Bearing on a great circle

Percentage Accurate: 78.6% → 99.7%

Time: 30.5s

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: 78.6% 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

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

FPCore

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

MATLAB

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

Derivation

1. Initial program 77.7%

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

   1. sin-diff 89.6%

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

4. Applied egg-rr 89.6%

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

   1. cos-diff 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ t_3 := \cos \phi_1 \cdot \sin \phi_2 - t\_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.05 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t\_1 - \mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)}{t\_3}\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-84}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2 - t\_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t\_1 - \mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda1) (sin lambda2)))
        (t_1 (* (sin lambda1) (cos lambda2)))
        (t_2 (* (cos phi2) (sin phi1)))
        (t_3 (- (* (cos phi1) (sin phi2)) (* t_2 (cos (- lambda1 lambda2))))))
   (if (<= phi2 -1.05e-13)
     (atan2 (* (cos phi2) (- t_1 (expm1 (log1p t_0)))) t_3)
     (if (<= phi2 3e-84)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
        (-
         (sin phi2)
         (*
          t_2
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda2) (cos lambda1))))))
       (atan2 (* (cos phi2) (- t_1 (log1p (expm1 t_0)))) t_3)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda1) * sin(lambda2);
	double t_1 = sin(lambda1) * cos(lambda2);
	double t_2 = cos(phi2) * sin(phi1);
	double t_3 = (cos(phi1) * sin(phi2)) - (t_2 * cos((lambda1 - lambda2)));
	double tmp;
	if (phi2 <= -1.05e-13) {
		tmp = atan2((cos(phi2) * (t_1 - expm1(log1p(t_0)))), t_3);
	} else if (phi2 <= 3e-84) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), (sin(phi2) - (t_2 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = atan2((cos(phi2) * (t_1 - log1p(expm1(t_0)))), t_3);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda1) * sin(lambda2))
	t_1 = Float64(sin(lambda1) * cos(lambda2))
	t_2 = Float64(cos(phi2) * sin(phi1))
	t_3 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_2 * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if (phi2 <= -1.05e-13)
		tmp = atan(Float64(cos(phi2) * Float64(t_1 - expm1(log1p(t_0)))), t_3);
	elseif (phi2 <= 3e-84)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), Float64(sin(phi2) - Float64(t_2 * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = atan(Float64(cos(phi2) * Float64(t_1 - log1p(expm1(t_0)))), t_3);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.05e-13], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 - N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3], $MachinePrecision], If[LessEqual[phi2, 3e-84], 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[Sin[phi2], $MachinePrecision] - N[(t$95$2 * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 - N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.04999999999999994e-13

   1. Initial program 70.9%

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

      1. sin-diff 88.3%

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

   4. Applied egg-rr 88.3%

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

      1. expm1-log1p-u 88.4%

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

      2. expm1-undefine 84.8%

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

   6. Applied egg-rr 84.8%

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

      1. expm1-define 88.4%

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

   8. Simplified 88.4%

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

   if -1.04999999999999994e-13 < phi2 < 3.0000000000000001e-84

   1. Initial program 82.6%

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

      1. sin-diff 88.9%

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

   4. Applied egg-rr 88.9%

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

      1. cos-diff 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. fma-define 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in phi1 around 0 99.8%

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

   if 3.0000000000000001e-84 < phi2

   1. Initial program 78.0%

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

      1. sin-diff 92.3%

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

   4. Applied egg-rr 92.3%

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

      1. log1p-expm1-u 92.3%

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

   6. Applied egg-rr 92.3%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.05 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 3 \cdot 10^{-84}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_2 := \cos \phi_2 \cdot \sin \phi_1\\ t_3 := \cos \phi_1 \cdot \sin \phi_2 - t\_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.02 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t\_1 - \mathsf{expm1}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)}{t\_3}\\ \mathbf{elif}\;\phi_2 \leq 3.4 \cdot 10^{-84}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(t\_1 - t\_0\right) \cdot \cos \phi_2}{\sin \phi_2 - t\_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t\_1 - \mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)\right)}{t\_3}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda1) (sin lambda2)))
        (t_1 (* (sin lambda1) (cos lambda2)))
        (t_2 (* (cos phi2) (sin phi1)))
        (t_3 (- (* (cos phi1) (sin phi2)) (* t_2 (cos (- lambda1 lambda2))))))
   (if (<= phi2 -1.02e-13)
     (atan2 (* (cos phi2) (- t_1 (expm1 (log1p t_0)))) t_3)
     (if (<= phi2 3.4e-84)
       (atan2
        (* (- t_1 t_0) (cos phi2))
        (-
         (sin phi2)
         (*
          t_2
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda2) (cos lambda1))))))
       (atan2 (* (cos phi2) (- t_1 (log1p (expm1 t_0)))) t_3)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda1) * sin(lambda2);
	double t_1 = sin(lambda1) * cos(lambda2);
	double t_2 = cos(phi2) * sin(phi1);
	double t_3 = (cos(phi1) * sin(phi2)) - (t_2 * cos((lambda1 - lambda2)));
	double tmp;
	if (phi2 <= -1.02e-13) {
		tmp = atan2((cos(phi2) * (t_1 - expm1(log1p(t_0)))), t_3);
	} else if (phi2 <= 3.4e-84) {
		tmp = atan2(((t_1 - t_0) * cos(phi2)), (sin(phi2) - (t_2 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = atan2((cos(phi2) * (t_1 - log1p(expm1(t_0)))), t_3);
	}
	return tmp;
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(lambda1) * Math.sin(lambda2);
	double t_1 = Math.sin(lambda1) * Math.cos(lambda2);
	double t_2 = Math.cos(phi2) * Math.sin(phi1);
	double t_3 = (Math.cos(phi1) * Math.sin(phi2)) - (t_2 * Math.cos((lambda1 - lambda2)));
	double tmp;
	if (phi2 <= -1.02e-13) {
		tmp = Math.atan2((Math.cos(phi2) * (t_1 - Math.expm1(Math.log1p(t_0)))), t_3);
	} else if (phi2 <= 3.4e-84) {
		tmp = Math.atan2(((t_1 - t_0) * Math.cos(phi2)), (Math.sin(phi2) - (t_2 * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1))))));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * (t_1 - Math.log1p(Math.expm1(t_0)))), t_3);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(lambda1) * math.sin(lambda2)
	t_1 = math.sin(lambda1) * math.cos(lambda2)
	t_2 = math.cos(phi2) * math.sin(phi1)
	t_3 = (math.cos(phi1) * math.sin(phi2)) - (t_2 * math.cos((lambda1 - lambda2)))
	tmp = 0
	if phi2 <= -1.02e-13:
		tmp = math.atan2((math.cos(phi2) * (t_1 - math.expm1(math.log1p(t_0)))), t_3)
	elif phi2 <= 3.4e-84:
		tmp = math.atan2(((t_1 - t_0) * math.cos(phi2)), (math.sin(phi2) - (t_2 * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1))))))
	else:
		tmp = math.atan2((math.cos(phi2) * (t_1 - math.log1p(math.expm1(t_0)))), t_3)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda1) * sin(lambda2))
	t_1 = Float64(sin(lambda1) * cos(lambda2))
	t_2 = Float64(cos(phi2) * sin(phi1))
	t_3 = Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_2 * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if (phi2 <= -1.02e-13)
		tmp = atan(Float64(cos(phi2) * Float64(t_1 - expm1(log1p(t_0)))), t_3);
	elseif (phi2 <= 3.4e-84)
		tmp = atan(Float64(Float64(t_1 - t_0) * cos(phi2)), Float64(sin(phi2) - Float64(t_2 * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = atan(Float64(cos(phi2) * Float64(t_1 - log1p(expm1(t_0)))), t_3);
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.02e-13], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 - N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3], $MachinePrecision], If[LessEqual[phi2, 3.4e-84], N[ArcTan[N[(N[(t$95$1 - t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(t$95$2 * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 - N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.0199999999999999e-13

