Midpoint on a great circle

Percentage Accurate: 98.7% → 98.7%

Time: 21.5s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. cos-neg 99.5%

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

   2. cos-neg 99.5%

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

   3. cos-neg 99.5%

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

   4. +-commutative 99.5%

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

   5. cos-neg 99.5%

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

   6. fma-define 99.5%

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

3. Simplified 99.5%

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

Alternative 2: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 21000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_0 + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{+201}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_2 \cdot t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi2 21000000.0)
     (+ lambda1 (atan2 t_1 (+ t_0 (cos phi1))))
     (if (<= phi2 8.8e+201)
       (+ lambda1 (atan2 t_1 (+ (cos phi1) (* (cos phi2) (cos lambda1)))))
       (+ lambda1 (atan2 t_1 (+ (* (cos phi2) t_0) 1.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 21000000.0) {
		tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)));
	} else if (phi2 <= 8.8e+201) {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2) * cos(lambda1))));
	} else {
		tmp = lambda1 + atan2(t_1, ((cos(phi2) * t_0) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi2 <= 21000000.0d0) then
        tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)))
    else if (phi2 <= 8.8d+201) then
        tmp = lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2) * cos(lambda1))))
    else
        tmp = lambda1 + atan2(t_1, ((cos(phi2) * t_0) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 21000000.0) {
		tmp = lambda1 + Math.atan2(t_1, (t_0 + Math.cos(phi1)));
	} else if (phi2 <= 8.8e+201) {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(phi1) + (Math.cos(phi2) * Math.cos(lambda1))));
	} else {
		tmp = lambda1 + Math.atan2(t_1, ((Math.cos(phi2) * t_0) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= 21000000.0:
		tmp = lambda1 + math.atan2(t_1, (t_0 + math.cos(phi1)))
	elif phi2 <= 8.8e+201:
		tmp = lambda1 + math.atan2(t_1, (math.cos(phi1) + (math.cos(phi2) * math.cos(lambda1))))
	else:
		tmp = lambda1 + math.atan2(t_1, ((math.cos(phi2) * t_0) + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= 21000000.0)
		tmp = Float64(lambda1 + atan(t_1, Float64(t_0 + cos(phi1))));
	elseif (phi2 <= 8.8e+201)
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + Float64(cos(phi2) * cos(lambda1)))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(Float64(cos(phi2) * t_0) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= 21000000.0)
		tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)));
	elseif (phi2 <= 8.8e+201)
		tmp = lambda1 + atan2(t_1, (cos(phi1) + (cos(phi2) * cos(lambda1))));
	else
		tmp = lambda1 + atan2(t_1, ((cos(phi2) * t_0) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, 21000000.0], N[(lambda1 + N[ArcTan[t$95$1 / N[(t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 8.8e+201], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.1e7

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0 84.6%

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

   if 2.1e7 < phi2 < 8.8e201

   1. Initial program 99.3%

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

   3. Taylor expanded in lambda2 around 0 90.9%

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

      1. +-commutative 90.9%

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

      2. *-commutative 90.9%

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

   5. Simplified 90.9%

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

   if 8.8e201 < phi2

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0 98.0%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 21000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 8.8 \cdot 10^{+201}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

Alternative 4: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in lambda1 around 0 97.8%

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

   1. +-commutative 97.8%

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

   2. cos-neg 97.8%

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

5. Simplified 97.8%

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

6. Final simplification 97.8%

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

Alternative 5: 83.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 21000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_0 + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 3.4 \cdot 10^{+201}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_2 \cdot t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi2 21000000.0)
     (+ lambda1 (atan2 t_1 (+ t_0 (cos phi1))))
     (if (<= phi2 3.4e+201)
       (+ lambda1 (atan2 t_1 (+ (cos phi2) (cos phi1))))
       (+ lambda1 (atan2 t_1 (+ (* (cos phi2) t_0) 1.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 21000000.0) {
		tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)));
	} else if (phi2 <= 3.4e+201) {
		tmp = lambda1 + atan2(t_1, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_1, ((cos(phi2) * t_0) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi2 <= 21000000.0d0) then
        tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)))
    else if (phi2 <= 3.4d+201) then
        tmp = lambda1 + atan2(t_1, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_1, ((cos(phi2) * t_0) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 21000000.0) {
		tmp = lambda1 + Math.atan2(t_1, (t_0 + Math.cos(phi1)));
	} else if (phi2 <= 3.4e+201) {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_1, ((Math.cos(phi2) * t_0) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= 21000000.0:
		tmp = lambda1 + math.atan2(t_1, (t_0 + math.cos(phi1)))
	elif phi2 <= 3.4e+201:
		tmp = lambda1 + math.atan2(t_1, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_1, ((math.cos(phi2) * t_0) + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= 21000000.0)
		tmp = Float64(lambda1 + atan(t_1, Float64(t_0 + cos(phi1))));
	elseif (phi2 <= 3.4e+201)
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(Float64(cos(phi2) * t_0) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= 21000000.0)
		tmp = lambda1 + atan2(t_1, (t_0 + cos(phi1)));
	elseif (phi2 <= 3.4e+201)
		tmp = lambda1 + atan2(t_1, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_1, ((cos(phi2) * t_0) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, 21000000.0], N[(lambda1 + N[ArcTan[t$95$1 / N[(t$95$0 + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.4e+201], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.1e7

