Midpoint on a great circle

Percentage Accurate: 98.6% → 99.6%

Time: 33.5s

Alternatives: 28

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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2((cos(phi2) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (cos(phi1) + (cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))))
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) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2)))), (Math.cos(phi1) + (Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Add Preprocessing
3. Step-by-step derivation

   1. sin-diff 98.4%

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

4. Applied egg-rr 98.4%

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

   1. cos-diff 99.5%

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

   2. +-commutative 99.5%

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

   3. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+
  lambda1
  (atan2
   (*
    (cos phi2)
    (- (* (sin lambda1) (cos lambda2)) (* (cos lambda1) (sin 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) * cos(lambda2)) - (cos(lambda1) * sin(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) * cos(lambda2)) - (cos(lambda1) * sin(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) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(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) * math.cos(lambda2)) - (math.cos(lambda1) * math.sin(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) * Float64(Float64(sin(lambda1) * cos(lambda2)) - Float64(cos(lambda1) * sin(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) * cos(lambda2)) - (cos(lambda1) * sin(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[(N[(N[Sin[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 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. Add Preprocessing
3. Step-by-step derivation

   1. sin-diff 98.4%

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

4. Applied egg-rr 98.4%

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

Alternative 3: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. sin-diff 98.4%

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

4. Applied egg-rr 98.4%

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

5. Taylor expanded in lambda1 around 0 98.2%

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

   1. cos-neg 98.2%

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

   2. mul-1-neg 98.2%

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

   3. distribute-rgt-neg-in 98.2%

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

   4. sin-neg 98.2%

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

   5. remove-double-neg 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in lambda1 around 0 98.2%

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

   1. +-commutative 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. *-commutative 98.2%

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

   5. distribute-lft-neg-in 98.2%

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

   6. sin-neg 98.2%

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

   7. distribute-rgt-in 98.2%

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

   8. sin-neg 98.2%

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

   9. sub-neg 98.2%

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

10. Simplified 98.2%

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

Alternative 4: 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 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. Step-by-step derivation

   1. cos-neg 98.1%

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

   2. cos-neg 98.1%

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

   3. cos-neg 98.1%

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

   4. +-commutative 98.1%

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

   5. cos-neg 98.1%

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

   6. fma-define 98.1%

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

3. Simplified 98.1%

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

Alternative 5: 96.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -650000 \lor \neg \left(\lambda_2 \leq 6 \cdot 10^{-142}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -650000.0) (not (<= lambda2 6e-142)))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda2)))
     (+ (cos phi1) (* (cos phi2) (cos lambda2)))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (+ (cos phi2) (cos phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -650000.0) || !(lambda2 <= 6e-142)) {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi1) + (cos(phi2) * cos(lambda2))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi2) + cos(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda2 <= (-650000.0d0)) .or. (.not. (lambda2 <= 6d-142))) then
        tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi1) + (cos(phi2) * cos(lambda2))))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi2) + cos(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -650000.0) || !(lambda2 <= 6e-142)) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos(lambda2))));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(phi2) + Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda2 <= -650000.0) or not (lambda2 <= 6e-142):
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin(-lambda2)), (math.cos(phi1) + (math.cos(phi2) * math.cos(lambda2))))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(phi2) + math.cos(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -650000.0) || !(lambda2 <= 6e-142))
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(cos(phi1) + Float64(cos(phi2) * cos(lambda2)))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi2) + cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda2 <= -650000.0) || ~((lambda2 <= 6e-142)))
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi1) + (cos(phi2) * cos(lambda2))));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi2) + cos(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -650000.0], N[Not[LessEqual[lambda2, 6e-142]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -6.5e5 or 6.0000000000000002e-142 < lambda2

   1. Initial program 97.4%

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

   3. Taylor expanded in lambda1 around 0 96.9%

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

      1. cos-neg 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in lambda1 around 0 95.4%

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

   if -6.5e5 < lambda2 < 6.0000000000000002e-142

   1. Initial program 99.2%

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

      1. add-cube-cbrt 99.2%

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

      2. pow3 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in lambda1 around 0 99.2%

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

      1. cos-neg 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in lambda2 around 0 99.2%

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

   9. Taylor expanded in phi1 around inf 99.2%

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

       1. +-commutative 99.2%

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

   11. Simplified 99.2%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -650000 \lor \neg \left(\lambda_2 \leq 6 \cdot 10^{-142}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.99:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.99)
     (+ lambda1 (atan2 t_0 (+ (cos phi2) (cos phi1))))
     (+
      lambda1
      (atan2 t_0 (+ (* (cos phi2) (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) <= 0.99) {
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, ((cos(phi2) * 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) <= 0.99d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, ((cos(phi2) * 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) <= 0.99) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, ((Math.cos(phi2) * 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) <= 0.99:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, ((math.cos(phi2) * 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) <= 0.99)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(Float64(cos(phi2) * 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) <= 0.99)
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, ((cos(phi2) * 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], 0.99], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.98999999999999999

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.0%

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

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in lambda1 around 0 98.0%

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

      1. cos-neg 98.1%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in lambda2 around 0 82.9%