   1. Initial program 70.9%

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

      1. sin-diff 88.3%

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

   4. Applied egg-rr 88.3%

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

      1. expm1-log1p-u 88.4%

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

      2. expm1-undefine 84.8%

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

   6. Applied egg-rr 84.8%

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

      1. expm1-define 88.4%

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

   8. Simplified 88.4%

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

   if -1.0199999999999999e-13 < phi2 < 3.40000000000000021e-84

   1. Initial program 82.6%

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

      1. sin-diff 88.9%

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

   4. Applied egg-rr 88.9%

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

      1. cos-diff 99.8%

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

      2. +-commutative 99.8%

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

      3. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in phi1 around 0 99.8%

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

   if 3.40000000000000021e-84 < phi2

   1. Initial program 78.0%

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

      1. sin-diff 92.3%

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

   4. Applied egg-rr 92.3%

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

      1. log1p-expm1-u 92.3%

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

   6. Applied egg-rr 92.3%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.02 \cdot 10^{-13}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 3.4 \cdot 10^{-84}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. sin-diff 89.6%

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

4. Applied egg-rr 89.6%

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

   1. expm1-log1p-u 89.6%

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

   2. expm1-undefine 82.2%

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

6. Applied egg-rr 82.2%

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

   1. expm1-define 89.6%

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

8. Simplified 89.6%

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

9. Final simplification 89.6%

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

Alternative 5: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -3000000 \lor \neg \left(\lambda_1 \leq 2.3 \cdot 10^{-47}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, t\_0 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin phi1))))
   (if (or (<= lambda1 -3000000.0) (not (<= lambda1 2.3e-47)))
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
       (cos phi2))
      (- (* (cos phi1) (sin phi2)) (* (cos lambda1) t_0)))
     (atan2
      (* (cos phi2) (sin (- lambda1 lambda2)))
      (fma (cos phi1) (sin phi2) (* t_0 (- (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin(phi1);
	double tmp;
	if ((lambda1 <= -3000000.0) || !(lambda1 <= 2.3e-47)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * t_0)));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), fma(cos(phi1), sin(phi2), (t_0 * -cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(phi1))
	tmp = 0.0
	if ((lambda1 <= -3000000.0) || !(lambda1 <= 2.3e-47))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda1) * t_0)));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), fma(cos(phi1), sin(phi2), Float64(t_0 * Float64(-cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -3000000.0], N[Not[LessEqual[lambda1, 2.3e-47]], $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[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(t$95$0 * (-N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3e6 or 2.29999999999999982e-47 < lambda1

   1. Initial program 57.6%

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

      1. sin-diff 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in lambda1 around inf 80.5%

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

   if -3e6 < lambda1 < 2.29999999999999982e-47

   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. Add Preprocessing
   3. Step-by-step derivation

      1. log1p-expm1-u 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in lambda1 around 0 99.5%

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

      1. fma-neg 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

      4. distribute-rgt-neg-in 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3000000 \lor \neg \left(\lambda_1 \leq 2.3 \cdot 10^{-47}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (atan2
  (*
   (- (* (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(Float64(cos(phi2) * 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[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.7%

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

   1. sin-diff 89.6%

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

4. Applied egg-rr 89.6%

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

5. Final simplification 89.6%

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

Alternative 7: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 1.25 \cdot 10^{-18}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(t\_0 - \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(t\_0 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot t\_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin lambda1) (cos lambda2))) (t_1 (cos (- lambda1 lambda2))))
   (if (or (<= phi1 -1.85e-6) (not (<= phi1 1.25e-18)))
     (atan2
      (* (cos phi2) (- t_0 (sin lambda2)))
      (- (* (cos phi1) (sin phi2)) (* (* (cos phi2) (sin phi1)) t_1)))
     (atan2
      (* (- t_0 (* (cos lambda1) (sin lambda2))) (cos phi2))
      (- (sin phi2) (* phi1 (* (cos phi2) t_1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(lambda1) * cos(lambda2);
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -1.85e-6) || !(phi1 <= 1.25e-18)) {
		tmp = atan2((cos(phi2) * (t_0 - sin(lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * t_1)));
	} else {
		tmp = atan2(((t_0 - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (sin(phi2) - (phi1 * (cos(phi2) * t_1))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(lambda1) * cos(lambda2)
    t_1 = cos((lambda1 - lambda2))
    if ((phi1 <= (-1.85d-6)) .or. (.not. (phi1 <= 1.25d-18))) then
        tmp = atan2((cos(phi2) * (t_0 - sin(lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * t_1)))
    else
        tmp = atan2(((t_0 - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (sin(phi2) - (phi1 * (cos(phi2) * t_1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(lambda1) * Math.cos(lambda2);
	double t_1 = Math.cos((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -1.85e-6) || !(phi1 <= 1.25e-18)) {
		tmp = Math.atan2((Math.cos(phi2) * (t_0 - Math.sin(lambda2))), ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos(phi2) * Math.sin(phi1)) * t_1)));
	} else {
		tmp = Math.atan2(((t_0 - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (Math.sin(phi2) - (phi1 * (Math.cos(phi2) * t_1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(lambda1) * math.cos(lambda2)
	t_1 = math.cos((lambda1 - lambda2))
	tmp = 0
	if (phi1 <= -1.85e-6) or not (phi1 <= 1.25e-18):
		tmp = math.atan2((math.cos(phi2) * (t_0 - math.sin(lambda2))), ((math.cos(phi1) * math.sin(phi2)) - ((math.cos(phi2) * math.sin(phi1)) * t_1)))
	else:
		tmp = math.atan2(((t_0 - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (math.sin(phi2) - (phi1 * (math.cos(phi2) * t_1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(lambda1) * cos(lambda2))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi1 <= -1.85e-6) || !(phi1 <= 1.25e-18))
		tmp = atan(Float64(cos(phi2) * Float64(t_0 - sin(lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * t_1)));
	else
		tmp = atan(Float64(Float64(t_0 - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(sin(phi2) - Float64(phi1 * Float64(cos(phi2) * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(lambda1) * cos(lambda2);
	t_1 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi1 <= -1.85e-6) || ~((phi1 <= 1.25e-18)))
		tmp = atan2((cos(phi2) * (t_0 - sin(lambda2))), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * t_1)));
	else
		tmp = atan2(((t_0 - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (sin(phi2) - (phi1 * (cos(phi2) * t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi1, -1.85e-6], N[Not[LessEqual[phi1, 1.25e-18]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$0 - N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(phi1 * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.8500000000000001e-6 or 1.25000000000000009e-18 < phi1