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0 84.6%

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

   if 2.1e7 < phi2 < 3.4e201

   1. Initial program 99.3%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. expm1-define 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in lambda1 around 0 95.4%

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

      1. cos-neg 95.4%

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

   9. Simplified 95.4%

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

   10. Taylor expanded in lambda2 around 0 87.1%

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

       1. +-commutative 87.1%

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

   12. Simplified 87.1%

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

   if 3.4e201 < phi2

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0 98.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 21000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 3.4 \cdot 10^{+201}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 21000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 1.95 \cdot 10^{+201}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi2 21000000.0)
     (+ lambda1 (atan2 t_0 (+ (cos (- lambda1 lambda2)) (cos phi1))))
     (if (<= phi2 1.95e+201)
       (+ lambda1 (atan2 t_0 (+ (cos phi2) (cos phi1))))
       (+ lambda1 (atan2 t_0 (+ (* (cos phi2) (cos lambda2)) 1.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 21000000.0) {
		tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + cos(phi1)));
	} else if (phi2 <= 1.95e+201) {
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, ((cos(phi2) * cos(lambda2)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi2 <= 21000000.0d0) then
        tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + cos(phi1)))
    else if (phi2 <= 1.95d+201) then
        tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, ((cos(phi2) * cos(lambda2)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 21000000.0) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos((lambda1 - lambda2)) + Math.cos(phi1)));
	} else if (phi2 <= 1.95e+201) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, ((Math.cos(phi2) * Math.cos(lambda2)) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= 21000000.0:
		tmp = lambda1 + math.atan2(t_0, (math.cos((lambda1 - lambda2)) + math.cos(phi1)))
	elif phi2 <= 1.95e+201:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, ((math.cos(phi2) * math.cos(lambda2)) + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= 21000000.0)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(Float64(lambda1 - lambda2)) + cos(phi1))));
	elseif (phi2 <= 1.95e+201)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(Float64(cos(phi2) * cos(lambda2)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= 21000000.0)
		tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + cos(phi1)));
	elseif (phi2 <= 1.95e+201)
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, ((cos(phi2) * cos(lambda2)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, 21000000.0], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.95e+201], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 2.1e7

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0 84.6%

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

   if 2.1e7 < phi2 < 1.95e201

   1. Initial program 99.3%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. expm1-define 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in lambda1 around 0 95.4%

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

      1. cos-neg 95.4%

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

   9. Simplified 95.4%

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

   10. Taylor expanded in lambda2 around 0 87.1%

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

       1. +-commutative 87.1%

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

   12. Simplified 87.1%

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

   if 1.95e201 < phi2

   1. Initial program 99.7%

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

      1. expm1-log1p-u 99.7%

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

      2. expm1-undefine 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. expm1-define 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in lambda1 around 0 97.6%

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

      1. cos-neg 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in phi1 around 0 97.7%