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

   9. Taylor expanded in phi1 around inf 82.9%

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

       1. +-commutative 82.9%

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

   11. Simplified 82.9%

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

   if 0.98999999999999999 < (cos.f64 phi1)

   1. Initial program 98.2%

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

   3. Taylor expanded in phi1 around 0 97.0%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.99:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.996:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.996)
     (+ lambda1 (atan2 t_0 (+ (cos phi2) (cos phi1))))
     (+ lambda1 (atan2 t_0 (+ (* (cos phi2) (cos lambda2)) 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.996) {
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, ((cos(phi2) * cos(lambda2)) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi1) <= 0.996d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, ((cos(phi2) * cos(lambda2)) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi1) <= 0.996) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, ((Math.cos(phi2) * Math.cos(lambda2)) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi1) <= 0.996:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, ((math.cos(phi2) * math.cos(lambda2)) + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= 0.996)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(Float64(cos(phi2) * cos(lambda2)) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi1) <= 0.996)
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, ((cos(phi2) * cos(lambda2)) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.996], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.996

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-cube-cbrt 98.0%

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

      2. pow3 98.0%

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

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in lambda1 around 0 98.0%

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

      1. cos-neg 98.1%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in lambda2 around 0 82.6%

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

   9. Taylor expanded in phi1 around inf 82.6%

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

       1. +-commutative 82.6%

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

   11. Simplified 82.6%

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

   if 0.996 < (cos.f64 phi1)

   1. Initial program 98.2%

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

   3. Taylor expanded in lambda1 around 0 97.5%

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

      1. cos-neg 97.5%

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

   5. Simplified 97.5%

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

   6. Taylor expanded in phi1 around 0 97.5%

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

      1. *-commutative 97.5%

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

   8. Simplified 97.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.996:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.852:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi2) 0.852)
     (+ lambda1 (atan2 (* (cos phi2) t_0) (+ (cos phi2) (cos phi1))))
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (* (cos phi2) (cos lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi2) <= 0.852) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * 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 = sin((lambda1 - lambda2))
    if (cos(phi2) <= 0.852d0) then
        tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * 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.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi2) <= 0.852) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * t_0), (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + (Math.cos(phi2) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi2) <= 0.852:
		tmp = lambda1 + math.atan2((math.cos(phi2) * t_0), (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + (math.cos(phi2) * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.852)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi2) <= 0.852)
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.852], 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[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.85199999999999998

   1. Initial program 97.8%

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

      1. add-cube-cbrt 97.8%

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

      2. pow3 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in lambda1 around 0 97.8%

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

      1. cos-neg 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in lambda2 around 0 76.3%

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

   9. Taylor expanded in phi1 around inf 76.3%

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

       1. +-commutative 76.3%

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

   11. Simplified 76.3%

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

   if 0.85199999999999998 < (cos.f64 phi2)

   1. Initial program 98.4%

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

   3. Taylor expanded in lambda1 around 0 97.8%

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

      1. cos-neg 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in phi2 around 0 93.7%

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

Alternative 9: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(-\lambda_2\right)\\ \mathbf{if}\;\cos \phi_2 \leq 0.94:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda2))))
   (if (<= (cos phi2) 0.94)
     (+ lambda1 (atan2 (* (cos phi2) t_0) (+ (cos phi2) (cos phi1))))
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (* (cos phi2) (cos lambda2))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(-lambda2);
	double tmp;
	if (cos(phi2) <= 0.94) {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * 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 = sin(-lambda2)
    if (cos(phi2) <= 0.94d0) then
        tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * 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.sin(-lambda2);
	double tmp;
	if (Math.cos(phi2) <= 0.94) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * t_0), (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + (Math.cos(phi2) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(-lambda2)
	tmp = 0
	if math.cos(phi2) <= 0.94:
		tmp = lambda1 + math.atan2((math.cos(phi2) * t_0), (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + (math.cos(phi2) * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-lambda2))
	tmp = 0.0
	if (cos(phi2) <= 0.94)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(-lambda2);
	tmp = 0.0;
	if (cos(phi2) <= 0.94)
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[(-lambda2)], $MachinePrecision]}, If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.94], 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[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.93999999999999995

   1. Initial program 97.6%

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

   3. Taylor expanded in lambda1 around 0 97.6%

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

      1. cos-neg 97.6%

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

   5. Simplified 97.6%

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

   6. Taylor expanded in lambda1 around 0 88.4%

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

   7. Taylor expanded in lambda2 around 0 67.1%

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

   if 0.93999999999999995 < (cos.f64 phi2)

   1. Initial program 98.6%

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

   3. Taylor expanded in lambda1 around 0 98.0%

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

      1. cos-neg 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in lambda1 around 0 87.5%