   1. Initial program 75.9%

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

      1. sin-diff 79.3%

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

   4. Applied egg-rr 79.3%

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

   5. Taylor expanded in lambda1 around 0 77.3%

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

   if -1.8500000000000001e-6 < phi1 < 1.25000000000000009e-18

   1. Initial program 79.5%

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

      1. sin-diff 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.85 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 1.25 \cdot 10^{-18}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \cos \lambda_2 - \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, t\_0 \cdot \left(-t\_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.25 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\cos \phi_1 \cdot \sin \phi_2 - t\_0 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t\_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin phi1)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.8e-6)
     (atan2 t_2 (fma (cos phi1) (sin phi2) (* t_0 (- t_1))))
     (if (<= phi1 1.25e-18)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (- (sin phi2) (* phi1 (* (cos phi2) t_1))))
       (atan2
        t_2
        (- (* (cos phi1) (sin phi2)) (* t_0 (log1p (expm1 t_1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin(phi1);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.8e-6) {
		tmp = atan2(t_2, fma(cos(phi1), sin(phi2), (t_0 * -t_1)));
	} else if (phi1 <= 1.25e-18) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (sin(phi2) - (phi1 * (cos(phi2) * t_1))));
	} else {
		tmp = atan2(t_2, ((cos(phi1) * sin(phi2)) - (t_0 * log1p(expm1(t_1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(phi1))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.8e-6)
		tmp = atan(t_2, fma(cos(phi1), sin(phi2), Float64(t_0 * Float64(-t_1))));
	elseif (phi1 <= 1.25e-18)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(sin(phi2) - Float64(phi1 * Float64(cos(phi2) * t_1))));
	else
		tmp = atan(t_2, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_0 * log1p(expm1(t_1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.8e-6], N[ArcTan[t$95$2 / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(t$95$0 * (-t$95$1)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.25e-18], 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[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Log[1 + N[(Exp[t$95$1] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.79999999999999987e-6

   1. Initial program 77.5%

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

      1. log1p-expm1-u 77.5%

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

   4. Applied egg-rr 77.5%

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

   5. Taylor expanded in lambda1 around 0 77.5%

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

      1. fma-neg 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*l* 77.6%

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

      4. distribute-rgt-neg-in 77.6%

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

   7. Simplified 77.6%

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

   if -2.79999999999999987e-6 < phi1 < 1.25000000000000009e-18

   1. Initial program 79.5%

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

      1. sin-diff 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

   if 1.25000000000000009e-18 < phi1

   1. Initial program 74.4%

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

      1. log1p-expm1-u 74.4%

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

   4. Applied egg-rr 74.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.25 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 87.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, t\_0 \cdot \left(-t\_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.25 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\cos \phi_1 \cdot \sin \phi_2 - t\_0 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin phi1)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -9e-6)
     (atan2 t_2 (fma (cos phi1) (sin phi2) (* t_0 (- t_1))))
     (if (<= phi1 1.25e-18)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (- (sin phi2) (* phi1 (* (cos phi2) t_1))))
       (atan2 t_2 (- (* (cos phi1) (sin phi2)) (* t_0 t_1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin(phi1);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -9e-6) {
		tmp = atan2(t_2, fma(cos(phi1), sin(phi2), (t_0 * -t_1)));
	} else if (phi1 <= 1.25e-18) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (sin(phi2) - (phi1 * (cos(phi2) * t_1))));
	} else {
		tmp = atan2(t_2, ((cos(phi1) * sin(phi2)) - (t_0 * t_1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(phi1))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -9e-6)
		tmp = atan(t_2, fma(cos(phi1), sin(phi2), Float64(t_0 * Float64(-t_1))));
	elseif (phi1 <= 1.25e-18)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(sin(phi2) - Float64(phi1 * Float64(cos(phi2) * t_1))));
	else
		tmp = atan(t_2, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_0 * t_1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -9e-6], N[ArcTan[t$95$2 / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(t$95$0 * (-t$95$1)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.25e-18], 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[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -9.00000000000000023e-6

