Midpoint on a great circle

Percentage Accurate: 98.6% → 98.6%

Time: 23.3s

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: 98.6% 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: 98.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2)))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (/
      (+ (* t_0 (pow t_0 2.0)) (pow (cos phi1) 3.0))
      (+ (pow (cos phi1) 2.0) (* t_0 (- t_0 (cos phi1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos((lambda1 - lambda2));
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (((t_0 * pow(t_0, 2.0)) + pow(cos(phi1), 3.0)) / (pow(cos(phi1), 2.0) + (t_0 * (t_0 - 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
    real(8) :: t_0
    t_0 = cos(phi2) * cos((lambda1 - lambda2))
    code = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (((t_0 * (t_0 ** 2.0d0)) + (cos(phi1) ** 3.0d0)) / ((cos(phi1) ** 2.0d0) + (t_0 * (t_0 - cos(phi1))))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.cos((lambda1 - lambda2));
	return lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (((t_0 * Math.pow(t_0, 2.0)) + Math.pow(Math.cos(phi1), 3.0)) / (Math.pow(Math.cos(phi1), 2.0) + (t_0 * (t_0 - Math.cos(phi1))))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.cos((lambda1 - lambda2))
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (((t_0 * math.pow(t_0, 2.0)) + math.pow(math.cos(phi1), 3.0)) / (math.pow(math.cos(phi1), 2.0) + (t_0 * (t_0 - math.cos(phi1))))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64(Float64(t_0 * (t_0 ^ 2.0)) + (cos(phi1) ^ 3.0)) / Float64((cos(phi1) ^ 2.0) + Float64(t_0 * Float64(t_0 - cos(phi1)))))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * cos((lambda1 - lambda2));
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (((t_0 * (t_0 ^ 2.0)) + (cos(phi1) ^ 3.0)) / ((cos(phi1) ^ 2.0) + (t_0 * (t_0 - cos(phi1))))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Cos[phi1], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$0 * N[(t$95$0 - N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.0%

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

   1. flip3-+ 99.0%

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

   2. pow3 99.0%

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

   3. pow3 99.1%

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

   4. +-commutative 99.1%

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

   5. pow3 99.0%

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

   6. pow3 99.0%

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

   7. *-commutative 99.0%

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

3. Applied egg-rr 99.0%

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

   1. unpow3 99.0%

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

   2. pow2 99.0%

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

5. Applied egg-rr 99.0%

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

6. Final simplification 99.0%

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

Alternative 2: 98.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2)))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (/
      (+ (pow (cos phi1) 3.0) (pow t_0 3.0))
      (+ (pow (cos phi1) 2.0) (* t_0 (- t_0 (cos phi1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos((lambda1 - lambda2));
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), ((pow(cos(phi1), 3.0) + pow(t_0, 3.0)) / (pow(cos(phi1), 2.0) + (t_0 * (t_0 - 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
    real(8) :: t_0
    t_0 = cos(phi2) * cos((lambda1 - lambda2))
    code = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (((cos(phi1) ** 3.0d0) + (t_0 ** 3.0d0)) / ((cos(phi1) ** 2.0d0) + (t_0 * (t_0 - cos(phi1))))))
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.cos((lambda1 - lambda2));
	return lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), ((Math.pow(Math.cos(phi1), 3.0) + Math.pow(t_0, 3.0)) / (Math.pow(Math.cos(phi1), 2.0) + (t_0 * (t_0 - Math.cos(phi1))))));
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.cos((lambda1 - lambda2))
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), ((math.pow(math.cos(phi1), 3.0) + math.pow(t_0, 3.0)) / (math.pow(math.cos(phi1), 2.0) + (t_0 * (t_0 - math.cos(phi1))))))

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(Float64((cos(phi1) ^ 3.0) + (t_0 ^ 3.0)) / Float64((cos(phi1) ^ 2.0) + Float64(t_0 * Float64(t_0 - cos(phi1)))))))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * cos((lambda1 - lambda2));
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (((cos(phi1) ^ 3.0) + (t_0 ^ 3.0)) / ((cos(phi1) ^ 2.0) + (t_0 * (t_0 - cos(phi1))))));
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[N[Cos[phi1], $MachinePrecision], 3.0], $MachinePrecision] + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$0 * N[(t$95$0 - N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.0%