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

   7. Taylor expanded in phi2 around 0 84.5%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.94:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.95:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(-\lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos phi2) 0.95)
   (+
    lambda1
    (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi2) 1.0)))
   (+
    lambda1
    (atan2 (sin (- lambda2)) (+ (cos phi1) (* (cos phi2) (cos lambda2)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos(phi2) <= 0.95) {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi2) + 1.0));
	} else {
		tmp = lambda1 + atan2(sin(-lambda2), (cos(phi1) + (cos(phi2) * 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) :: tmp
    if (cos(phi2) <= 0.95d0) then
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi2) + 1.0d0))
    else
        tmp = lambda1 + atan2(sin(-lambda2), (cos(phi1) + (cos(phi2) * cos(lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.cos(phi2) <= 0.95) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(phi2) + 1.0));
	} else {
		tmp = lambda1 + Math.atan2(Math.sin(-lambda2), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.cos(phi2) <= 0.95:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(phi2) + 1.0))
	else:
		tmp = lambda1 + math.atan2(math.sin(-lambda2), (math.cos(phi1) + (math.cos(phi2) * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(phi2) <= 0.95)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi2) + 1.0)));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(-lambda2)), Float64(cos(phi1) + Float64(cos(phi2) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (cos(phi2) <= 0.95)
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi2) + 1.0));
	else
		tmp = lambda1 + atan2(sin(-lambda2), (cos(phi1) + (cos(phi2) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.95], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[(-lambda2)], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.94999999999999996

   1. Initial program 97.6%

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

      1. add-cube-cbrt 97.6%

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

      2. pow3 97.6%

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in lambda1 around 0 97.6%

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

      1. cos-neg 97.6%

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

   7. Simplified 97.6%

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

   8. Taylor expanded in lambda2 around 0 76.5%

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

   9. Taylor expanded in phi1 around 0 65.0%

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

       1. +-commutative 65.0%

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

   11. Simplified 65.0%

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

   if 0.94999999999999996 < (cos.f64 phi2)

   1. Initial program 98.6%

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

   3. Taylor expanded in lambda1 around 0 98.0%

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

      1. cos-neg 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in lambda1 around 0 87.4%

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

   7. Taylor expanded in phi2 around 0 85.0%

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

Alternative 11: 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]

Derivation

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. Add Preprocessing
3. Add Preprocessing

Alternative 12: 97.8% 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 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. Add Preprocessing

3. Taylor expanded in lambda1 around 0 97.8%

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

   1. cos-neg 97.8%

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

5. Simplified 97.8%

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

Alternative 13: 87.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.8 \cdot 10^{+31} \lor \neg \left(\lambda_2 \leq 2.2 \cdot 10^{-14}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -1.8e+31) (not (<= lambda2 2.2e-14)))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda2)))
     (+ (* (cos phi2) (cos lambda2)) 1.0)))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (+ (cos phi2) (cos phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -1.8e+31) || !(lambda2 <= 2.2e-14)) {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), ((cos(phi2) * cos(lambda2)) + 1.0));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi2) + cos(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((lambda2 <= (-1.8d+31)) .or. (.not. (lambda2 <= 2.2d-14))) then
        tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), ((cos(phi2) * cos(lambda2)) + 1.0d0))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi2) + cos(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -1.8e+31) || !(lambda2 <= 2.2e-14)) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), ((Math.cos(phi2) * Math.cos(lambda2)) + 1.0));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(phi2) + Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda2 <= -1.8e+31) or not (lambda2 <= 2.2e-14):
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin(-lambda2)), ((math.cos(phi2) * math.cos(lambda2)) + 1.0))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(phi2) + math.cos(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -1.8e+31) || !(lambda2 <= 2.2e-14))
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(Float64(cos(phi2) * cos(lambda2)) + 1.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi2) + cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda2 <= -1.8e+31) || ~((lambda2 <= 2.2e-14)))
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), ((cos(phi2) * cos(lambda2)) + 1.0));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi2) + cos(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -1.8e+31], N[Not[LessEqual[lambda2, 2.2e-14]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.79999999999999998e31 or 2.2000000000000001e-14 < lambda2

   1. Initial program 97.0%

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

      1. add-cube-cbrt 96.7%

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

      2. pow3 96.7%

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

   4. Applied egg-rr 96.7%

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

   5. Taylor expanded in lambda1 around 0 96.7%

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

      1. cos-neg 97.0%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in lambda1 around 0 96.7%

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

   9. Taylor expanded in phi1 around 0 79.6%

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

       1. *-commutative 79.6%

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

   11. Simplified 79.6%

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

   if -1.79999999999999998e31 < lambda2 < 2.2000000000000001e-14

   1. Initial program 99.2%

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

      1. add-cube-cbrt 99.2%

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

      2. pow3 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in lambda1 around 0 98.6%

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

      1. cos-neg 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in lambda2 around 0 96.2%

