Bearing on a great circle

Percentage Accurate: 79.4% → 99.7%

Time: 43.3s

Alternatives: 28

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sin \lambda_1 \cdot \sin \lambda_2}\\ \tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left({t_0}^{2}, t_0, \cos \lambda_2 \cdot \cos \lambda_1\right)} \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cbrt (* (sin lambda1) (sin lambda2)))))
   (atan2
    (*
     (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
     (cos phi2))
    (-
     (* (cos phi1) (sin phi2))
     (*
      (* (cos phi2) (sin phi1))
      (fma (pow t_0 2.0) t_0 (* (cos lambda2) (cos lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cbrt((sin(lambda1) * sin(lambda2)));
	return atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), ((cos(phi1) * sin(phi2)) - ((cos(phi2) * sin(phi1)) * fma(pow(t_0, 2.0), t_0, (cos(lambda2) * cos(lambda1))))));
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cbrt(Float64(sin(lambda1) * sin(lambda2)))
	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)) * fma((t_0 ^ 2.0), t_0, Float64(cos(lambda2) * cos(lambda1))))))
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision], 1/3], $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[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$0 + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.7%

   \[\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\sin \lambda_1 \cdot \sin \lambda_2}\right)}^{2}, \sqrt[3]{\sin \lambda_1 \cdot \sin \lambda_2}, \cos \lambda_2 \cdot \cos \lambda_1\right)} \]

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

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))
    (fma (sin lambda2) (sin lambda1) (* (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)) * fma(sin(lambda2), sin(lambda1), (cos(lambda2) * 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)) * fma(sin(lambda2), sin(lambda1), Float64(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[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.6%

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

Alternative 3: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \sin \lambda_1 \cdot \cos \lambda_2\\ t_4 := \left(t_3 - t_0\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{t_2}\\ \mathbf{elif}\;\phi_2 \leq 5.2 \cdot 10^{-37}:\\ \;\;\;\;\tan^{-1}_* \frac{t_4}{t_1 - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(t_3\right)\right) - t_0\right)}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda1) (sin lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (- t_1 (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
        (t_3 (* (sin lambda1) (cos lambda2)))
        (t_4 (* (- t_3 t_0) (cos phi2))))
   (if (<= phi2 -1.6e+18)
     (atan2 t_4 t_2)
     (if (<= phi2 5.2e-37)
       (atan2
        t_4
        (-
         t_1
         (*
          (sin phi1)
          (fma (sin lambda1) (sin lambda2) (* (cos lambda2) (cos lambda1))))))
       (atan2 (* (cos phi2) (- (log1p (expm1 t_3)) t_0)) t_2)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda1) * sin(lambda2);
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = t_1 - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)));
	double t_3 = sin(lambda1) * cos(lambda2);
	double t_4 = (t_3 - t_0) * cos(phi2);
	double tmp;
	if (phi2 <= -1.6e+18) {
		tmp = atan2(t_4, t_2);
	} else if (phi2 <= 5.2e-37) {
		tmp = atan2(t_4, (t_1 - (sin(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = atan2((cos(phi2) * (log1p(expm1(t_3)) - t_0)), t_2);
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda1) * sin(lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = Float64(t_1 - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2))))
	t_3 = Float64(sin(lambda1) * cos(lambda2))
	t_4 = Float64(Float64(t_3 - t_0) * cos(phi2))
	tmp = 0.0
	if (phi2 <= -1.6e+18)
		tmp = atan(t_4, t_2);
	elseif (phi2 <= 5.2e-37)
		tmp = atan(t_4, Float64(t_1 - Float64(sin(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = atan(Float64(cos(phi2) * Float64(log1p(expm1(t_3)) - t_0)), t_2);
	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[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 - t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e+18], N[ArcTan[t$95$4 / t$95$2], $MachinePrecision], If[LessEqual[phi2, 5.2e-37], N[ArcTan[t$95$4 / N[(t$95$1 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Log[1 + N[(Exp[t$95$3] - 1), $MachinePrecision]], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2], $MachinePrecision]]]]]]]]

Derivation

1. Initial program 94.8%

   \[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{+18}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 5.2 \cdot 10^{-37}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_1 \cdot \cos \lambda_2\right)\right) - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 4: 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 99.6%

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

Alternative 5: 89.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 90.9%

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

Alternative 6: 89.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ \mathbf{if}\;\lambda_1 \leq -0.00125 \lor \neg \left(\lambda_1 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \cos \lambda_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \cos \lambda_2\right)}{t_0 - t_1 \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (* (cos phi2) (sin phi1))))
   (if (or (<= lambda1 -0.00125) (not (<= lambda1 2e-6)))
     (atan2
      (*
       (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
       (cos phi2))
      (- t_0 (* (cos lambda1) t_1)))
     (atan2
      (* (cos phi2) (+ (sin (- lambda2)) (* lambda1 (cos lambda2))))
      (- t_0 (* t_1 (+ (cos lambda2) (* lambda1 (sin lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin(phi1);
	double tmp;
	if ((lambda1 <= -0.00125) || !(lambda1 <= 2e-6)) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos(lambda1) * t_1)));
	} else {
		tmp = atan2((cos(phi2) * (sin(-lambda2) + (lambda1 * cos(lambda2)))), (t_0 - (t_1 * (cos(lambda2) + (lambda1 * sin(lambda2))))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin(phi1)
    if ((lambda1 <= (-0.00125d0)) .or. (.not. (lambda1 <= 2d-6))) then
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos(lambda1) * t_1)))
    else
        tmp = atan2((cos(phi2) * (sin(-lambda2) + (lambda1 * cos(lambda2)))), (t_0 - (t_1 * (cos(lambda2) + (lambda1 * sin(lambda2))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin(phi1);
	double tmp;
	if ((lambda1 <= -0.00125) || !(lambda1 <= 2e-6)) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (t_0 - (Math.cos(lambda1) * t_1)));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * (Math.sin(-lambda2) + (lambda1 * Math.cos(lambda2)))), (t_0 - (t_1 * (Math.cos(lambda2) + (lambda1 * Math.sin(lambda2))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos(phi2) * math.sin(phi1)
	tmp = 0
	if (lambda1 <= -0.00125) or not (lambda1 <= 2e-6):
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (t_0 - (math.cos(lambda1) * t_1)))
	else:
		tmp = math.atan2((math.cos(phi2) * (math.sin(-lambda2) + (lambda1 * math.cos(lambda2)))), (t_0 - (t_1 * (math.cos(lambda2) + (lambda1 * math.sin(lambda2))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(phi1))
	tmp = 0.0
	if ((lambda1 <= -0.00125) || !(lambda1 <= 2e-6))
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(cos(lambda1) * t_1)));
	else
		tmp = atan(Float64(cos(phi2) * Float64(sin(Float64(-lambda2)) + Float64(lambda1 * cos(lambda2)))), Float64(t_0 - Float64(t_1 * Float64(cos(lambda2) + Float64(lambda1 * sin(lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = cos(phi2) * sin(phi1);
	tmp = 0.0;
	if ((lambda1 <= -0.00125) || ~((lambda1 <= 2e-6)))
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (cos(lambda1) * t_1)));
	else
		tmp = atan2((cos(phi2) * (sin(-lambda2) + (lambda1 * cos(lambda2)))), (t_0 - (t_1 * (cos(lambda2) + (lambda1 * sin(lambda2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[lambda1, -0.00125], N[Not[LessEqual[lambda1, 2e-6]], $MachinePrecision]], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[(-lambda2)], $MachinePrecision] + N[(lambda1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[(N[Cos[lambda2], $MachinePrecision] + N[(lambda1 * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Initial program 90.9%