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

   1. flip3-+ 99.0%

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

   2. pow3 99.0%

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

   3. pow3 99.1%

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

   4. +-commutative 99.1%

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

   5. pow3 99.0%

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

   6. pow3 99.0%

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

   7. *-commutative 99.0%

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

3. Applied egg-rr 99.0%

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

4. Final simplification 99.0%

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

Alternative 3: 98.6% 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 99.0%

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

   1. cos-neg 99.0%

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

   2. cos-neg 99.0%

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

   3. cos-neg 99.0%

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

   4. +-commutative 99.0%

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

   5. cos-neg 99.0%

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

   6. fma-def 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 4: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.999999999999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.999999999999)
     (+ lambda1 (atan2 (* (cos phi2) t_0) (+ (cos phi2) (cos phi1))))
     (+
      lambda1
      (atan2 t_0 (+ (* (cos phi2) (cos (- lambda1 lambda2))) (cos phi1)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.999999999999) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, ((cos(phi2) * cos((lambda1 - lambda2))) + 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 (cos(phi2) <= 0.999999999999d0) then
        tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, ((cos(phi2) * cos((lambda1 - lambda2))) + 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 (Math.cos(phi2) <= 0.999999999999) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * t_0), (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, ((Math.cos(phi2) * Math.cos((lambda1 - lambda2))) + Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.999999999999:
		tmp = lambda1 + math.atan2((math.cos(phi2) * t_0), (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, ((math.cos(phi2) * math.cos((lambda1 - lambda2))) + math.cos(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.999999999999)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))) + cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.999999999999)
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, ((cos(phi2) * cos((lambda1 - lambda2))) + 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[N[Cos[phi2], $MachinePrecision], 0.999999999999], 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], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.999999999999000022

   1. Initial program 99.0%

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

   2. Taylor expanded in lambda1 around 0 97.7%

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

      1. mul-1-neg 97.7%

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

      2. unsub-neg 97.7%

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

      3. sin-neg 97.7%

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

      4. distribute-rgt-neg-out 97.7%

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

      5. distribute-lft-neg-in 97.7%

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

      6. cancel-sign-sub 97.7%

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

      7. cos-neg 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in lambda2 around 0 79.6%

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

      1. +-commutative 79.6%

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

   7. Simplified 79.6%

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

   if 0.999999999999000022 < (cos.f64 phi2)

   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. Step-by-step derivation

      1. add-cbrt-cube 89.9%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sqrt[3]{\left(\left(\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      2. pow3 89.9%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sqrt[3]{\color{blue}{{\left(\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\right)}^{3}}}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

      3. *-commutative 89.9%

         \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sqrt[3]{{\color{blue}{\left(\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}}^{3}}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   3. Applied egg-rr 89.9%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sqrt[3]{{\left(\sin \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)}^{3}}}}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)} \]

   4. Taylor expanded in phi2 around 0 98.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.999999999999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \end{array} \]

Alternative 5: 88.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.999999999999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi2) 0.999999999999)
     (+ lambda1 (atan2 t_0 (+ (cos phi2) (cos phi1))))
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.999999999999) {
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi2) <= 0.999999999999d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.999999999999) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.999999999999:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + math.cos((lambda2 - lambda1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi2) <= 0.999999999999)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.999999999999)
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda2 - lambda1))));
	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[N[Cos[phi2], $MachinePrecision], 0.999999999999], 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[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.999999999999000022

   1. Initial program 99.0%

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

   2. Taylor expanded in lambda1 around 0 97.7%

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

      1. mul-1-neg 97.7%

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

      2. unsub-neg 97.7%

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

      3. sin-neg 97.7%

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

      4. distribute-rgt-neg-out 97.7%

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

      5. distribute-lft-neg-in 97.7%

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

      6. cancel-sign-sub 97.7%

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

      7. cos-neg 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in lambda2 around 0 79.6%

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

      1. +-commutative 79.6%

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

   7. Simplified 79.6%

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

   if 0.999999999999000022 < (cos.f64 phi2)

   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. Taylor expanded in phi2 around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. sub-neg 98.9%

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

      3. remove-double-neg 98.9%

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

      4. mul-1-neg 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. +-commutative 98.9%

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

      7. cos-neg 98.9%

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

      8. mul-1-neg 98.9%

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

      9. unsub-neg 98.9%

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

   4. Simplified 98.9%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.999999999999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]