   1. Initial program 77.5%

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

      1. log1p-expm1-u 77.5%

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

   4. Applied egg-rr 77.5%

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

   5. Taylor expanded in lambda1 around 0 77.5%

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

      1. fma-neg 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*l* 77.6%

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

      4. distribute-rgt-neg-in 77.6%

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

   7. Simplified 77.6%

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

   if -9.00000000000000023e-6 < phi1 < 1.25000000000000009e-18

   1. Initial program 79.5%

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

      1. sin-diff 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0 99.7%

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

      1. mul-1-neg 99.7%

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

      2. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

   if 1.25000000000000009e-18 < phi1

   1. Initial program 74.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.25 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2 - \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, t\_0 \cdot \left(-t\_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.8 \cdot 10^{-60}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\cos \phi_1 \cdot \sin \phi_2 - t\_0 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin phi1)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.1e-17)
     (atan2 t_2 (fma (cos phi1) (sin phi2) (* t_0 (- t_1))))
     (if (<= phi1 3.8e-60)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (sin phi2))
       (atan2 t_2 (- (* (cos phi1) (sin phi2)) (* t_0 t_1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin(phi1);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.1e-17) {
		tmp = atan2(t_2, fma(cos(phi1), sin(phi2), (t_0 * -t_1)));
	} else if (phi1 <= 3.8e-60) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_2, ((cos(phi1) * sin(phi2)) - (t_0 * t_1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(phi1))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.1e-17)
		tmp = atan(t_2, fma(cos(phi1), sin(phi2), Float64(t_0 * Float64(-t_1))));
	elseif (phi1 <= 3.8e-60)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_2, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(t_0 * t_1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.1e-17], N[ArcTan[t$95$2 / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(t$95$0 * (-t$95$1)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.8e-60], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.09999999999999992e-17

   1. Initial program 78.4%

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

      1. log1p-expm1-u 78.4%

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

   4. Applied egg-rr 78.4%

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

   5. Taylor expanded in lambda1 around 0 78.4%

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

      1. fma-neg 78.5%

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

      2. *-commutative 78.5%

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

      3. associate-*l* 78.5%

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

      4. distribute-rgt-neg-in 78.5%

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

   7. Simplified 78.5%

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

   if -2.09999999999999992e-17 < phi1 < 3.79999999999999994e-60

   1. Initial program 78.9%

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

      1. sin-diff 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0 98.2%

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

   if 3.79999999999999994e-60 < phi1

   1. Initial program 75.1%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-17}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.8 \cdot 10^{-60}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-27} \lor \neg \left(\phi_1 \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -7.2e-27) (not (<= phi1 5e-60)))
   (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))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -7.2e-27) || !(phi1 <= 5e-60)) {
		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));
	}
	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 <= (-7.2d-27)) .or. (.not. (phi1 <= 5d-60))) 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))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -7.2e-27) || !(phi1 <= 5e-60)) {
		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));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -7.2e-27) or not (phi1 <= 5e-60):
		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))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -7.2e-27) || !(phi1 <= 5e-60))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -7.2e-27) || ~((phi1 <= 5e-60)))
		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));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -7.2e-27], N[Not[LessEqual[phi1, 5e-60]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(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[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.1999999999999997e-27 or 5.0000000000000001e-60 < phi1

   1. Initial program 76.7%

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

   if -7.1999999999999997e-27 < phi1 < 5.0000000000000001e-60

   1. Initial program 78.9%

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

      1. sin-diff 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0 98.2%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-27} \lor \neg \left(\phi_1 \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.0136 \lor \neg \left(\lambda_2 \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -0.0136) (not (<= lambda2 3.2e-7)))
   (atan2
    (*
     (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
     (cos phi2))
    (sin phi2))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (cos lambda1) (* (cos phi2) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -0.0136) || !(lambda2 <= 3.2e-7)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (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) :: tmp
    if ((lambda2 <= (-0.0136d0)) .or. (.not. (lambda2 <= 3.2d-7))) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -0.0136) || !(lambda2 <= 3.2e-7)) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), ((Math.cos(phi1) * Math.sin(phi2)) - (Math.cos(lambda1) * (Math.cos(phi2) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda2 <= -0.0136) or not (lambda2 <= 3.2e-7):
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), ((math.cos(phi1) * math.sin(phi2)) - (math.cos(lambda1) * (math.cos(phi2) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -0.0136) || !(lambda2 <= 3.2e-7))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(lambda1) * Float64(cos(phi2) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda2 <= -0.0136) || ~((lambda2 <= 3.2e-7)))
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -0.0136], N[Not[LessEqual[lambda2, 3.2e-7]], $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[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -0.0135999999999999992 or 3.2000000000000001e-7 < lambda2

   1. Initial program 57.8%

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

      1. sin-diff 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in phi1 around 0 63.5%

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

   if -0.0135999999999999992 < lambda2 < 3.2000000000000001e-7

   1. Initial program 99.3%

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

   3. Taylor expanded in lambda1 around inf 99.3%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.0136 \lor \neg \left(\lambda_2 \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -25500000 \lor \neg \left(\lambda_1 \leq 0.0024\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda1 -25500000.0) (not (<= lambda1 0.0024)))
   (atan2
    (*
     (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
     (cos phi2))
    (sin phi2))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (cos phi2) (* (cos lambda2) (sin phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 <= -25500000.0) || !(lambda1 <= 0.0024)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda1 <= (-25500000.0d0)) .or. (.not. (lambda1 <= 0.0024d0))) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2))
    else
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (cos(lambda2) * sin(phi1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 <= -25500000.0) || !(lambda1 <= 0.0024)) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), ((Math.cos(phi1) * Math.sin(phi2)) - (Math.cos(phi2) * (Math.cos(lambda2) * Math.sin(phi1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 <= -25500000.0) or not (lambda1 <= 0.0024):
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), ((math.cos(phi1) * math.sin(phi2)) - (math.cos(phi2) * (math.cos(lambda2) * math.sin(phi1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda1 <= -25500000.0) || !(lambda1 <= 0.0024))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(cos(phi2) * Float64(cos(lambda2) * sin(phi1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 <= -25500000.0) || ~((lambda1 <= 0.0024)))
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(phi2) * (cos(lambda2) * sin(phi1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda1, -25500000.0], N[Not[LessEqual[lambda1, 0.0024]], $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[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.55e7 or 0.00239999999999999979 < lambda1

   1. Initial program 55.9%

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

      1. sin-diff 79.8%

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

   4. Applied egg-rr 79.8%

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

   5. Taylor expanded in phi1 around 0 62.1%

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

   if -2.55e7 < lambda1 < 0.00239999999999999979

   1. Initial program 98.6%

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

   3. Taylor expanded in lambda1 around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. cos-neg 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -25500000 \lor \neg \left(\lambda_1 \leq 0.0024\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \sin \phi_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{-24} \lor \neg \left(\phi_1 \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -1e-24) (not (<= phi1 5e-60)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (- (sin phi2) (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
   (atan2
    (*
     (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
     (cos phi2))
    (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -1e-24) || !(phi1 <= 5e-60)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi1 <= (-1d-24)) .or. (.not. (phi1 <= 5d-60))) then
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))))
    else
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -1e-24) || !(phi1 <= 5e-60)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (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));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -1e-24) or not (phi1 <= 5e-60):
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (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))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -1e-24) || !(phi1 <= 5e-60))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(sin(phi2) - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -1e-24) || ~((phi1 <= 5e-60)))
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi2) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	else
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -1e-24], N[Not[LessEqual[phi1, 5e-60]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi2], $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(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[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.99999999999999924e-25 or 5.0000000000000001e-60 < phi1