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

   9. Taylor expanded in phi1 around inf 96.2%

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

       1. +-commutative 96.2%

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

   11. Simplified 96.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.8 \cdot 10^{+31} \lor \neg \left(\lambda_2 \leq 2.2 \cdot 10^{-14}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_2 \cdot \cos \lambda_2 + 1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -650000 \lor \neg \left(\lambda_2 \leq 1.8 \cdot 10^{-171}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \lambda_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= lambda2 -650000.0) (not (<= lambda2 1.8e-171)))
   (+
    lambda1
    (atan2 (* (cos phi2) (sin (- lambda2))) (+ (cos lambda2) (cos phi1))))
   (+
    lambda1
    (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ (cos phi1) 1.0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -650000.0) || !(lambda2 <= 1.8e-171)) {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(lambda2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + 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) :: tmp
    if ((lambda2 <= (-650000.0d0)) .or. (.not. (lambda2 <= 1.8d-171))) then
        tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(lambda2) + cos(phi1)))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda2 <= -650000.0) || !(lambda2 <= 1.8e-171)) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (Math.cos(lambda2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(phi1) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda2 <= -650000.0) or not (lambda2 <= 1.8e-171):
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin(-lambda2)), (math.cos(lambda2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(phi1) + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((lambda2 <= -650000.0) || !(lambda2 <= 1.8e-171))
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(cos(lambda2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(phi1) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda2 <= -650000.0) || ~((lambda2 <= 1.8e-171)))
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(lambda2) + cos(phi1)));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(phi1) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[lambda2, -650000.0], N[Not[LessEqual[lambda2, 1.8e-171]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda2], $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[(N[Cos[phi1], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -6.5e5 or 1.80000000000000002e-171 < lambda2

   1. Initial program 97.5%

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

   3. Taylor expanded in lambda1 around 0 97.0%

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

      1. cos-neg 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in lambda1 around 0 95.0%

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

   7. Taylor expanded in phi2 around 0 75.1%

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

   if -6.5e5 < lambda2 < 1.80000000000000002e-171

   1. Initial program 99.2%

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

      1. add-cube-cbrt 99.2%

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

      2. pow3 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in lambda1 around 0 99.2%

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

      1. cos-neg 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in lambda2 around 0 99.2%

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

   9. Taylor expanded in phi2 around 0 79.2%

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

       1. +-commutative 79.2%

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

   11. Simplified 79.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -650000 \lor \neg \left(\lambda_2 \leq 1.8 \cdot 10^{-171}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \lambda_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 8.5e-7)
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (+ (cos lambda2) (cos phi1))))
   (+
    lambda1
    (atan2 (* (cos phi2) (sin (- lambda2))) (+ (cos phi2) (cos phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.5e-7) {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi2) + cos(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 8.5d-7) then
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + cos(phi1)))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi2) + cos(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.5e-7) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(lambda2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (Math.cos(phi2) + Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 8.5e-7:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(lambda2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin(-lambda2)), (math.cos(phi2) + math.cos(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 8.5e-7)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(lambda2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(cos(phi2) + cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 8.5e-7)
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + cos(phi1)));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi2) + cos(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 8.5e-7], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.50000000000000014e-7

   1. Initial program 98.0%

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

   3. Taylor expanded in phi2 around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. sub-neg 82.3%

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

      3. remove-double-neg 82.3%

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

      4. mul-1-neg 82.3%

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

      5. distribute-neg-in 82.3%

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

      6. +-commutative 82.3%

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

      7. cos-neg 82.3%

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

      8. mul-1-neg 82.3%

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

      9. unsub-neg 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in lambda1 around 0 82.0%

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

   if 8.50000000000000014e-7 < phi2

   1. Initial program 98.5%

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

   3. Taylor expanded in lambda1 around 0 98.5%

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

      1. cos-neg 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in lambda1 around 0 85.6%

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

   7. Taylor expanded in lambda2 around 0 64.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.5 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.00085:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.00085)
   (+
    lambda1
    (atan2
     (sin (- lambda1 lambda2))
     (+ (cos phi1) (* (cos phi2) (cos lambda2)))))
   (+
    lambda1
    (atan2 (* (cos phi2) (sin (- lambda2))) (+ (cos phi2) (cos phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.00085) {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + (cos(phi2) * cos(lambda2))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi2) + cos(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 0.00085d0) then
        tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + (cos(phi2) * cos(lambda2))))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi2) + cos(phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.00085) {
		tmp = lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos(lambda2))));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin(-lambda2)), (Math.cos(phi2) + Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.00085:
		tmp = lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos(phi1) + (math.cos(phi2) * math.cos(lambda2))))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin(-lambda2)), (math.cos(phi2) + math.cos(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.00085)
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(phi1) + Float64(cos(phi2) * cos(lambda2)))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(-lambda2))), Float64(cos(phi2) + cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.00085)
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(phi1) + (cos(phi2) * cos(lambda2))));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin(-lambda2)), (cos(phi2) + cos(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.00085], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[(-lambda2)], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.49999999999999953e-4

   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. Add Preprocessing

   3. Taylor expanded in lambda1 around 0 97.7%

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

      1. cos-neg 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in phi2 around 0 81.1%

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

   if 8.49999999999999953e-4 < phi2

   1. Initial program 98.4%

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

   3. Taylor expanded in lambda1 around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in lambda1 around 0 85.1%

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

   7. Taylor expanded in lambda2 around 0 64.6%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.00085:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(-\lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 70.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.44:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\cos \lambda_2 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= (cos phi1) 0.44)
     (+ lambda1 (atan2 t_0 (+ (cos phi2) (cos phi1))))
     (+ lambda1 (atan2 (* (cos phi2) t_0) (+ (cos lambda2) 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.44) {
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(lambda2) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((lambda1 - lambda2))
    if (cos(phi1) <= 0.44d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(lambda2) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi1) <= 0.44) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * t_0), (Math.cos(lambda2) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi1) <= 0.44:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * t_0), (math.cos(lambda2) + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (cos(phi1) <= 0.44)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(cos(lambda2) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi1) <= 0.44)
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(lambda2) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.44], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.440000000000000002