Alternative 6: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

2. Final simplification 99.0%

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

Alternative 7: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.936:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_2 \cdot t_0 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{t_0 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= (cos phi1) 0.936)
     (+ lambda1 (atan2 (sin lambda1) (+ (* (cos phi2) t_0) (cos phi1))))
     (+
      lambda1
      (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ t_0 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.936) {
		tmp = lambda1 + atan2(sin(lambda1), ((cos(phi2) * t_0) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (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) :: tmp
    t_0 = cos((lambda1 - lambda2))
    if (cos(phi1) <= 0.936d0) then
        tmp = lambda1 + atan2(sin(lambda1), ((cos(phi2) * t_0) + cos(phi1)))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (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 tmp;
	if (Math.cos(phi1) <= 0.936) {
		tmp = lambda1 + Math.atan2(Math.sin(lambda1), ((Math.cos(phi2) * t_0) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (t_0 + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi1) <= 0.936:
		tmp = lambda1 + math.atan2(math.sin(lambda1), ((math.cos(phi2) * t_0) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (t_0 + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi1) <= 0.936)
		tmp = Float64(lambda1 + atan(sin(lambda1), Float64(Float64(cos(phi2) * t_0) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(t_0 + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi1) <= 0.936)
		tmp = lambda1 + atan2(sin(lambda1), ((cos(phi2) * t_0) + cos(phi1)));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (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]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.936], N[(lambda1 + N[ArcTan[N[Sin[lambda1], $MachinePrecision] / N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.936000000000000054

   1. Initial program 98.1%

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

   2. Taylor expanded in lambda2 around 0 61.7%

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

      1. *-commutative 61.7%

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

   4. Simplified 61.7%

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

   5. Taylor expanded in phi2 around 0 57.0%

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

   if 0.936000000000000054 < (cos.f64 phi1)

   1. Initial program 99.6%

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

   2. Taylor expanded in phi2 around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. sub-neg 80.5%

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

      3. remove-double-neg 80.5%

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

      4. mul-1-neg 80.5%

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

      5. distribute-neg-in 80.5%

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

      6. +-commutative 80.5%

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

      7. cos-neg 80.5%

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

      8. mul-1-neg 80.5%

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

      9. unsub-neg 80.5%

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

   4. Simplified 80.5%

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

   5. Taylor expanded in phi1 around 0 78.6%

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

      1. +-commutative 78.6%

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

      2. cos-neg 78.6%

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

      3. sub-neg 78.6%

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

      4. +-commutative 78.6%

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

      5. distribute-neg-in 78.6%

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

      6. remove-double-neg 78.6%

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

      7. sub-neg 78.6%

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

   7. Simplified 78.6%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.936:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \lambda_1}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]

Alternative 8: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.936:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.936)
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (cos lambda1))))
     (+ lambda1 (atan2 t_0 (+ (cos (- lambda1 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 (cos(phi1) <= 0.936) {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(lambda1)));
	} else {
		tmp = lambda1 + atan2(t_0, (cos((lambda1 - 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 (cos(phi1) <= 0.936d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(lambda1)))
    else
        tmp = lambda1 + atan2(t_0, (cos((lambda1 - 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 (Math.cos(phi1) <= 0.936) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + Math.cos(lambda1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos((lambda1 - lambda2)) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi1) <= 0.936:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + math.cos(lambda1)))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos((lambda1 - 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 (cos(phi1) <= 0.936)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + cos(lambda1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(Float64(lambda1 - 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 (cos(phi1) <= 0.936)
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(lambda1)));
	else
		tmp = lambda1 + atan2(t_0, (cos((lambda1 - 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[N[Cos[phi1], $MachinePrecision], 0.936], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.936000000000000054

   1. Initial program 98.1%

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

   2. Taylor expanded in phi2 around 0 74.2%

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

      1. +-commutative 74.2%

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

      2. sub-neg 74.2%

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

      3. remove-double-neg 74.2%

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

      4. mul-1-neg 74.2%

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

      5. distribute-neg-in 74.2%

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

      6. +-commutative 74.2%

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

      7. cos-neg 74.2%

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

      8. mul-1-neg 74.2%

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

      9. unsub-neg 74.2%

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

   4. Simplified 74.2%

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

   5. Taylor expanded in lambda2 around 0 64.6%

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

      1. cos-neg 64.6%

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

   7. Simplified 64.6%

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

   if 0.936000000000000054 < (cos.f64 phi1)