   \[\begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.00125 \lor \neg \left(\lambda_1 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \left(-\lambda_2\right) + \lambda_1 \cdot \cos \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)}\\ \end{array} \]

Alternative 7: 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 90.9%

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

Alternative 8: 88.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 0.000112\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_0\right)}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - t_0\right) \cdot \cos \phi_2}{t_1 - t_2 \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda1) (sin lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (cos (- lambda1 lambda2))))
   (if (or (<= phi1 -2.8e-5) (not (<= phi1 0.000112)))
     (atan2
      (* (cos phi2) (- (sin lambda1) t_0))
      (- t_1 (* (* (cos phi2) (sin phi1)) t_2)))
     (atan2
      (* (- (* (sin lambda1) (cos lambda2)) t_0) (cos phi2))
      (- t_1 (* t_2 (* (cos phi2) phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda1) * sin(lambda2);
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = cos((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -2.8e-5) || !(phi1 <= 0.000112)) {
		tmp = atan2((cos(phi2) * (sin(lambda1) - t_0)), (t_1 - ((cos(phi2) * sin(phi1)) * t_2)));
	} else {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - t_0) * cos(phi2)), (t_1 - (t_2 * (cos(phi2) * phi1))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(lambda1) * sin(lambda2)
    t_1 = cos(phi1) * sin(phi2)
    t_2 = cos((lambda1 - lambda2))
    if ((phi1 <= (-2.8d-5)) .or. (.not. (phi1 <= 0.000112d0))) then
        tmp = atan2((cos(phi2) * (sin(lambda1) - t_0)), (t_1 - ((cos(phi2) * sin(phi1)) * t_2)))
    else
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - t_0) * cos(phi2)), (t_1 - (t_2 * (cos(phi2) * phi1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(lambda1) * Math.sin(lambda2);
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double t_2 = Math.cos((lambda1 - lambda2));
	double tmp;
	if ((phi1 <= -2.8e-5) || !(phi1 <= 0.000112)) {
		tmp = Math.atan2((Math.cos(phi2) * (Math.sin(lambda1) - t_0)), (t_1 - ((Math.cos(phi2) * Math.sin(phi1)) * t_2)));
	} else {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - t_0) * Math.cos(phi2)), (t_1 - (t_2 * (Math.cos(phi2) * phi1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(lambda1) * math.sin(lambda2)
	t_1 = math.cos(phi1) * math.sin(phi2)
	t_2 = math.cos((lambda1 - lambda2))
	tmp = 0
	if (phi1 <= -2.8e-5) or not (phi1 <= 0.000112):
		tmp = math.atan2((math.cos(phi2) * (math.sin(lambda1) - t_0)), (t_1 - ((math.cos(phi2) * math.sin(phi1)) * t_2)))
	else:
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - t_0) * math.cos(phi2)), (t_1 - (t_2 * (math.cos(phi2) * phi1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda1) * sin(lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi1 <= -2.8e-5) || !(phi1 <= 0.000112))
		tmp = atan(Float64(cos(phi2) * Float64(sin(lambda1) - t_0)), Float64(t_1 - Float64(Float64(cos(phi2) * sin(phi1)) * t_2)));
	else
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - t_0) * cos(phi2)), Float64(t_1 - Float64(t_2 * Float64(cos(phi2) * phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(lambda1) * sin(lambda2);
	t_1 = cos(phi1) * sin(phi2);
	t_2 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi1 <= -2.8e-5) || ~((phi1 <= 0.000112)))
		tmp = atan2((cos(phi2) * (sin(lambda1) - t_0)), (t_1 - ((cos(phi2) * sin(phi1)) * t_2)));
	else
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - t_0) * cos(phi2)), (t_1 - (t_2 * (cos(phi2) * phi1))));
	end
	tmp_2 = 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[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi1, -2.8e-5], N[Not[LessEqual[phi1, 0.000112]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(t$95$2 * N[(N[Cos[phi2], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Initial program 89.9%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 0.000112\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_2 \cdot \phi_1\right)}\\ \end{array} \]