   1. Initial program 97.7%

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

      1. add-cube-cbrt 97.7%

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

      2. pow3 97.7%

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

   4. Applied egg-rr 97.7%

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

   5. Taylor expanded in lambda1 around 0 97.7%

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

      1. cos-neg 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in lambda2 around 0 80.7%

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

   9. Taylor expanded in phi2 around 0 73.0%

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

   if 0.440000000000000002 < (cos.f64 phi1)

   1. Initial program 98.4%

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

   3. Taylor expanded in phi2 around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. sub-neg 75.8%

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

      3. remove-double-neg 75.8%

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

      4. mul-1-neg 75.8%

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

      5. distribute-neg-in 75.8%

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

      6. +-commutative 75.8%

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

      7. cos-neg 75.8%

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

      8. mul-1-neg 75.8%

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

      9. unsub-neg 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in phi1 around 0 72.8%

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

      1. +-commutative 72.8%

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

   8. Simplified 72.8%

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

   9. Taylor expanded in lambda1 around 0 72.3%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.44:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.54:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\cos \phi_1 + \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos phi1) 0.54)
   (+
    lambda1
    (atan2 lambda1 (+ (cos phi1) (+ (cos lambda2) (* lambda1 (sin lambda2))))))
   (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) 2.0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos(phi1) <= 0.54) {
		tmp = lambda1 + atan2(lambda1, (cos(phi1) + (cos(lambda2) + (lambda1 * sin(lambda2)))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0);
	}
	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) :: tmp
    if (cos(phi1) <= 0.54d0) then
        tmp = lambda1 + atan2(lambda1, (cos(phi1) + (cos(lambda2) + (lambda1 * sin(lambda2)))))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.cos(phi1) <= 0.54) {
		tmp = lambda1 + Math.atan2(lambda1, (Math.cos(phi1) + (Math.cos(lambda2) + (lambda1 * Math.sin(lambda2)))));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), 2.0);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.cos(phi1) <= 0.54:
		tmp = lambda1 + math.atan2(lambda1, (math.cos(phi1) + (math.cos(lambda2) + (lambda1 * math.sin(lambda2)))))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), 2.0)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(phi1) <= 0.54)
		tmp = Float64(lambda1 + atan(lambda1, Float64(cos(phi1) + Float64(cos(lambda2) + Float64(lambda1 * sin(lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (cos(phi1) <= 0.54)
		tmp = lambda1 + atan2(lambda1, (cos(phi1) + (cos(lambda2) + (lambda1 * sin(lambda2)))));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.54], N[(lambda1 + N[ArcTan[lambda1 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] + N[(lambda1 * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.54000000000000004

   1. Initial program 97.7%

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

   3. Taylor expanded in phi2 around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. sub-neg 79.8%

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

      3. remove-double-neg 79.8%

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

      4. mul-1-neg 79.8%

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

      5. distribute-neg-in 79.8%

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

      6. +-commutative 79.8%

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

      7. cos-neg 79.8%

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

      8. mul-1-neg 79.8%

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

      9. unsub-neg 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in lambda1 around 0 80.1%

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

      1. +-commutative 80.1%

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

      2. sin-neg 80.1%

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

      3. unsub-neg 80.1%

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

      4. *-commutative 80.1%

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

      5. cos-neg 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in lambda2 around 0 61.0%

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

       1. *-commutative 61.0%

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

   11. Simplified 61.0%

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

   12. Taylor expanded in phi2 around 0 61.4%

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

   13. Taylor expanded in lambda1 around 0 61.4%

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

   if 0.54000000000000004 < (cos.f64 phi1)

   1. Initial program 98.5%

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

   3. Taylor expanded in phi2 around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. sub-neg 76.0%

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

      3. remove-double-neg 76.0%

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

      4. mul-1-neg 76.0%

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

      5. distribute-neg-in 76.0%

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

      6. +-commutative 76.0%

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

      7. cos-neg 76.0%

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

      8. mul-1-neg 76.0%

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

      9. unsub-neg 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in phi1 around 0 74.0%

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

      1. +-commutative 74.0%

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

   8. Simplified 74.0%

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

   9. Taylor expanded in lambda2 around 0 66.4%

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

       1. cos-neg 66.4%

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

   11. Simplified 66.4%

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

   12. Taylor expanded in lambda1 around 0 65.9%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.54:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\cos \phi_1 + \left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 68.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{-35}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot t\_0}{\cos \phi_2 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (- lambda1 lambda2))))
   (if (<= phi2 4.8e-35)
     (+ lambda1 (atan2 t_0 (+ (cos phi2) (cos phi1))))
     (+ lambda1 (atan2 (* (cos phi2) t_0) (+ (cos phi2) 1.0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 4.8e-35) {
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + 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 = sin((lambda1 - lambda2))
    if (phi2 <= 4.8d-35) then
        tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)))
    else
        tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi2 <= 4.8e-35) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * t_0), (Math.cos(phi2) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((lambda1 - lambda2))
	tmp = 0
	if phi2 <= 4.8e-35:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * t_0), (math.cos(phi2) + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= 4.8e-35)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * t_0), Float64(cos(phi2) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= 4.8e-35)
		tmp = lambda1 + atan2(t_0, (cos(phi2) + cos(phi1)));
	else
		tmp = lambda1 + atan2((cos(phi2) * t_0), (cos(phi2) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 4.8e-35], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] / N[(N[Cos[phi2], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.8000000000000003e-35