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

       1. *-commutative 97.7%

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

   12. Simplified 97.7%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 21000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{elif}\;\phi_2 \leq 1.95 \cdot 10^{+201}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot t\_0\\ \mathbf{if}\;\phi_2 \leq 11.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 5.3 \cdot 10^{+201}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))) (t_1 (* (cos phi2) t_0)))
   (if (<= phi2 11.5)
     (+
      lambda1
      (atan2 t_0 (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
     (if (<= phi2 5.3e+201)
       (+ lambda1 (atan2 t_1 (+ (cos phi2) (cos phi1))))
       (+ lambda1 (atan2 t_1 (+ (* (cos phi2) (cos lambda2)) 1.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double t_1 = cos(phi2) * t_0;
	double tmp;
	if (phi2 <= 11.5) {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	} else if (phi2 <= 5.3e+201) {
		tmp = lambda1 + atan2(t_1, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_1, ((cos(phi2) * cos(lambda2)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    t_1 = cos(phi2) * t_0
    if (phi2 <= 11.5d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
    else if (phi2 <= 5.3d+201) then
        tmp = lambda1 + atan2(t_1, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_1, ((cos(phi2) * cos(lambda2)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * t_0;
	double tmp;
	if (phi2 <= 11.5) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
	} else if (phi2 <= 5.3e+201) {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_1, ((Math.cos(phi2) * Math.cos(lambda2)) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	t_1 = math.cos(phi2) * t_0
	tmp = 0
	if phi2 <= 11.5:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))
	elif phi2 <= 5.3e+201:
		tmp = lambda1 + math.atan2(t_1, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_1, ((math.cos(phi2) * math.cos(lambda2)) + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi2) * t_0)
	tmp = 0.0
	if (phi2 <= 11.5)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	elseif (phi2 <= 5.3e+201)
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(Float64(cos(phi2) * cos(lambda2)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	t_1 = cos(phi2) * t_0;
	tmp = 0.0;
	if (phi2 <= 11.5)
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	elseif (phi2 <= 5.3e+201)
		tmp = lambda1 + atan2(t_1, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_1, ((cos(phi2) * cos(lambda2)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[phi2, 11.5], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5.3e+201], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 11.5

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0 84.6%

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

   if 11.5 < phi2 < 5.30000000000000035e201

   1. Initial program 99.2%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.1%

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

   4. Applied egg-rr 99.1%

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

      1. expm1-define 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in lambda1 around 0 95.5%

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

      1. cos-neg 95.5%

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

   9. Simplified 95.5%

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

   10. Taylor expanded in lambda2 around 0 83.9%

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

       1. +-commutative 83.9%

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

   12. Simplified 83.9%

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

   if 5.30000000000000035e201 < phi2

   1. Initial program 99.7%

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

      1. expm1-log1p-u 99.7%

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

      2. expm1-undefine 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. expm1-define 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in lambda1 around 0 97.6%

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

      1. cos-neg 97.6%

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

   9. Simplified 97.6%

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

   10. Taylor expanded in phi1 around 0 97.7%

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

       1. *-commutative 97.7%

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

   12. Simplified 97.7%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 11.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\phi_2 \leq 5.3 \cdot 10^{+201}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 11.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi2 11.5)
     (+
      lambda1
      (atan2 t_0 (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
     (+ lambda1 (atan2 (* (cos phi2) t_0) (+ (cos phi2) (cos phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 11.5) {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(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 = sin((lambda1 - lambda2))
    if (phi2 <= 11.5d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
    else
        tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 11.5) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * t_0), (Math.cos(phi2) + Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= 11.5:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * t_0), (math.cos(phi2) + math.cos(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= 11.5)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(cos(phi2) + cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= 11.5)
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	else
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 11.5], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 11.5

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0 84.6%

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

   if 11.5 < phi2

   1. Initial program 99.4%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-undefine 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. expm1-define 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in lambda1 around 0 96.2%

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

      1. cos-neg 96.2%

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

   9. Simplified 96.2%

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

   10. Taylor expanded in lambda2 around 0 84.3%

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

       1. +-commutative 84.3%

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

   12. Simplified 84.3%

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

Alternative 9: 82.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi2 1.15e-11)
     (+ lambda1 (atan2 t_0 (+ (cos (- lambda1 lambda2)) (cos phi1))))
     (+ lambda1 (atan2 (* (cos phi2) t_0) (+ (cos phi2) (cos phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 1.15e-11) {
		tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(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 = sin((lambda1 - lambda2))
    if (phi2 <= 1.15d-11) then
        tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + cos(phi1)))
    else
        tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 1.15e-11) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos((lambda1 - lambda2)) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * t_0), (Math.cos(phi2) + Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= 1.15e-11:
		tmp = lambda1 + math.atan2(t_0, (math.cos((lambda1 - lambda2)) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * t_0), (math.cos(phi2) + math.cos(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= 1.15e-11)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(Float64(lambda1 - lambda2)) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(cos(phi2) + cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= 1.15e-11)
		tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + cos(phi1)));
	else
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 1.15e-11], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.15000000000000007e-11

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0 84.5%

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

   4. Taylor expanded in phi2 around 0 84.0%

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

   if 1.15000000000000007e-11 < phi2

   1. Initial program 99.4%

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

      1. expm1-log1p-u 99.3%

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

      2. expm1-undefine 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. expm1-define 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in lambda1 around 0 96.3%

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

      1. cos-neg 96.3%

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

   9. Simplified 96.3%

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

   10. Taylor expanded in lambda2 around 0 84.8%

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

       1. +-commutative 84.8%

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

   12. Simplified 84.8%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. expm1-log1p-u 99.4%

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

   2. expm1-undefine 99.4%

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

4. Applied egg-rr 99.4%

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

   1. expm1-define 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in lambda1 around 0 97.8%