Alternative 9: 88.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \lambda_1 \cdot \sin \lambda_2\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 3.65 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - t_0\right)}{t_1 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - t_0\right) \cdot \cos \phi_2}{t_1 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda1) (sin lambda2))) (t_1 (* (cos phi1) (sin phi2))))
   (if (or (<= phi1 -9e-14) (not (<= phi1 3.65e-9)))
     (atan2
      (* (cos phi2) (- (sin lambda1) t_0))
      (- t_1 (* (* (cos phi2) (sin phi1)) (cos (- lambda1 lambda2)))))
     (atan2
      (* (- (* (sin lambda1) (cos lambda2)) t_0) (cos phi2))
      (- t_1 (* phi1 (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda1) * sin(lambda2);
	double t_1 = cos(phi1) * sin(phi2);
	double tmp;
	if ((phi1 <= -9e-14) || !(phi1 <= 3.65e-9)) {
		tmp = atan2((cos(phi2) * (sin(lambda1) - t_0)), (t_1 - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - t_0) * cos(phi2)), (t_1 - (phi1 * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(lambda1) * sin(lambda2)
    t_1 = cos(phi1) * sin(phi2)
    if ((phi1 <= (-9d-14)) .or. (.not. (phi1 <= 3.65d-9))) then
        tmp = atan2((cos(phi2) * (sin(lambda1) - t_0)), (t_1 - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))))
    else
        tmp = atan2((((sin(lambda1) * cos(lambda2)) - t_0) * cos(phi2)), (t_1 - (phi1 * cos((lambda2 - lambda1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(lambda1) * Math.sin(lambda2);
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((phi1 <= -9e-14) || !(phi1 <= 3.65e-9)) {
		tmp = Math.atan2((Math.cos(phi2) * (Math.sin(lambda1) - t_0)), (t_1 - ((Math.cos(phi2) * Math.sin(phi1)) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - t_0) * Math.cos(phi2)), (t_1 - (phi1 * Math.cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(lambda1) * math.sin(lambda2)
	t_1 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (phi1 <= -9e-14) or not (phi1 <= 3.65e-9):
		tmp = math.atan2((math.cos(phi2) * (math.sin(lambda1) - t_0)), (t_1 - ((math.cos(phi2) * math.sin(phi1)) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - t_0) * math.cos(phi2)), (t_1 - (phi1 * math.cos((lambda2 - lambda1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda1) * sin(lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((phi1 <= -9e-14) || !(phi1 <= 3.65e-9))
		tmp = atan(Float64(cos(phi2) * Float64(sin(lambda1) - t_0)), Float64(t_1 - Float64(Float64(cos(phi2) * sin(phi1)) * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - t_0) * cos(phi2)), Float64(t_1 - Float64(phi1 * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(lambda1) * sin(lambda2);
	t_1 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((phi1 <= -9e-14) || ~((phi1 <= 3.65e-9)))
		tmp = atan2((cos(phi2) * (sin(lambda1) - t_0)), (t_1 - ((cos(phi2) * sin(phi1)) * cos((lambda1 - lambda2)))));
	else
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - t_0) * cos(phi2)), (t_1 - (phi1 * cos((lambda2 - lambda1)))));
	end
	tmp_2 = 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[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -9e-14], N[Not[LessEqual[phi1, 3.65e-9]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - 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] - t$95$0), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Initial program 89.9%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 3.65 \cdot 10^{-9}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\sin \lambda_1 - \cos \lambda_1 \cdot \sin \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]

Alternative 10: 87.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t_1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -7.2e-14)
     (atan2 t_2 (- t_0 (* (sin phi1) (* (cos phi2) t_1))))
     (if (<= phi1 3e-8)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (- t_0 (* phi1 (cos (- lambda2 lambda1)))))
       (atan2
        t_2
        (- t_0 (* (* (cos phi2) (sin phi1)) (expm1 (log1p t_1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -7.2e-14) {
		tmp = atan2(t_2, (t_0 - (sin(phi1) * (cos(phi2) * t_1))));
	} else if (phi1 <= 3e-8) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (phi1 * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2(t_2, (t_0 - ((cos(phi2) * sin(phi1)) * expm1(log1p(t_1)))));
	}
	return tmp;
}

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos((lambda1 - lambda2));
	double t_2 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -7.2e-14) {
		tmp = Math.atan2(t_2, (t_0 - (Math.sin(phi1) * (Math.cos(phi2) * t_1))));
	} else if (phi1 <= 3e-8) {
		tmp = Math.atan2((((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2))) * Math.cos(phi2)), (t_0 - (phi1 * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = Math.atan2(t_2, (t_0 - ((Math.cos(phi2) * Math.sin(phi1)) * Math.expm1(Math.log1p(t_1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos((lambda1 - lambda2))
	t_2 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -7.2e-14:
		tmp = math.atan2(t_2, (t_0 - (math.sin(phi1) * (math.cos(phi2) * t_1))))
	elif phi1 <= 3e-8:
		tmp = math.atan2((((math.sin(lambda1) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(lambda2))) * math.cos(phi2)), (t_0 - (phi1 * math.cos((lambda2 - lambda1)))))
	else:
		tmp = math.atan2(t_2, (t_0 - ((math.cos(phi2) * math.sin(phi1)) * math.expm1(math.log1p(t_1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -7.2e-14)
		tmp = atan(t_2, Float64(t_0 - Float64(sin(phi1) * Float64(cos(phi2) * t_1))));
	elseif (phi1 <= 3e-8)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(phi1 * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(t_2, Float64(t_0 - Float64(Float64(cos(phi2) * sin(phi1)) * expm1(log1p(t_1)))));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -7.2e-14], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3e-8], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[(Exp[N[Log[1 + t$95$1], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Initial program 89.4%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\\ \end{array} \]