   1. Initial program 76.7%

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

   3. Taylor expanded in phi1 around 0 56.6%

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

   if -9.99999999999999924e-25 < phi1 < 5.0000000000000001e-60

   1. Initial program 78.9%

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

      1. sin-diff 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0 98.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{-24} \lor \neg \left(\phi_1 \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{-20}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot t\_0}\\ \mathbf{elif}\;\phi_1 \leq 6 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -8.6e-20)
     (atan2 t_1 (- (* (cos phi1) (sin phi2)) (* (sin phi1) t_0)))
     (if (<= phi1 6e-61)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (sin phi2))
       (atan2 t_1 (- (sin phi2) (* (* (cos phi2) (sin phi1)) t_0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -8.6e-20) {
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (sin(phi1) * t_0)));
	} else if (phi1 <= 6e-61) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_1, (sin(phi2) - ((cos(phi2) * sin(phi1)) * t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= (-8.6d-20)) then
        tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (sin(phi1) * t_0)))
    else if (phi1 <= 6d-61) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2))
    else
        tmp = atan2(t_1, (sin(phi2) - ((cos(phi2) * sin(phi1)) * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -8.6e-20) {
		tmp = Math.atan2(t_1, ((Math.cos(phi1) * Math.sin(phi2)) - (Math.sin(phi1) * t_0)));
	} else if (phi1 <= 6e-61) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2(t_1, (Math.sin(phi2) - ((Math.cos(phi2) * Math.sin(phi1)) * t_0)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -8.6e-20:
		tmp = math.atan2(t_1, ((math.cos(phi1) * math.sin(phi2)) - (math.sin(phi1) * t_0)))
	elif phi1 <= 6e-61:
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = math.atan2(t_1, (math.sin(phi2) - ((math.cos(phi2) * math.sin(phi1)) * t_0)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -8.6e-20)
		tmp = atan(t_1, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * t_0)));
	elseif (phi1 <= 6e-61)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_1, Float64(sin(phi2) - Float64(Float64(cos(phi2) * sin(phi1)) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -8.6e-20)
		tmp = atan2(t_1, ((cos(phi1) * sin(phi2)) - (sin(phi1) * t_0)));
	elseif (phi1 <= 6e-61)
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan2(t_1, (sin(phi2) - ((cos(phi2) * sin(phi1)) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8.6e-20], N[ArcTan[t$95$1 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 6e-61], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(N[Sin[phi2], $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -8.60000000000000022e-20

   1. Initial program 78.4%

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

   3. Taylor expanded in phi2 around 0 56.4%

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

   if -8.60000000000000022e-20 < phi1 < 6.00000000000000024e-61

   1. Initial program 78.9%

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

      1. sin-diff 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0 98.2%

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

   if 6.00000000000000024e-61 < phi1

   1. Initial program 75.1%

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

   3. Taylor expanded in phi1 around 0 57.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.6 \cdot 10^{-20}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_1 \leq 6 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.059:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.75 \cdot 10^{-59}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi1 -0.059)
     (atan2 (* (cos phi2) t_0) (* (sin phi1) (- (cos (- lambda2 lambda1)))))
     (if (<= phi1 1.75e-59)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (sin phi2))
       (atan2
        t_0
        (-
         (* (cos phi1) (sin phi2))
         (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.059) {
		tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))));
	} else if (phi1 <= 1.75e-59) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if (phi1 <= (-0.059d0)) then
        tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))))
    else if (phi1 <= 1.75d-59) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2))
    else
        tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.059) {
		tmp = Math.atan2((Math.cos(phi2) * t_0), (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else if (phi1 <= 1.75e-59) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2(t_0, ((Math.cos(phi1) * Math.sin(phi2)) - ((Math.cos(phi2) * Math.sin(phi1)) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -0.059:
		tmp = math.atan2((math.cos(phi2) * t_0), (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	elif phi1 <= 1.75e-59:
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = math.atan2(t_0, ((math.cos(phi1) * math.sin(phi2)) - ((math.cos(phi2) * math.sin(phi1)) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -0.059)
		tmp = atan(Float64(cos(phi2) * t_0), Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	elseif (phi1 <= 1.75e-59)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -0.059)
		tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))));
	elseif (phi1 <= 1.75e-59)
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.059], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.75e-59], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.058999999999999997