   1. Initial program 98.0%

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

      1. add-cube-cbrt 97.9%

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

      2. pow3 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in lambda1 around 0 97.5%

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

      1. cos-neg 97.6%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in lambda2 around 0 79.7%

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

   9. Taylor expanded in phi2 around 0 71.1%

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

   if 4.8000000000000003e-35 < phi2

   1. Initial program 98.5%

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

      1. add-cube-cbrt 98.5%

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

      2. pow3 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in lambda1 around 0 98.4%

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

      1. cos-neg 98.5%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in lambda2 around 0 78.4%

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

   9. Taylor expanded in phi1 around 0 66.4%

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

       1. +-commutative 66.4%

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

   11. Simplified 66.4%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{-35}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 62.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.54:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\cos \lambda_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos phi1) 0.54)
   (+ lambda1 (atan2 lambda1 (+ (cos lambda2) (cos phi1))))
   (+ lambda1 (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) 2.0))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos(phi1) <= 0.54) {
		tmp = lambda1 + atan2(lambda1, (cos(lambda2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0);
	}
	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) :: tmp
    if (cos(phi1) <= 0.54d0) then
        tmp = lambda1 + atan2(lambda1, (cos(lambda2) + cos(phi1)))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.cos(phi1) <= 0.54) {
		tmp = lambda1 + Math.atan2(lambda1, (Math.cos(lambda2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), 2.0);
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.cos(phi1) <= 0.54:
		tmp = lambda1 + math.atan2(lambda1, (math.cos(lambda2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), 2.0)
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(phi1) <= 0.54)
		tmp = Float64(lambda1 + atan(lambda1, Float64(cos(lambda2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (cos(phi1) <= 0.54)
		tmp = lambda1 + atan2(lambda1, (cos(lambda2) + cos(phi1)));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.54], N[(lambda1 + N[ArcTan[lambda1 / N[(N[Cos[lambda2], $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] / 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.54000000000000004

   1. Initial program 97.7%

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

   3. Taylor expanded in phi2 around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. sub-neg 79.8%

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

      3. remove-double-neg 79.8%

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

      4. mul-1-neg 79.8%

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

      5. distribute-neg-in 79.8%

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

      6. +-commutative 79.8%

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

      7. cos-neg 79.8%

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

      8. mul-1-neg 79.8%

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

      9. unsub-neg 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in lambda1 around 0 80.1%

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

      1. +-commutative 80.1%

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

      2. sin-neg 80.1%

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

      3. unsub-neg 80.1%

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

      4. *-commutative 80.1%

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

      5. cos-neg 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in lambda2 around 0 61.0%

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

       1. *-commutative 61.0%

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

   11. Simplified 61.0%

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

   12. Taylor expanded in phi2 around 0 61.4%

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

   13. Taylor expanded in lambda1 around 0 61.4%

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

   if 0.54000000000000004 < (cos.f64 phi1)

   1. Initial program 98.5%

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

   3. Taylor expanded in phi2 around 0 76.0%

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

      1. +-commutative 76.0%

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

      2. sub-neg 76.0%

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

      3. remove-double-neg 76.0%

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

      4. mul-1-neg 76.0%

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

      5. distribute-neg-in 76.0%

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

      6. +-commutative 76.0%

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

      7. cos-neg 76.0%

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

      8. mul-1-neg 76.0%

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

      9. unsub-neg 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in phi1 around 0 74.0%

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

      1. +-commutative 74.0%

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

   8. Simplified 74.0%

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

   9. Taylor expanded in lambda2 around 0 66.4%

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

       1. cos-neg 66.4%

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

   11. Simplified 66.4%

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

   12. Taylor expanded in lambda1 around 0 65.9%

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

Alternative 21: 61.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.44:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\cos \lambda_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos phi1) 0.44)
   (+ lambda1 (atan2 lambda1 (+ (cos lambda2) (cos phi1))))
   (+ lambda1 (atan2 (sin (- lambda1 lambda2)) (+ (cos lambda1) 1.0)))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos(phi1) <= 0.44) {
		tmp = lambda1 + atan2(lambda1, (cos(lambda2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda1) + 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) :: tmp
    if (cos(phi1) <= 0.44d0) then
        tmp = lambda1 + atan2(lambda1, (cos(lambda2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda1) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.cos(phi1) <= 0.44) {
		tmp = lambda1 + Math.atan2(lambda1, (Math.cos(lambda2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(Math.sin((lambda1 - lambda2)), (Math.cos(lambda1) + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.cos(phi1) <= 0.44:
		tmp = lambda1 + math.atan2(lambda1, (math.cos(lambda2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(math.sin((lambda1 - lambda2)), (math.cos(lambda1) + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(phi1) <= 0.44)
		tmp = Float64(lambda1 + atan(lambda1, Float64(cos(lambda2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(sin(Float64(lambda1 - lambda2)), Float64(cos(lambda1) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (cos(phi1) <= 0.44)
		tmp = lambda1 + atan2(lambda1, (cos(lambda2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(sin((lambda1 - lambda2)), (cos(lambda1) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.44], N[(lambda1 + N[ArcTan[lambda1 / N[(N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] / N[(N[Cos[lambda1], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.440000000000000002