Alternative 11: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 4.45 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{t_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\mathsf{fma}\left(\cos \phi_2, t_1 \cdot \left(-\sin \phi_1\right), t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -9e-14)
     (atan2 t_2 (- t_0 (* (sin phi1) (* (cos phi2) t_1))))
     (if (<= phi1 4.45e-10)
       (atan2
        (*
         (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin lambda2)))
         (cos phi2))
        (- t_0 (* phi1 (cos (- lambda2 lambda1)))))
       (atan2 t_2 (fma (cos phi2) (* t_1 (- (sin phi1))) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -9e-14) {
		tmp = atan2(t_2, (t_0 - (sin(phi1) * (cos(phi2) * t_1))));
	} else if (phi1 <= 4.45e-10) {
		tmp = atan2((((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2))) * cos(phi2)), (t_0 - (phi1 * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2(t_2, fma(cos(phi2), (t_1 * -sin(phi1)), t_0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -9e-14)
		tmp = atan(t_2, Float64(t_0 - Float64(sin(phi1) * Float64(cos(phi2) * t_1))));
	elseif (phi1 <= 4.45e-10)
		tmp = atan(Float64(Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(lambda2))) * cos(phi2)), Float64(t_0 - Float64(phi1 * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(t_2, fma(cos(phi2), Float64(t_1 * Float64(-sin(phi1))), t_0));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(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-14], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 4.45e-10], N[ArcTan[N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Initial program 89.4%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 4.45 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin \lambda_1 \cdot \cos \lambda_2 - \cos \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \end{array} \]

Alternative 12: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-53}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\mathsf{fma}\left(\cos \phi_2, t_1 \cdot \left(-\sin \phi_1\right), t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.7e-53)
     (atan2 t_2 (- t_0 (* (sin phi1) (* (cos phi2) t_1))))
     (if (<= phi1 3.2e-61)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
        (sin phi2))
       (atan2 t_2 (fma (cos phi2) (* t_1 (- (sin phi1))) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.7e-53) {
		tmp = atan2(t_2, (t_0 - (sin(phi1) * (cos(phi2) * t_1))));
	} else if (phi1 <= 3.2e-61) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), sin(phi2));
	} else {
		tmp = atan2(t_2, fma(cos(phi2), (t_1 * -sin(phi1)), t_0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.7e-53)
		tmp = atan(t_2, Float64(t_0 - Float64(sin(phi1) * Float64(cos(phi2) * t_1))));
	elseif (phi1 <= 3.2e-61)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), sin(phi2));
	else
		tmp = atan(t_2, fma(cos(phi2), Float64(t_1 * Float64(-sin(phi1))), t_0));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.7e-53], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.2e-61], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Initial program 88.2%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-53}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.2 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \end{array} \]

Alternative 13: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2 - \phi_1 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{\mathsf{fma}\left(\cos \phi_2, t_1 \cdot \left(-\sin \phi_1\right), t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -9e-14)
     (atan2 t_2 (- t_0 (* (sin phi1) (* (cos phi2) t_1))))
     (if (<= phi1 2.4e-10)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
        (- (sin phi2) (* phi1 t_1)))
       (atan2 t_2 (fma (cos phi2) (* t_1 (- (sin phi1))) t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -9e-14) {
		tmp = atan2(t_2, (t_0 - (sin(phi1) * (cos(phi2) * t_1))));
	} else if (phi1 <= 2.4e-10) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), (sin(phi2) - (phi1 * t_1)));
	} else {
		tmp = atan2(t_2, fma(cos(phi2), (t_1 * -sin(phi1)), t_0));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -9e-14)
		tmp = atan(t_2, Float64(t_0 - Float64(sin(phi1) * Float64(cos(phi2) * t_1))));
	elseif (phi1 <= 2.4e-10)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), Float64(sin(phi2) - Float64(phi1 * t_1)));
	else
		tmp = atan(t_2, fma(cos(phi2), Float64(t_1 * Float64(-sin(phi1))), t_0));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(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-14], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 2.4e-10], 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[(phi1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Initial program 89.4%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 2.4 \cdot 10^{-10}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\mathsf{fma}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right) \cdot \left(-\sin \phi_1\right), \cos \phi_1 \cdot \sin \phi_2\right)}\\ \end{array} \]

Alternative 14: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -2.4 \cdot 10^{+153}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq -0.0052:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{t_2 - t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \lambda_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \lambda_2 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin phi1)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (* (cos phi1) (sin phi2))))
   (if (<= lambda2 -2.4e+153)
     (atan2
      (*
       (cos phi2)
       (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
      (sin phi2))
     (if (<= lambda2 -0.0052)
       (atan2
        (* (cos phi2) (sin (- lambda2)))
        (- t_2 (* t_0 (cos (- lambda1 lambda2)))))
       (if (<= lambda2 5.7e-5)
         (atan2 t_1 (- t_2 (* (cos lambda1) t_0)))
         (atan2 t_1 (- t_2 (* (cos lambda2) t_0))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin(phi1);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double t_2 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -2.4e+153) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), sin(phi2));
	} else if (lambda2 <= -0.0052) {
		tmp = atan2((cos(phi2) * sin(-lambda2)), (t_2 - (t_0 * cos((lambda1 - lambda2)))));
	} else if (lambda2 <= 5.7e-5) {
		tmp = atan2(t_1, (t_2 - (cos(lambda1) * t_0)));
	} else {
		tmp = atan2(t_1, (t_2 - (cos(lambda2) * t_0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(phi1))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_2 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -2.4e+153)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), sin(phi2));
	elseif (lambda2 <= -0.0052)
		tmp = atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(t_2 - Float64(t_0 * cos(Float64(lambda1 - lambda2)))));
	elseif (lambda2 <= 5.7e-5)
		tmp = atan(t_1, Float64(t_2 - Float64(cos(lambda1) * t_0)));
	else
		tmp = atan(t_1, Float64(t_2 - Float64(cos(lambda2) * t_0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -2.4e+153], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, -0.0052], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 - N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 5.7e-5], N[ArcTan[t$95$1 / N[(t$95$2 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$2 - N[(N[Cos[lambda2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Initial program 82.1%