   1. Initial program 76.8%

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

      1. sin-cos-mult 44.2%

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

      2. clear-num 44.2%

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

      3. associate--l- 44.2%

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

      4. associate-+l- 44.2%

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

   4. Applied egg-rr 44.2%

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

   5. Taylor expanded in phi2 around 0 42.7%

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

      1. associate-*r* 42.7%

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

      2. neg-mul-1 42.7%

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

   7. Simplified 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. clear-num 42.7%

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

      3. associate--r+ 42.7%

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

      4. associate--r- 42.7%

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

      5. sin-cos-mult 51.6%

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

      6. *-commutative 51.6%

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

   9. Applied egg-rr 51.6%

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

       1. *-lft-identity 51.6%

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

       2. remove-double-div 51.6%

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

       3. *-commutative 51.6%

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

       4. sub-neg 51.6%

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

       5. remove-double-neg 51.6%

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

       6. mul-1-neg 51.6%

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

       7. distribute-neg-in 51.6%

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

       8. +-commutative 51.6%

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

       9. cos-neg 51.6%

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

       10. mul-1-neg 51.6%

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

       11. unsub-neg 51.6%

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

   11. Simplified 51.6%

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

   if -0.058999999999999997 < phi1 < 1.75e-59

   1. Initial program 79.8%

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

      1. sin-diff 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0 97.2%

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

   if 1.75e-59 < phi1

   1. Initial program 74.7%

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

   3. Taylor expanded in phi2 around 0 52.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.059:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.75 \cdot 10^{-59}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 69.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.062:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.75 \cdot 10^{-59}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi1 -0.062)
     (atan2 (* (cos phi2) t_0) (* (sin phi1) (- (cos (- lambda2 lambda1)))))
     (if (<= phi1 1.75e-59)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (sin phi2))
       (atan2
        t_0
        (-
         (* (cos phi1) (sin phi2))
         (* (sin phi1) (cos (- lambda1 lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.062) {
		tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))));
	} else if (phi1 <= 1.75e-59) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	} else {
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if (phi1 <= (-0.062d0)) then
        tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))))
    else if (phi1 <= 1.75d-59) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2))
    else
        tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.062) {
		tmp = Math.atan2((Math.cos(phi2) * t_0), (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else if (phi1 <= 1.75e-59) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), Math.sin(phi2));
	} else {
		tmp = Math.atan2(t_0, ((Math.cos(phi1) * Math.sin(phi2)) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -0.062:
		tmp = math.atan2((math.cos(phi2) * t_0), (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	elif phi1 <= 1.75e-59:
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), math.sin(phi2))
	else:
		tmp = math.atan2(t_0, ((math.cos(phi1) * math.sin(phi2)) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -0.062)
		tmp = atan(Float64(cos(phi2) * t_0), Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	elseif (phi1 <= 1.75e-59)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan(t_0, Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -0.062)
		tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))));
	elseif (phi1 <= 1.75e-59)
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), sin(phi2));
	else
		tmp = atan2(t_0, ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.062], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 1.75e-59], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.062

   1. Initial program 76.8%

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

      1. sin-cos-mult 44.2%

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

      2. clear-num 44.2%

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

      3. associate--l- 44.2%

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

      4. associate-+l- 44.2%

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

   4. Applied egg-rr 44.2%

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

   5. Taylor expanded in phi2 around 0 42.7%

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

      1. associate-*r* 42.7%

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

      2. neg-mul-1 42.7%

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

   7. Simplified 42.7%

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

      1. *-un-lft-identity 42.7%

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

      2. clear-num 42.7%

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

      3. associate--r+ 42.7%

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

      4. associate--r- 42.7%

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

      5. sin-cos-mult 51.6%

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

      6. *-commutative 51.6%

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

   9. Applied egg-rr 51.6%

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

       1. *-lft-identity 51.6%

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

       2. remove-double-div 51.6%

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

       3. *-commutative 51.6%

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

       4. sub-neg 51.6%

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

       5. remove-double-neg 51.6%

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

       6. mul-1-neg 51.6%

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

       7. distribute-neg-in 51.6%

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

       8. +-commutative 51.6%

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

       9. cos-neg 51.6%

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

       10. mul-1-neg 51.6%

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

       11. unsub-neg 51.6%

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

   11. Simplified 51.6%

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

   if -0.062 < phi1 < 1.75e-59

   1. Initial program 79.8%

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

      1. sin-diff 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in phi1 around 0 97.2%

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

   if 1.75e-59 < phi1

   1. Initial program 74.7%

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

   3. Taylor expanded in phi2 around 0 52.6%

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

   4. Taylor expanded in phi2 around 0 52.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.062:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 1.75 \cdot 10^{-59}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 52.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-48} \lor \neg \left(\phi_1 \leq 1.9 \cdot 10^{-61}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -2.9e-48) (not (<= phi1 1.9e-61)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (* (sin phi1) (- (cos (- lambda2 lambda1)))))
   (atan2
    (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
    (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -2.9e-48) || !(phi1 <= 1.9e-61)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi1) * -cos((lambda2 - lambda1))));
	} else {
		tmp = atan2(((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))), sin(phi2));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi1 <= (-2.9d-48)) .or. (.not. (phi1 <= 1.9d-61))) then
        tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi1) * -cos((lambda2 - lambda1))))
    else
        tmp = atan2(((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -2.9e-48) || !(phi1 <= 1.9e-61)) {
		tmp = Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else {
		tmp = Math.atan2(((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -2.9e-48) or not (phi1 <= 1.9e-61):
		tmp = math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	else:
		tmp = math.atan2(((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -2.9e-48) || !(phi1 <= 1.9e-61))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -2.9e-48) || ~((phi1 <= 1.9e-61)))
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), (sin(phi1) * -cos((lambda2 - lambda1))));
	else
		tmp = atan2(((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -2.9e-48], N[Not[LessEqual[phi1, 1.9e-61]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.9000000000000003e-48 or 1.8999999999999999e-61 < phi1

   1. Initial program 76.0%

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

      1. sin-cos-mult 47.0%

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

      2. clear-num 47.0%

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

      3. associate--l- 47.0%

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

      4. associate-+l- 47.0%

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

   4. Applied egg-rr 47.0%

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

   5. Taylor expanded in phi2 around 0 43.7%

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

      1. associate-*r* 43.7%

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

      2. neg-mul-1 43.7%

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

   7. Simplified 43.7%

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

      1. *-un-lft-identity 43.7%

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

      2. clear-num 43.7%

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

      3. associate--r+ 43.7%

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

      4. associate--r- 43.7%

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

      5. sin-cos-mult 49.6%

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

      6. *-commutative 49.6%

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

   9. Applied egg-rr 49.6%

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

       1. *-lft-identity 49.6%

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

       2. remove-double-div 49.6%

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

       3. *-commutative 49.6%

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

       4. sub-neg 49.6%

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

       5. remove-double-neg 49.6%

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

       6. mul-1-neg 49.6%

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

       7. distribute-neg-in 49.6%

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

       8. +-commutative 49.6%

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

       9. cos-neg 49.6%

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

       10. mul-1-neg 49.6%

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

       11. unsub-neg 49.6%

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

   11. Simplified 49.6%

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

   if -2.9000000000000003e-48 < phi1 < 1.8999999999999999e-61

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0 51.8%

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

   4. Taylor expanded in phi1 around 0 51.0%

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

      1. sin-diff 99.7%

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

   6. Applied egg-rr 57.0%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{-48} \lor \neg \left(\phi_1 \leq 1.9 \cdot 10^{-61}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 48.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in phi2 around 0 50.4%