   1. Initial program 97.7%

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

   3. Taylor expanded in phi2 around 0 80.4%

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

      1. +-commutative 80.4%

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

      2. sub-neg 80.4%

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

      3. remove-double-neg 80.4%

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

      4. mul-1-neg 80.4%

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

      5. distribute-neg-in 80.4%

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

      6. +-commutative 80.4%

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

      7. cos-neg 80.4%

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

      8. mul-1-neg 80.4%

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

      9. unsub-neg 80.4%

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

   5. Simplified 80.4%

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

   6. Taylor expanded in lambda1 around 0 80.7%

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

      1. +-commutative 80.7%

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

      2. sin-neg 80.7%

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

      3. unsub-neg 80.7%

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

      4. *-commutative 80.7%

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

      5. cos-neg 80.7%

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

   8. Simplified 80.7%

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

   9. Taylor expanded in lambda2 around 0 61.6%

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

       1. *-commutative 61.6%

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

   11. Simplified 61.6%

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

   12. Taylor expanded in phi2 around 0 62.0%

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

   13. Taylor expanded in lambda1 around 0 62.0%

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

   if 0.440000000000000002 < (cos.f64 phi1)

   1. Initial program 98.4%

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

   3. Taylor expanded in phi2 around 0 75.8%

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

      1. +-commutative 75.8%

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

      2. sub-neg 75.8%

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

      3. remove-double-neg 75.8%

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

      4. mul-1-neg 75.8%

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

      5. distribute-neg-in 75.8%

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

      6. +-commutative 75.8%

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

      7. cos-neg 75.8%

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

      8. mul-1-neg 75.8%

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

      9. unsub-neg 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in phi1 around 0 72.8%

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

      1. +-commutative 72.8%

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

   8. Simplified 72.8%

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

   9. Taylor expanded in lambda2 around 0 65.6%

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

       1. cos-neg 65.6%

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

   11. Simplified 65.6%

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

   12. Taylor expanded in phi2 around 0 63.8%

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

Alternative 22: 66.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. add-cube-cbrt 98.0%

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

   2. pow3 98.0%

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in lambda1 around 0 97.7%

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

   1. cos-neg 97.8%

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

7. Simplified 97.7%

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

8. Taylor expanded in lambda2 around 0 79.4%

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

9. Taylor expanded in phi2 around 0 68.1%

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

10. Final simplification 68.1%

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

Alternative 23: 56.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in phi2 around 0 77.6%

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

   1. +-commutative 77.6%

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

   2. sub-neg 77.6%

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

   3. remove-double-neg 77.6%

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

   4. mul-1-neg 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. +-commutative 77.6%

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

   7. cos-neg 77.6%

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

   8. mul-1-neg 77.6%

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

   9. unsub-neg 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in lambda1 around 0 77.4%

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

   1. +-commutative 77.4%

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

   2. sin-neg 77.4%

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

   3. unsub-neg 77.4%

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

   4. *-commutative 77.4%

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

   5. cos-neg 77.4%

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

8. Simplified 77.4%

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

9. Taylor expanded in lambda2 around 0 59.7%

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

    1. *-commutative 59.7%

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

11. Simplified 59.7%

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

12. Taylor expanded in phi2 around 0 59.8%

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

13. Final simplification 59.8%

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

Alternative 24: 56.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in phi2 around 0 77.6%

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

   1. +-commutative 77.6%

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

   2. sub-neg 77.6%

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

   3. remove-double-neg 77.6%

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

   4. mul-1-neg 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. +-commutative 77.6%

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

   7. cos-neg 77.6%

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

   8. mul-1-neg 77.6%

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

   9. unsub-neg 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in lambda1 around 0 77.4%

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

   1. +-commutative 77.4%

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

   2. sin-neg 77.4%

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

   3. unsub-neg 77.4%

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

   4. *-commutative 77.4%

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

   5. cos-neg 77.4%

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

8. Simplified 77.4%

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

9. Taylor expanded in lambda2 around 0 59.7%

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

    1. *-commutative 59.7%

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

11. Simplified 59.7%

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

12. Taylor expanded in phi2 around 0 59.8%

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

13. Taylor expanded in lambda1 around 0 59.8%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\color{blue}{\cos \lambda_2} + \cos \phi_1} \]
14. Add Preprocessing

Alternative 25: 53.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = lambda1 + atan2(lambda1, (1.0d0 + cos((lambda2 - lambda1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in phi2 around 0 77.6%