   \[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.4 \cdot 10^{+153}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq -0.0052:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\lambda_2 \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 15: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.5 \cdot 10^{-55} \lor \neg \left(\phi_1 \leq 3.8 \cdot 10^{-61}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -9.5e-55) (not (<= phi1 3.8e-61)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (-
     (* (cos phi1) (sin phi2))
     (* (sin phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
   (atan2
    (*
     (cos phi2)
     (fma (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 <= -9.5e-55) || !(phi1 <= 3.8e-61)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -9.5e-55) || !(phi1 <= 3.8e-61))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -9.5e-55], N[Not[LessEqual[phi1, 3.8e-61]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 88.2%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.5 \cdot 10^{-55} \lor \neg \left(\phi_1 \leq 3.8 \cdot 10^{-61}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 16: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-53}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot t_1\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.5 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2}{t_0 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (cos (- lambda1 lambda2)))
        (t_2 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 -2.7e-53)
     (atan2 t_2 (- t_0 (* (sin phi1) (* (cos phi2) t_1))))
     (if (<= phi1 3.5e-61)
       (atan2
        (*
         (cos phi2)
         (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
        (sin phi2))
       (atan2 t_2 (- t_0 (* (* (cos phi2) (sin phi1)) t_1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos((lambda1 - lambda2));
	double t_2 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2.7e-53) {
		tmp = atan2(t_2, (t_0 - (sin(phi1) * (cos(phi2) * t_1))));
	} else if (phi1 <= 3.5e-61) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), sin(phi2));
	} else {
		tmp = atan2(t_2, (t_0 - ((cos(phi2) * sin(phi1)) * t_1)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = cos(Float64(lambda1 - lambda2))
	t_2 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -2.7e-53)
		tmp = atan(t_2, Float64(t_0 - Float64(sin(phi1) * Float64(cos(phi2) * t_1))));
	elseif (phi1 <= 3.5e-61)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), sin(phi2));
	else
		tmp = atan(t_2, Float64(t_0 - Float64(Float64(cos(phi2) * sin(phi1)) * t_1)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2.7e-53], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.5e-61], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$2 / N[(t$95$0 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Initial program 88.2%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-53}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.5 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 17: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.024 \lor \neg \left(\lambda_2 \leq 5.1 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -0.024) (not (<= lambda2 5.1e-5)))
   (atan2
    (*
     (cos phi2)
     (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
    (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 <= -0.024) || !(lambda2 <= 5.1e-5)) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), sin(phi2));
	} else {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (cos(lambda1) * (cos(phi2) * sin(phi1)))));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -0.024) || !(lambda2 <= 5.1e-5))
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), 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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -0.024], N[Not[LessEqual[lambda2, 5.1e-5]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[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. Initial program 78.9%

   \[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.024 \lor \neg \left(\lambda_2 \leq 5.1 \cdot 10^{-5}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 18: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{if}\;\phi_1 \leq -2.15 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\phi_1 \leq -9 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.8 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1
         (atan2
          (* (cos phi2) (sin (- lambda1 lambda2)))
          (- t_0 (* (sin phi1) (cos (- lambda2 lambda1)))))))
   (if (<= phi1 -2.15e+52)
     t_1
     (if (<= phi1 -9e-14)
       (atan2
        (* (sin lambda1) (cos phi2))
        (- t_0 (* (sin phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
       (if (<= phi1 3.8e-61)
         (atan2
          (*
           (cos phi2)
           (fma
            (sin lambda1)
            (cos lambda2)
            (* (cos lambda1) (- (sin lambda2)))))
          (sin phi2))
         t_1)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = atan2((cos(phi2) * sin((lambda1 - lambda2))), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))));
	double tmp;
	if (phi1 <= -2.15e+52) {
		tmp = t_1;
	} else if (phi1 <= -9e-14) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (sin(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	} else if (phi1 <= 3.8e-61) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), sin(phi2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))))
	tmp = 0.0
	if (phi1 <= -2.15e+52)
		tmp = t_1;
	elseif (phi1 <= -9e-14)
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(sin(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	elseif (phi1 <= 3.8e-61)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), sin(phi2));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -2.15e+52], t$95$1, If[LessEqual[phi1, -9e-14], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[phi1, 3.8e-61], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Initial program 73.3%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.15 \cdot 10^{+52}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{elif}\;\phi_1 \leq -9 \cdot 10^{-14}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 3.8 \cdot 10^{-61}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]

Alternative 19: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \phi_1\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ t_2 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -0.027:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 0.00024:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \lambda_1 \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_2 - \cos \lambda_2 \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin phi1)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2))))
        (t_2 (* (cos phi1) (sin phi2))))
   (if (<= lambda2 -0.027)
     (atan2
      (*
       (cos phi2)
       (fma (sin lambda1) (cos lambda2) (* (cos lambda1) (- (sin lambda2)))))
      (sin phi2))
     (if (<= lambda2 0.00024)
       (atan2 t_1 (- t_2 (* (cos lambda1) t_0)))
       (atan2 t_1 (- t_2 (* (cos lambda2) t_0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin(phi1);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double t_2 = cos(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -0.027) {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), sin(phi2));
	} else if (lambda2 <= 0.00024) {
		tmp = atan2(t_1, (t_2 - (cos(lambda1) * t_0)));
	} else {
		tmp = atan2(t_1, (t_2 - (cos(lambda2) * t_0)));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(phi1))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	t_2 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -0.027)
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), sin(phi2));
	elseif (lambda2 <= 0.00024)
		tmp = atan(t_1, Float64(t_2 - Float64(cos(lambda1) * t_0)));
	else
		tmp = atan(t_1, Float64(t_2 - Float64(cos(lambda2) * t_0)));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -0.027], 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[Sin[phi2], $MachinePrecision]], $MachinePrecision], If[LessEqual[lambda2, 0.00024], N[ArcTan[t$95$1 / N[(t$95$2 - N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$2 - N[(N[Cos[lambda2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Initial program 80.5%

   \[\begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.027:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \mathbf{elif}\;\lambda_2 \leq 0.00024:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_2 \cdot \left(\cos \phi_2 \cdot \sin \phi_1\right)}\\ \end{array} \]