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

4. Taylor expanded in phi2 around 0 50.5%

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

5. Final simplification 50.5%

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

Alternative 20: 48.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.75:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi2 -1.35e+16)
     (atan2 (* (cos phi2) t_0) (* (sin phi1) (- (cos (- lambda2 lambda1)))))
     (if (<= phi2 1.75)
       (atan2
        t_0
        (- (* phi2 (cos phi1)) (* (sin phi1) (cos (- lambda1 lambda2)))))
       (atan2 (sin lambda1) (sin phi2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -1.35e+16) {
		tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))));
	} else if (phi2 <= 1.75) {
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2(sin(lambda1), sin(phi2));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if (phi2 <= (-1.35d+16)) then
        tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))))
    else if (phi2 <= 1.75d0) then
        tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda1 - lambda2)))))
    else
        tmp = atan2(sin(lambda1), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -1.35e+16) {
		tmp = Math.atan2((Math.cos(phi2) * t_0), (Math.sin(phi1) * -Math.cos((lambda2 - lambda1))));
	} else if (phi2 <= 1.75) {
		tmp = Math.atan2(t_0, ((phi2 * Math.cos(phi1)) - (Math.sin(phi1) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2(Math.sin(lambda1), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= -1.35e+16:
		tmp = math.atan2((math.cos(phi2) * t_0), (math.sin(phi1) * -math.cos((lambda2 - lambda1))))
	elif phi2 <= 1.75:
		tmp = math.atan2(t_0, ((phi2 * math.cos(phi1)) - (math.sin(phi1) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2(math.sin(lambda1), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= -1.35e+16)
		tmp = atan(Float64(cos(phi2) * t_0), Float64(sin(phi1) * Float64(-cos(Float64(lambda2 - lambda1)))));
	elseif (phi2 <= 1.75)
		tmp = atan(t_0, Float64(Float64(phi2 * cos(phi1)) - Float64(sin(phi1) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(sin(lambda1), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= -1.35e+16)
		tmp = atan2((cos(phi2) * t_0), (sin(phi1) * -cos((lambda2 - lambda1))));
	elseif (phi2 <= 1.75)
		tmp = atan2(t_0, ((phi2 * cos(phi1)) - (sin(phi1) * cos((lambda1 - lambda2)))));
	else
		tmp = atan2(sin(lambda1), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -1.35e+16], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi2, 1.75], N[ArcTan[t$95$0 / N[(N[(phi2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.35e16

   1. Initial program 69.5%

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

      1. sin-cos-mult 15.5%

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

      2. clear-num 15.5%

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

      3. associate--l- 15.5%

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

      4. associate-+l- 15.5%

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

   4. Applied egg-rr 15.5%

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

   5. Taylor expanded in phi2 around 0 10.7%

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

      1. associate-*r* 10.7%

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

      2. neg-mul-1 10.7%

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

   7. Simplified 10.7%

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

      1. *-un-lft-identity 10.7%

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

      2. clear-num 10.7%

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

      3. associate--r+ 10.7%

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

      4. associate--r- 10.7%

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

      5. sin-cos-mult 20.4%

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

      6. *-commutative 20.4%

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

   9. Applied egg-rr 20.4%

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

       1. *-lft-identity 20.4%

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

       2. remove-double-div 20.4%

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

       3. *-commutative 20.4%

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

       4. sub-neg 20.4%

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

       5. remove-double-neg 20.4%

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

       6. mul-1-neg 20.4%

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

       7. distribute-neg-in 20.4%

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

       8. +-commutative 20.4%

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

       9. cos-neg 20.4%

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

       10. mul-1-neg 20.4%

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

       11. unsub-neg 20.4%

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

   11. Simplified 20.4%

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

   if -1.35e16 < phi2 < 1.75

   1. Initial program 83.9%

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

   3. Taylor expanded in phi2 around 0 82.4%

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

   4. Taylor expanded in phi2 around 0 82.7%

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

   if 1.75 < 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. Add Preprocessing

   3. Taylor expanded in phi2 around 0 18.1%

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

   4. Taylor expanded in phi1 around 0 15.7%

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

   5. Taylor expanded in lambda2 around 0 19.2%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\sin \phi_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.35 \cdot 10^{+16}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_2 - \lambda_1\right)\right)}\\ \mathbf{elif}\;\phi_2 \leq 1.75:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2 \cdot \cos \phi_1 - \sin \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 48.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. sin-cos-mult 47.6%

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

   2. clear-num 47.6%

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

   3. associate--l- 47.6%

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

   4. associate-+l- 47.6%

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

4. Applied egg-rr 47.6%

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

5. Taylor expanded in phi2 around 0 43.5%

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

   1. associate-*r* 43.5%

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

   2. neg-mul-1 43.5%

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

7. Simplified 43.5%

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

   1. *-un-lft-identity 43.5%

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

   2. clear-num 43.5%

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

   3. associate--r+ 43.5%

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

   4. associate--r- 43.5%

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

   5. sin-cos-mult 48.1%

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

   6. *-commutative 48.1%

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

9. Applied egg-rr 48.1%

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

    1. *-lft-identity 48.1%

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

    2. remove-double-div 48.1%

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

    3. *-commutative 48.1%

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

    4. sub-neg 48.1%

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

    5. remove-double-neg 48.1%

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

    6. mul-1-neg 48.1%

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

    7. distribute-neg-in 48.1%

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

    8. +-commutative 48.1%

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

    9. cos-neg 48.1%

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

    10. mul-1-neg 48.1%

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

    11. unsub-neg 48.1%

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

11. Simplified 48.1%

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

Alternative 22: 47.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-52} \lor \neg \left(\phi_1 \leq 7.8 \cdot 10^{-60}\right):\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_0}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (or (<= phi1 -2.1e-52) (not (<= phi1 7.8e-60)))
     (atan2 t_0 (* (sin phi1) (- (cos (- lambda1 lambda2)))))
     (atan2 t_0 (sin phi2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -2.1e-52) || !(phi1 <= 7.8e-60)) {
		tmp = atan2(t_0, (sin(phi1) * -cos((lambda1 - lambda2))));
	} else {
		tmp = atan2(t_0, sin(phi2));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if ((phi1 <= (-2.1d-52)) .or. (.not. (phi1 <= 7.8d-60))) then
        tmp = atan2(t_0, (sin(phi1) * -cos((lambda1 - lambda2))))
    else
        tmp = atan2(t_0, sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -2.1e-52) || !(phi1 <= 7.8e-60)) {
		tmp = Math.atan2(t_0, (Math.sin(phi1) * -Math.cos((lambda1 - lambda2))));
	} else {
		tmp = Math.atan2(t_0, Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi1 <= -2.1e-52) or not (phi1 <= 7.8e-60):
		tmp = math.atan2(t_0, (math.sin(phi1) * -math.cos((lambda1 - lambda2))))
	else:
		tmp = math.atan2(t_0, math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi1 <= -2.1e-52) || !(phi1 <= 7.8e-60))
		tmp = atan(t_0, Float64(sin(phi1) * Float64(-cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(t_0, sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi1 <= -2.1e-52) || ~((phi1 <= 7.8e-60)))
		tmp = atan2(t_0, (sin(phi1) * -cos((lambda1 - lambda2))));
	else
		tmp = atan2(t_0, sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi1, -2.1e-52], N[Not[LessEqual[phi1, 7.8e-60]], $MachinePrecision]], N[ArcTan[t$95$0 / N[(N[Sin[phi1], $MachinePrecision] * (-N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$0 / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.0999999999999999e-52 or 7.8000000000000004e-60 < phi1