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

   1. +-commutative 77.6%

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

   2. sub-neg 77.6%

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

   3. remove-double-neg 77.6%

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

   4. mul-1-neg 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. +-commutative 77.6%

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

   7. cos-neg 77.6%

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

   8. mul-1-neg 77.6%

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

   9. unsub-neg 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in lambda1 around 0 77.4%

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

   1. +-commutative 77.4%

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

   2. sin-neg 77.4%

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

   3. unsub-neg 77.4%

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

   4. *-commutative 77.4%

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

   5. cos-neg 77.4%

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

8. Simplified 77.4%

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

9. Taylor expanded in lambda2 around 0 59.7%

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

    1. *-commutative 59.7%

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

11. Simplified 59.7%

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

12. Taylor expanded in phi2 around 0 59.8%

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

13. Taylor expanded in phi1 around 0 56.3%

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

    1. +-commutative 67.3%

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

15. Simplified 56.3%

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

16. Final simplification 56.3%

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

Alternative 26: 53.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2(lambda1, (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(lambda1, (cos(lambda1) + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in phi2 around 0 77.6%

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

   1. +-commutative 77.6%

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

   2. sub-neg 77.6%

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

   3. remove-double-neg 77.6%

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

   4. mul-1-neg 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. +-commutative 77.6%

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

   7. cos-neg 77.6%

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

   8. mul-1-neg 77.6%

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

   9. unsub-neg 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in lambda1 around 0 77.4%

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

   1. +-commutative 77.4%

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

   2. sin-neg 77.4%

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

   3. unsub-neg 77.4%

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

   4. *-commutative 77.4%

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

   5. cos-neg 77.4%

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

8. Simplified 77.4%

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

9. Taylor expanded in lambda2 around 0 59.7%

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

    1. *-commutative 59.7%

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

11. Simplified 59.7%

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

12. Taylor expanded in phi2 around 0 59.8%

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

13. Taylor expanded in phi1 around 0 56.3%

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

    1. +-commutative 67.3%

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

15. Simplified 56.3%

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

16. Taylor expanded in lambda2 around 0 56.3%

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

    1. cos-neg 62.7%

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

18. Simplified 56.3%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\color{blue}{\cos \lambda_1} + 1} \]
19. Add Preprocessing

Alternative 27: 53.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\lambda_1}{2} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (+ lambda1 (atan2 lambda1 2.0)))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + atan2(lambda1, 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(lambda1, 2.0d0)
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 + Math.atan2(lambda1, 2.0);
}

Python

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

Julia

function code(lambda1, lambda2, phi1, phi2)
	return Float64(lambda1 + atan(lambda1, 2.0))
end

MATLAB

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

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 + N[ArcTan[lambda1 / 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in phi2 around 0 77.6%

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

   1. +-commutative 77.6%

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

   2. sub-neg 77.6%

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

   3. remove-double-neg 77.6%

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

   4. mul-1-neg 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. +-commutative 77.6%

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

   7. cos-neg 77.6%

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

   8. mul-1-neg 77.6%

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

   9. unsub-neg 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in lambda1 around 0 77.4%

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

   1. +-commutative 77.4%

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

   2. sin-neg 77.4%

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

   3. unsub-neg 77.4%

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

   4. *-commutative 77.4%

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

   5. cos-neg 77.4%

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

8. Simplified 77.4%

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

9. Taylor expanded in lambda2 around 0 59.7%

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

    1. *-commutative 59.7%

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

11. Simplified 59.7%

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

12. Taylor expanded in phi2 around 0 59.8%

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

13. Taylor expanded in phi1 around 0 56.3%

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

    1. +-commutative 67.3%

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

15. Simplified 56.3%

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

16. Taylor expanded in lambda2 around 0 56.3%

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

    1. cos-neg 62.7%

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

18. Simplified 56.3%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\color{blue}{\cos \lambda_1} + 1} \]

19. Taylor expanded in lambda1 around 0 56.3%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\lambda_1}{\color{blue}{2}} \]
20. Add Preprocessing

Alternative 28: 51.4% accurate, 614.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Add Preprocessing

3. Taylor expanded in phi2 around 0 77.6%

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

   1. +-commutative 77.6%

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

   2. sub-neg 77.6%

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

   3. remove-double-neg 77.6%

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

   4. mul-1-neg 77.6%

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

   5. distribute-neg-in 77.6%

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

   6. +-commutative 77.6%

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

   7. cos-neg 77.6%

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

   8. mul-1-neg 77.6%

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

   9. unsub-neg 77.6%

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

5. Simplified 77.6%

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

6. Taylor expanded in lambda1 around 0 77.4%

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

   1. +-commutative 77.4%

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

   2. sin-neg 77.4%

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

   3. unsub-neg 77.4%

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

   4. *-commutative 77.4%

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

   5. cos-neg 77.4%

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

8. Simplified 77.4%

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

9. Taylor expanded in lambda2 around 0 59.7%

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

    1. *-commutative 59.7%

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

11. Simplified 59.7%

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

12. Taylor expanded in phi2 around 0 59.8%

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

13. Taylor expanded in lambda1 around inf 54.2%

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

Reproduce

herbie shell --seed 2024105 
(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)))))))