Alternative 20: 72.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.55 \cdot 10^{-52} \lor \neg \left(\phi_1 \leq 3.5 \cdot 10^{-61}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -2.55e-52) (not (<= phi1 3.5e-61)))
   (atan2
    (* (cos phi2) (sin (- lambda1 lambda2)))
    (- (* (cos phi1) (sin phi2)) (* (sin phi1) (cos (- lambda2 lambda1)))))
   (atan2
    (*
     (cos phi2)
     (fma (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 <= -2.55e-52) || !(phi1 <= 3.5e-61)) {
		tmp = atan2((cos(phi2) * sin((lambda1 - lambda2))), ((cos(phi1) * sin(phi2)) - (sin(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2((cos(phi2) * fma(sin(lambda1), cos(lambda2), (cos(lambda1) * -sin(lambda2)))), sin(phi2));
	}
	return tmp;
}

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -2.55e-52) || !(phi1 <= 3.5e-61))
		tmp = atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(cos(phi1) * sin(phi2)) - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(Float64(cos(phi2) * fma(sin(lambda1), cos(lambda2), Float64(cos(lambda1) * Float64(-sin(lambda2))))), sin(phi2));
	end
	return tmp
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -2.55e-52], N[Not[LessEqual[phi1, 3.5e-61]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * (-N[Sin[lambda2], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 71.7%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.55 \cdot 10^{-52} \lor \neg \left(\phi_1 \leq 3.5 \cdot 10^{-61}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \cos \lambda_2, \cos \lambda_1 \cdot \left(-\sin \lambda_2\right)\right)}{\sin \phi_2}\\ \end{array} \]

Alternative 21: 69.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{+18} \lor \neg \left(\phi_2 \leq 0.5\right):\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1}{t_0 - \left(\sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (or (<= phi2 -1.6e+18) (not (<= phi2 0.5)))
     (atan2 t_1 (- t_0 (* (cos phi2) (sin phi1))))
     (atan2
      t_1
      (-
       t_0
       (*
        (* (sin phi1) (cos (- lambda2 lambda1)))
        (+ 1.0 (* -0.5 (* phi2 phi2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -1.6e+18) || !(phi2 <= 0.5)) {
		tmp = atan2(t_1, (t_0 - (cos(phi2) * sin(phi1))));
	} else {
		tmp = atan2(t_1, (t_0 - ((sin(phi1) * cos((lambda2 - lambda1))) * (1.0 + (-0.5 * (phi2 * 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) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if ((phi2 <= (-1.6d+18)) .or. (.not. (phi2 <= 0.5d0))) then
        tmp = atan2(t_1, (t_0 - (cos(phi2) * sin(phi1))))
    else
        tmp = atan2(t_1, (t_0 - ((sin(phi1) * cos((lambda2 - lambda1))) * (1.0d0 + ((-0.5d0) * (phi2 * phi2))))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -1.6e+18) || !(phi2 <= 0.5)) {
		tmp = Math.atan2(t_1, (t_0 - (Math.cos(phi2) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2(t_1, (t_0 - ((Math.sin(phi1) * Math.cos((lambda2 - lambda1))) * (1.0 + (-0.5 * (phi2 * phi2))))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -1.6e+18) or not (phi2 <= 0.5):
		tmp = math.atan2(t_1, (t_0 - (math.cos(phi2) * math.sin(phi1))))
	else:
		tmp = math.atan2(t_1, (t_0 - ((math.sin(phi1) * math.cos((lambda2 - lambda1))) * (1.0 + (-0.5 * (phi2 * phi2))))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if ((phi2 <= -1.6e+18) || !(phi2 <= 0.5))
		tmp = atan(t_1, Float64(t_0 - Float64(cos(phi2) * sin(phi1))));
	else
		tmp = atan(t_1, Float64(t_0 - Float64(Float64(sin(phi1) * cos(Float64(lambda2 - lambda1))) * Float64(1.0 + Float64(-0.5 * Float64(phi2 * phi2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -1.6e+18) || ~((phi2 <= 0.5)))
		tmp = atan2(t_1, (t_0 - (cos(phi2) * sin(phi1))));
	else
		tmp = atan2(t_1, (t_0 - ((sin(phi1) * cos((lambda2 - lambda1))) * (1.0 + (-0.5 * (phi2 * phi2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -1.6e+18], N[Not[LessEqual[phi2, 0.5]], $MachinePrecision]], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[t$95$1 / N[(t$95$0 - N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Initial program 66.8%

   \[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{+18} \lor \neg \left(\phi_2 \leq 0.5\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}\\ \end{array} \]

Alternative 22: 69.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_2 \cdot \sin \phi_1\\ t_2 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -62000000000 \lor \neg \left(\phi_2 \leq 0.048\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_2}{t_0 - t_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_2 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{t_0 - t_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2)))
        (t_1 (* (cos phi2) (sin phi1)))
        (t_2 (sin (- lambda1 lambda2))))
   (if (or (<= phi2 -62000000000.0) (not (<= phi2 0.048)))
     (atan2 (* (cos phi2) t_2) (- t_0 t_1))
     (atan2
      (* t_2 (+ 1.0 (* -0.5 (* phi2 phi2))))
      (- t_0 (* t_1 (cos (- lambda1 lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = cos(phi2) * sin(phi1);
	double t_2 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -62000000000.0) || !(phi2 <= 0.048)) {
		tmp = atan2((cos(phi2) * t_2), (t_0 - t_1));
	} else {
		tmp = atan2((t_2 * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = cos(phi2) * sin(phi1)
    t_2 = sin((lambda1 - lambda2))
    if ((phi2 <= (-62000000000.0d0)) .or. (.not. (phi2 <= 0.048d0))) then
        tmp = atan2((cos(phi2) * t_2), (t_0 - t_1))
    else
        tmp = atan2((t_2 * (1.0d0 + ((-0.5d0) * (phi2 * phi2)))), (t_0 - (t_1 * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.cos(phi2) * Math.sin(phi1);
	double t_2 = Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -62000000000.0) || !(phi2 <= 0.048)) {
		tmp = Math.atan2((Math.cos(phi2) * t_2), (t_0 - t_1));
	} else {
		tmp = Math.atan2((t_2 * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (t_1 * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.cos(phi2) * math.sin(phi1)
	t_2 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -62000000000.0) or not (phi2 <= 0.048):
		tmp = math.atan2((math.cos(phi2) * t_2), (t_0 - t_1))
	else:
		tmp = math.atan2((t_2 * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (t_1 * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = Float64(cos(phi2) * sin(phi1))
	t_2 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi2 <= -62000000000.0) || !(phi2 <= 0.048))
		tmp = atan(Float64(cos(phi2) * t_2), Float64(t_0 - t_1));
	else
		tmp = atan(Float64(t_2 * Float64(1.0 + Float64(-0.5 * Float64(phi2 * phi2)))), Float64(t_0 - Float64(t_1 * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = cos(phi2) * sin(phi1);
	t_2 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -62000000000.0) || ~((phi2 <= 0.048)))
		tmp = atan2((cos(phi2) * t_2), (t_0 - t_1));
	else
		tmp = atan2((t_2 * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (t_1 * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -62000000000.0], N[Not[LessEqual[phi2, 0.048]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(t$95$2 * N[(1.0 + N[(-0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Initial program 66.9%