   1. Initial program 76.4%

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

   3. Taylor expanded in phi2 around 0 49.7%

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

   4. Taylor expanded in phi2 around 0 47.4%

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

      1. associate-*r* 44.1%

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

      2. neg-mul-1 44.1%

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

   6. Simplified 47.4%

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

   if -2.0999999999999999e-52 < phi1 < 7.8000000000000004e-60

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0 51.2%

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

   4. Taylor expanded in phi1 around 0 50.4%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-52} \lor \neg \left(\phi_1 \leq 7.8 \cdot 10^{-60}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_1 \cdot \left(-\cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\sin \phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 30.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.9)
   (atan2 (sin (- lambda1 lambda2)) phi2)
   (atan2 (sin lambda1) (sin phi2))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.9) {
		tmp = atan2(sin((lambda1 - lambda2)), phi2);
	} else {
		tmp = atan2(sin(lambda1), sin(phi2));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.9d0) then
        tmp = atan2(sin((lambda1 - lambda2)), phi2)
    else
        tmp = atan2(sin(lambda1), sin(phi2))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.9) {
		tmp = Math.atan2(Math.sin((lambda1 - lambda2)), phi2);
	} else {
		tmp = Math.atan2(Math.sin(lambda1), Math.sin(phi2));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.9:
		tmp = math.atan2(math.sin((lambda1 - lambda2)), phi2)
	else:
		tmp = math.atan2(math.sin(lambda1), math.sin(phi2))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.9)
		tmp = atan(sin(Float64(lambda1 - lambda2)), phi2);
	else
		tmp = atan(sin(lambda1), sin(phi2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.9)
		tmp = atan2(sin((lambda1 - lambda2)), phi2);
	else
		tmp = atan2(sin(lambda1), sin(phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.9], N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.8999999999999999

   1. Initial program 78.7%

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

   3. Taylor expanded in phi2 around 0 58.8%

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

   4. Taylor expanded in phi1 around 0 36.3%

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

   5. Taylor expanded in phi2 around 0 36.7%

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

   if 1.8999999999999999 < 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. Add Preprocessing

   3. Taylor expanded in phi2 around 0 18.1%

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

   4. Taylor expanded in phi1 around 0 15.7%

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

   5. Taylor expanded in lambda2 around 0 19.2%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\sin \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 31.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in phi2 around 0 50.4%

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

4. Taylor expanded in phi1 around 0 32.0%

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

Alternative 25: 26.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.00105 \lor \neg \left(\lambda_2 \leq 3.8 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -0.00105) (not (<= lambda2 3.8e-9)))
   (atan2 (sin (- lambda2)) phi2)
   (atan2 (sin lambda1) phi2)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -0.00105) || !(lambda2 <= 3.8e-9)) {
		tmp = atan2(sin(-lambda2), phi2);
	} else {
		tmp = atan2(sin(lambda1), phi2);
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda2 <= (-0.00105d0)) .or. (.not. (lambda2 <= 3.8d-9))) then
        tmp = atan2(sin(-lambda2), phi2)
    else
        tmp = atan2(sin(lambda1), phi2)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -0.00105) || !(lambda2 <= 3.8e-9)) {
		tmp = Math.atan2(Math.sin(-lambda2), phi2);
	} else {
		tmp = Math.atan2(Math.sin(lambda1), phi2);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda2 <= -0.00105) or not (lambda2 <= 3.8e-9):
		tmp = math.atan2(math.sin(-lambda2), phi2)
	else:
		tmp = math.atan2(math.sin(lambda1), phi2)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -0.00105) || !(lambda2 <= 3.8e-9))
		tmp = atan(sin(Float64(-lambda2)), phi2);
	else
		tmp = atan(sin(lambda1), phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda2 <= -0.00105) || ~((lambda2 <= 3.8e-9)))
		tmp = atan2(sin(-lambda2), phi2);
	else
		tmp = atan2(sin(lambda1), phi2);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -0.00105], N[Not[LessEqual[lambda2, 3.8e-9]], $MachinePrecision]], N[ArcTan[N[Sin[(-lambda2)], $MachinePrecision] / phi2], $MachinePrecision], N[ArcTan[N[Sin[lambda1], $MachinePrecision] / phi2], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -0.00104999999999999994 or 3.80000000000000011e-9 < lambda2

   1. Initial program 57.9%

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

   3. Taylor expanded in phi2 around 0 37.2%

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

   4. Taylor expanded in phi1 around 0 29.0%

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

   5. Taylor expanded in phi2 around 0 26.7%

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

   6. Taylor expanded in lambda1 around 0 27.7%

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

   if -0.00104999999999999994 < lambda2 < 3.80000000000000011e-9

   1. Initial program 99.6%

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

   3. Taylor expanded in phi2 around 0 64.9%

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

   4. Taylor expanded in phi1 around 0 35.4%

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

   5. Taylor expanded in phi2 around 0 33.5%

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

   6. Taylor expanded in lambda2 around 0 28.3%

      \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.00105 \lor \neg \left(\lambda_2 \leq 3.8 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1}{\phi_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 28.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in phi2 around 0 50.4%

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

4. Taylor expanded in phi1 around 0 32.0%

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

5. Taylor expanded in phi2 around 0 29.9%

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

Alternative 27: 21.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sin \lambda_1}{\phi_2} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(lambda1, lambda2, phi1, phi2)
	return atan(sin(lambda1), phi2)
end

MATLAB

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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[ArcTan[N[Sin[lambda1], $MachinePrecision] / phi2], $MachinePrecision]

Derivation

1. Initial program 77.7%

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

3. Taylor expanded in phi2 around 0 50.4%

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

4. Taylor expanded in phi1 around 0 32.0%

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

5. Taylor expanded in phi2 around 0 29.9%

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

6. Taylor expanded in lambda2 around 0 21.4%

   \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \lambda_1}}{\phi_2} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(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))))))