   1. Initial program 99.6%

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

   2. Taylor expanded in phi2 around 0 80.5%

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

      1. +-commutative 80.5%

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

      2. sub-neg 80.5%

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

      3. remove-double-neg 80.5%

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

      4. mul-1-neg 80.5%

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

      5. distribute-neg-in 80.5%

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

      6. +-commutative 80.5%

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

      7. cos-neg 80.5%

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

      8. mul-1-neg 80.5%

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

      9. unsub-neg 80.5%

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

   4. Simplified 80.5%

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

   5. Taylor expanded in phi1 around 0 78.6%

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

      1. +-commutative 78.6%

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

      2. cos-neg 78.6%

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

      3. sub-neg 78.6%

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

      4. +-commutative 78.6%

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

      5. distribute-neg-in 78.6%

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

      6. remove-double-neg 78.6%

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

      7. sub-neg 78.6%

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

   7. Simplified 78.6%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.936:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]

Alternative 9: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.999999999999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t_0}{\cos \phi_1 + \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi2) 0.999999999999)
     (+ lambda1 (atan2 t_0 (+ (cos phi2) (cos phi1))))
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (cos lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.999999999999) {
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(lambda2)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi2) <= 0.999999999999d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(lambda2)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.999999999999) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + Math.cos(lambda2)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.999999999999:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + math.cos(lambda2)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi2) <= 0.999999999999)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + cos(lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.999999999999)
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.999999999999], 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[Cos[phi1], $MachinePrecision] + N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.999999999999000022

   1. Initial program 99.0%

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

   2. Taylor expanded in lambda1 around 0 97.7%

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

      1. mul-1-neg 97.7%

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

      2. unsub-neg 97.7%

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

      3. sin-neg 97.7%

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

      4. distribute-rgt-neg-out 97.7%

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

      5. distribute-lft-neg-in 97.7%

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

      6. cancel-sign-sub 97.7%

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

      7. cos-neg 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in lambda2 around 0 79.6%

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

      1. +-commutative 79.6%

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

   7. Simplified 79.6%

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

   if 0.999999999999000022 < (cos.f64 phi2)

   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. Taylor expanded in phi2 around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. sub-neg 98.9%

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

      3. remove-double-neg 98.9%

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

      4. mul-1-neg 98.9%

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

      5. distribute-neg-in 98.9%

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

      6. +-commutative 98.9%

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

      7. cos-neg 98.9%

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

      8. mul-1-neg 98.9%

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

      9. unsub-neg 98.9%

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

   4. Simplified 98.9%

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

   5. Taylor expanded in lambda1 around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.999999999999:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \lambda_2}\\ \end{array} \]

Alternative 10: 97.9% 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 99.0%

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

2. Taylor expanded in lambda1 around 0 98.1%

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

   1. cos-neg 67.3%

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

4. Simplified 98.1%

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

5. Final simplification 98.1%

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

Alternative 11: 77.3% 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 99.0%

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

2. Taylor expanded in phi2 around 0 77.8%

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

   1. +-commutative 77.8%

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

   2. sub-neg 77.8%

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

   3. remove-double-neg 77.8%

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

   4. mul-1-neg 77.8%

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

   5. distribute-neg-in 77.8%

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

   6. +-commutative 77.8%

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

   7. cos-neg 77.8%

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

   8. mul-1-neg 77.8%

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

   9. unsub-neg 77.8%

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

4. Simplified 77.8%

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

5. Taylor expanded in lambda1 around 0 77.6%

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

   1. +-commutative 77.6%

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

7. Simplified 77.6%

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

8. Final simplification 77.6%

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

Alternative 12: 67.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos((lambda1 - lambda2)) + 1.0));
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[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in phi2 around 0 77.8%

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

   1. +-commutative 77.8%

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

   2. sub-neg 77.8%

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

   3. remove-double-neg 77.8%

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

   4. mul-1-neg 77.8%

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

   5. distribute-neg-in 77.8%

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

   6. +-commutative 77.8%

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

   7. cos-neg 77.8%

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

   8. mul-1-neg 77.8%

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

   9. unsub-neg 77.8%

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

4. Simplified 77.8%

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

5. Taylor expanded in phi1 around 0 67.5%

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

   1. +-commutative 67.5%

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

   2. cos-neg 67.5%

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

   3. sub-neg 67.5%

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

   4. +-commutative 67.5%

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

   5. distribute-neg-in 67.5%

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

   6. remove-double-neg 67.5%

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

   7. sub-neg 67.5%

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

7. Simplified 67.5%

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

8. Final simplification 67.5%

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

Alternative 13: 62.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda1) + 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((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda1) + 1.0d0))
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(lambda1) + 1.0));
}