   \[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -62000000000 \lor \neg \left(\phi_2 \leq 0.048\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \left(\cos \phi_2 \cdot \sin \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \end{array} \]

Alternative 23: 69.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -62000000000 \lor \neg \left(\phi_2 \leq 0.0102\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_1}{t_0 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t_1 \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{t_0 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))) (t_1 (sin (- lambda1 lambda2))))
   (if (or (<= phi2 -62000000000.0) (not (<= phi2 0.0102)))
     (atan2 (* (cos phi2) t_1) (- t_0 (* (cos phi2) (sin phi1))))
     (atan2
      (* t_1 (+ 1.0 (* -0.5 (* phi2 phi2))))
      (- t_0 (* (sin phi1) (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double t_1 = sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -62000000000.0) || !(phi2 <= 0.0102)) {
		tmp = atan2((cos(phi2) * t_1), (t_0 - (cos(phi2) * sin(phi1))));
	} else {
		tmp = atan2((t_1 * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    t_1 = sin((lambda1 - lambda2))
    if ((phi2 <= (-62000000000.0d0)) .or. (.not. (phi2 <= 0.0102d0))) then
        tmp = atan2((cos(phi2) * t_1), (t_0 - (cos(phi2) * sin(phi1))))
    else
        tmp = atan2((t_1 * (1.0d0 + ((-0.5d0) * (phi2 * phi2)))), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double t_1 = Math.sin((lambda1 - lambda2));
	double tmp;
	if ((phi2 <= -62000000000.0) || !(phi2 <= 0.0102)) {
		tmp = Math.atan2((Math.cos(phi2) * t_1), (t_0 - (Math.cos(phi2) * Math.sin(phi1))));
	} else {
		tmp = Math.atan2((t_1 * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (Math.sin(phi1) * Math.cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	t_1 = math.sin((lambda1 - lambda2))
	tmp = 0
	if (phi2 <= -62000000000.0) or not (phi2 <= 0.0102):
		tmp = math.atan2((math.cos(phi2) * t_1), (t_0 - (math.cos(phi2) * math.sin(phi1))))
	else:
		tmp = math.atan2((t_1 * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (math.sin(phi1) * math.cos((lambda2 - lambda1)))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	t_1 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if ((phi2 <= -62000000000.0) || !(phi2 <= 0.0102))
		tmp = atan(Float64(cos(phi2) * t_1), Float64(t_0 - Float64(cos(phi2) * sin(phi1))));
	else
		tmp = atan(Float64(t_1 * Float64(1.0 + Float64(-0.5 * Float64(phi2 * phi2)))), Float64(t_0 - Float64(sin(phi1) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	t_1 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if ((phi2 <= -62000000000.0) || ~((phi2 <= 0.0102)))
		tmp = atan2((cos(phi2) * t_1), (t_0 - (cos(phi2) * sin(phi1))));
	else
		tmp = atan2((t_1 * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (sin(phi1) * cos((lambda2 - lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -62000000000.0], N[Not[LessEqual[phi2, 0.0102]], $MachinePrecision]], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(t$95$1 * N[(1.0 + N[(-0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Sin[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Initial program 66.8%

   \[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -62000000000 \lor \neg \left(\phi_2 \leq 0.0102\right):\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \phi_2 \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]

Alternative 24: 63.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \sin \phi_2\\ t_2 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.00024 \lor \neg \left(\phi_1 \leq 135\right):\\ \;\;\;\;\tan^{-1}_* \frac{\frac{2 \cdot t_0}{2}}{t_1 - \sin \phi_1 \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{t_1 - \phi_1 \cdot t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (sin phi2)))
        (t_2 (cos (- lambda2 lambda1))))
   (if (or (<= phi1 -0.00024) (not (<= phi1 135.0)))
     (atan2 (/ (* 2.0 t_0) 2.0) (- t_1 (* (sin phi1) t_2)))
     (atan2 (* (cos phi2) t_0) (- t_1 (* phi1 t_2))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos(phi1) * sin(phi2);
	double t_2 = cos((lambda2 - lambda1));
	double tmp;
	if ((phi1 <= -0.00024) || !(phi1 <= 135.0)) {
		tmp = atan2(((2.0 * t_0) / 2.0), (t_1 - (sin(phi1) * t_2)));
	} else {
		tmp = atan2((cos(phi2) * t_0), (t_1 - (phi1 * t_2)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = cos(phi1) * sin(phi2)
    t_2 = cos((lambda2 - lambda1))
    if ((phi1 <= (-0.00024d0)) .or. (.not. (phi1 <= 135.0d0))) then
        tmp = atan2(((2.0d0 * t_0) / 2.0d0), (t_1 - (sin(phi1) * t_2)))
    else
        tmp = atan2((cos(phi2) * t_0), (t_1 - (phi1 * t_2)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.cos(phi1) * Math.sin(phi2);
	double t_2 = Math.cos((lambda2 - lambda1));
	double tmp;
	if ((phi1 <= -0.00024) || !(phi1 <= 135.0)) {
		tmp = Math.atan2(((2.0 * t_0) / 2.0), (t_1 - (Math.sin(phi1) * t_2)));
	} else {
		tmp = Math.atan2((Math.cos(phi2) * t_0), (t_1 - (phi1 * t_2)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.cos(phi1) * math.sin(phi2)
	t_2 = math.cos((lambda2 - lambda1))
	tmp = 0
	if (phi1 <= -0.00024) or not (phi1 <= 135.0):
		tmp = math.atan2(((2.0 * t_0) / 2.0), (t_1 - (math.sin(phi1) * t_2)))
	else:
		tmp = math.atan2((math.cos(phi2) * t_0), (t_1 - (phi1 * t_2)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * sin(phi2))
	t_2 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if ((phi1 <= -0.00024) || !(phi1 <= 135.0))
		tmp = atan(Float64(Float64(2.0 * t_0) / 2.0), Float64(t_1 - Float64(sin(phi1) * t_2)));
	else
		tmp = atan(Float64(cos(phi2) * t_0), Float64(t_1 - Float64(phi1 * t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = cos(phi1) * sin(phi2);
	t_2 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if ((phi1 <= -0.00024) || ~((phi1 <= 135.0)))
		tmp = atan2(((2.0 * t_0) / 2.0), (t_1 - (sin(phi1) * t_2)));
	else
		tmp = atan2((cos(phi2) * t_0), (t_1 - (phi1 * t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi1, -0.00024], N[Not[LessEqual[phi1, 135.0]], $MachinePrecision]], N[ArcTan[N[(N[(2.0 * t$95$0), $MachinePrecision] / 2.0), $MachinePrecision] / N[(t$95$1 - N[(N[Sin[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(t$95$1 - N[(phi1 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Initial program 61.8%

