47.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-*l* 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in phi1 around 0 50.0%

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

   1. sub-neg 50.0%

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

   2. remove-double-neg 50.0%

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

   3. mul-1-neg 50.0%

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

   4. distribute-neg-in 50.0%

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

   5. +-commutative 50.0%

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

   6. cos-neg 50.0%

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

   7. mul-1-neg 50.0%

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

   8. unsub-neg 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in phi1 around 0 50.0%

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

9. Taylor expanded in lambda2 around 0 49.9%

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

    1. cos-neg 49.9%

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

    2. associate-*r* 49.9%

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

    3. *-commutative 49.9%

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

11. Simplified 49.9%

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

12. Final simplification 49.9%

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

Alternative 26: 31.1% 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 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-*l* 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in phi1 around 0 50.0%

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

   1. sub-neg 50.0%

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

   2. remove-double-neg 50.0%

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

   3. mul-1-neg 50.0%

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

   4. distribute-neg-in 50.0%

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

   5. +-commutative 50.0%

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

   6. cos-neg 50.0%

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

   7. mul-1-neg 50.0%

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

   8. unsub-neg 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in phi1 around 0 50.0%

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

9. Taylor expanded in phi2 around 0 48.5%

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

10. Taylor expanded in lambda2 around 0 32.9%

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

11. Final simplification 32.9%

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

Alternative 27: 47.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

   1. *-commutative 82.7%

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

   2. associate-*l* 82.7%

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

3. Simplified 82.7%

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

5. Taylor expanded in phi1 around 0 50.0%

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

   1. sub-neg 50.0%

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

   2. remove-double-neg 50.0%

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

   3. mul-1-neg 50.0%

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

   4. distribute-neg-in 50.0%

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

   5. +-commutative 50.0%

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

   6. cos-neg 50.0%

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

   7. mul-1-neg 50.0%

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

   8. unsub-neg 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in phi1 around 0 50.0%

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

9. Taylor expanded in phi2 around 0 48.5%

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

10. Taylor expanded in lambda2 around 0 48.7%

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

    1. cos-neg 48.7%

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

12. Simplified 48.7%

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

13. Final simplification 48.7%

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

Reproduce

herbie shell --seed 2024026 
(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))))))