Python

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

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda1) + 1.0));
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[lambda1], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in phi2 around 0 77.8%

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

   1. +-commutative 77.8%

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

   2. sub-neg 77.8%

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

   3. remove-double-neg 77.8%

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

   4. mul-1-neg 77.8%

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

   5. distribute-neg-in 77.8%

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

   6. +-commutative 77.8%

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

   7. cos-neg 77.8%

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

   8. mul-1-neg 77.8%

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

   9. unsub-neg 77.8%

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

4. Simplified 77.8%

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

5. Taylor expanded in phi1 around 0 67.5%

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

   1. +-commutative 67.5%

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

   2. cos-neg 67.5%

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

   3. sub-neg 67.5%

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

   4. +-commutative 67.5%

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

   5. distribute-neg-in 67.5%

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

   6. remove-double-neg 67.5%

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

   7. sub-neg 67.5%

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

7. Simplified 67.5%

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

8. Taylor expanded in lambda2 around 0 62.9%

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

   1. +-commutative 62.9%

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

10. Simplified 62.9%

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

11. Final simplification 62.9%

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

Alternative 14: 67.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(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((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + 1.0d0))
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(lambda2) + 1.0));
}

Python

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

Julia

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

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + 1.0));
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[lambda2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in phi2 around 0 77.8%

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

   1. +-commutative 77.8%

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

   2. sub-neg 77.8%

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

   3. remove-double-neg 77.8%

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

   4. mul-1-neg 77.8%

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

   5. distribute-neg-in 77.8%

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

   6. +-commutative 77.8%

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

   7. cos-neg 77.8%

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

   8. mul-1-neg 77.8%

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

   9. unsub-neg 77.8%

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

4. Simplified 77.8%

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

5. Taylor expanded in phi1 around 0 67.5%

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

   1. +-commutative 67.5%

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

   2. cos-neg 67.5%

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

   3. sub-neg 67.5%

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

   4. +-commutative 67.5%

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

   5. distribute-neg-in 67.5%

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

   6. remove-double-neg 67.5%

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

   7. sub-neg 67.5%

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

7. Simplified 67.5%

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

8. Taylor expanded in lambda1 around 0 67.3%

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

   1. cos-neg 67.3%

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

10. Simplified 67.3%

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

11. Final simplification 67.3%

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

Alternative 15: 62.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) 2.0)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0);
}

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((cos(phi2) * sin((lambda1 - lambda2))), 2.0d0)
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))), 2.0);
}

Python

def code(lambda1, lambda2, phi1, phi2):
	return lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), 2.0)

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), 2.0))
end

MATLAB

function tmp = code(lambda1, lambda2, phi1, phi2)
	tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0);
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

2. Taylor expanded in phi2 around 0 77.8%

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

   1. +-commutative 77.8%

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

   2. sub-neg 77.8%

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

   3. remove-double-neg 77.8%

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

   4. mul-1-neg 77.8%

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

   5. distribute-neg-in 77.8%

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

   6. +-commutative 77.8%

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

   7. cos-neg 77.8%

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

   8. mul-1-neg 77.8%

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

   9. unsub-neg 77.8%

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

4. Simplified 77.8%

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

5. Taylor expanded in phi1 around 0 67.5%

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

   1. +-commutative 67.5%

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

   2. cos-neg 67.5%

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

   3. sub-neg 67.5%

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

   4. +-commutative 67.5%

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

   5. distribute-neg-in 67.5%

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

   6. remove-double-neg 67.5%

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

   7. sub-neg 67.5%

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

7. Simplified 67.5%

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

8. Taylor expanded in lambda2 around 0 62.9%

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

   1. +-commutative 62.9%

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

10. Simplified 62.9%

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

11. Taylor expanded in lambda1 around 0 62.7%

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

12. Final simplification 62.7%

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

Reproduce

herbie shell --seed 2024024 
(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)))))))