   \[\begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00024 \lor \neg \left(\phi_1 \leq 135\right):\\ \;\;\;\;\tan^{-1}_* \frac{\frac{2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{2}}{\cos \phi_1 \cdot \sin \phi_2 - \sin \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]

Alternative 25: 38.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -450000000000 \lor \neg \left(\phi_2 \leq 3.1 \cdot 10^{-8}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{t_0 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{t_0 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (sin phi2))))
   (if (or (<= phi2 -450000000000.0) (not (<= phi2 3.1e-8)))
     (atan2
      (* (sin lambda1) (cos phi2))
      (- t_0 (* phi1 (cos (- lambda2 lambda1)))))
     (atan2
      (* (sin (- lambda1 lambda2)) (+ 1.0 (* -0.5 (* phi2 phi2))))
      (- t_0 (* (cos lambda1) phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * sin(phi2);
	double tmp;
	if ((phi2 <= -450000000000.0) || !(phi2 <= 3.1e-8)) {
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (phi1 * cos((lambda2 - lambda1)))));
	} else {
		tmp = atan2((sin((lambda1 - lambda2)) * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (cos(lambda1) * phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * sin(phi2)
    if ((phi2 <= (-450000000000.0d0)) .or. (.not. (phi2 <= 3.1d-8))) then
        tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (phi1 * cos((lambda2 - lambda1)))))
    else
        tmp = atan2((sin((lambda1 - lambda2)) * (1.0d0 + ((-0.5d0) * (phi2 * phi2)))), (t_0 - (cos(lambda1) * phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.sin(phi2);
	double tmp;
	if ((phi2 <= -450000000000.0) || !(phi2 <= 3.1e-8)) {
		tmp = Math.atan2((Math.sin(lambda1) * Math.cos(phi2)), (t_0 - (phi1 * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = Math.atan2((Math.sin((lambda1 - lambda2)) * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (Math.cos(lambda1) * phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.sin(phi2)
	tmp = 0
	if (phi2 <= -450000000000.0) or not (phi2 <= 3.1e-8):
		tmp = math.atan2((math.sin(lambda1) * math.cos(phi2)), (t_0 - (phi1 * math.cos((lambda2 - lambda1)))))
	else:
		tmp = math.atan2((math.sin((lambda1 - lambda2)) * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (math.cos(lambda1) * phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * sin(phi2))
	tmp = 0.0
	if ((phi2 <= -450000000000.0) || !(phi2 <= 3.1e-8))
		tmp = atan(Float64(sin(lambda1) * cos(phi2)), Float64(t_0 - Float64(phi1 * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = atan(Float64(sin(Float64(lambda1 - lambda2)) * Float64(1.0 + Float64(-0.5 * Float64(phi2 * phi2)))), Float64(t_0 - Float64(cos(lambda1) * phi1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * sin(phi2);
	tmp = 0.0;
	if ((phi2 <= -450000000000.0) || ~((phi2 <= 3.1e-8)))
		tmp = atan2((sin(lambda1) * cos(phi2)), (t_0 - (phi1 * cos((lambda2 - lambda1)))));
	else
		tmp = atan2((sin((lambda1 - lambda2)) * (1.0 + (-0.5 * (phi2 * phi2)))), (t_0 - (cos(lambda1) * phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -450000000000.0], N[Not[LessEqual[phi2, 3.1e-8]], $MachinePrecision]], N[ArcTan[N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(phi1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[ArcTan[N[(N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(phi2 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - N[(N[Cos[lambda1], $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 37.7%

   \[\begin{array}{l} \mathbf{if}\;\phi_2 \leq -450000000000 \lor \neg \left(\phi_2 \leq 3.1 \cdot 10^{-8}\right):\\ \;\;\;\;\tan^{-1}_* \frac{\sin \lambda_1 \cdot \cos \phi_2}{\cos \phi_1 \cdot \sin \phi_2 - \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right) \cdot \left(1 + -0.5 \cdot \left(\phi_2 \cdot \phi_2\right)\right)}{\cos \phi_1 \cdot \sin \phi_2 - \cos \lambda_1 \cdot \phi_1}\\ \end{array} \]

Alternative 26: 47.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.3%

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

Alternative 27: 47.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.5%

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

Alternative 28: 31.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 28.8%

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

Reproduce

herbie shell --seed 2023297 
(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))))))