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

   1. cos-neg 97.8%

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

9. Simplified 97.8%

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

10. Taylor expanded in phi2 around 0 79.1%

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

Alternative 11: 69.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 35000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \lambda_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi1 35000000000.0)
     (+ lambda1 (atan2 t_0 (+ (cos (- lambda1 lambda2)) 1.0)))
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (cos lambda1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 35000000000.0) {
		tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + 1.0));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(lambda1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if (phi1 <= 35000000000.0d0) then
        tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + 1.0d0))
    else
        tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(lambda1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 35000000000.0) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos((lambda1 - lambda2)) + 1.0));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + Math.cos(lambda1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= 35000000000.0:
		tmp = lambda1 + math.atan2(t_0, (math.cos((lambda1 - lambda2)) + 1.0))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + math.cos(lambda1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= 35000000000.0)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(Float64(lambda1 - lambda2)) + 1.0)));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + cos(lambda1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= 35000000000.0)
		tmp = lambda1 + atan2(t_0, (cos((lambda1 - lambda2)) + 1.0));
	else
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, 35000000000.0], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 3.5e10

   1. Initial program 99.7%

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

   3. Taylor expanded in phi2 around 0 79.3%

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

      1. add-sqr-sqrt 66.5%

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

      2. pow2 66.5%

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

      3. +-commutative 66.5%

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

      4. fma-undefine 66.5%

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

   5. Applied egg-rr 66.5%

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

   6. Taylor expanded in phi2 around 0 78.3%

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

      1. +-commutative 78.3%

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

   8. Simplified 78.3%

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

   9. Taylor expanded in phi1 around 0 70.2%

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

       1. +-commutative 70.2%

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

   11. Simplified 70.2%

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

   if 3.5e10 < phi1

   1. Initial program 98.9%

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

   3. Taylor expanded in phi2 around 0 78.7%

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

      1. add-sqr-sqrt 45.4%

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

      2. pow2 45.4%

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

      3. +-commutative 45.4%

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

      4. fma-undefine 45.4%

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

   5. Applied egg-rr 45.4%

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

   6. Taylor expanded in phi2 around 0 78.8%

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

      1. +-commutative 78.8%

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

   8. Simplified 78.8%

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

   9. Taylor expanded in lambda2 around 0 67.3%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 35000000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \lambda_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in phi2 around 0 79.1%

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

4. Taylor expanded in phi2 around 0 78.4%

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

5. Final simplification 78.4%

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

Alternative 13: 77.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in phi2 around 0 79.1%

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

   1. add-sqr-sqrt 60.7%

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

   2. pow2 60.7%

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

   3. +-commutative 60.7%

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

   4. fma-undefine 60.7%

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

5. Applied egg-rr 60.7%

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

6. Taylor expanded in phi2 around 0 78.4%

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

   1. +-commutative 78.4%

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

8. Simplified 78.4%

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

9. Taylor expanded in lambda1 around 0 77.7%

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

    1. cos-neg 77.7%

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

11. Simplified 77.7%

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

Alternative 14: 67.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2 (sin (- lambda1 lambda2)) (+ (cos (- lambda1 lambda2)) 1.0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2(sin((lambda1 - lambda2)), (cos((lambda1 - lambda2)) + 1.0));
}

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 = lambda1 + atan2(sin((lambda1 - lambda2)), (cos((lambda1 - lambda2)) + 1.0d0))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos((lambda1 - lambda2)) + 1.0));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos((lambda1 - lambda2)) + 1.0))

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(Float64(lambda1 - lambda2)) + 1.0)))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos((lambda1 - lambda2)) + 1.0));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in phi2 around 0 79.1%

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

   1. add-sqr-sqrt 60.7%

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

   2. pow2 60.7%

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

   3. +-commutative 60.7%

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

   4. fma-undefine 60.7%

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

5. Applied egg-rr 60.7%

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

6. Taylor expanded in phi2 around 0 78.4%

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

   1. +-commutative 78.4%

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

8. Simplified 78.4%

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

9. Taylor expanded in phi1 around 0 66.2%

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

    1. +-commutative 66.2%

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

11. Simplified 66.2%

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

Alternative 15: 53.3% accurate, 614.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 \end{array} \]

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return 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 = lambda1
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1

Julia

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

MATLAB

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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := lambda1

Derivation

1. Initial program 99.5%

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

   1. log1p-expm1-u 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in lambda1 around inf 52.2%

   \[\leadsto \color{blue}{\lambda_1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024148 
(FPCore (lambda1 lambda2 phi1 phi2)
  :name "Midpoint on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))