Midpoint on a great circle

Percentage Accurate: 98.7% → 99.7%

Time: 21.3s

Alternatives: 22

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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)))), fma(cos(phi2), fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))), cos(phi1))))
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[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. sin-diff 98.9%

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

4. Applied egg-rr 98.9%

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

   1. cos-diff 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in phi1 around inf 99.7%

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

   1. +-commutative 99.7%

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

   2. fma-define 99.7%

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

   3. fma-define 99.7%

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

9. Simplified 99.7%

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

Alternative 2: 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 98.9%

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

   1. sin-diff 98.9%

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

4. Applied egg-rr 98.9%

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

   1. cos-diff 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

Alternative 3: 99.0% accurate, 0.5× speedup

Math

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

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 lambda2) (cos lambda1)) (* lambda1 (sin 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(lambda2) * cos(lambda1)) + (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) * ((sin(lambda1) * cos(lambda2)) - (cos(lambda1) * sin(lambda2)))), (cos(phi1) + (cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (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) * ((Math.sin(lambda1) * Math.cos(lambda2)) - (Math.cos(lambda1) * Math.sin(lambda2)))), (Math.cos(phi1) + (Math.cos(phi2) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (lambda1 * Math.sin(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(lambda2) * math.cos(lambda1)) + (lambda1 * math.sin(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) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(lambda1 * sin(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(lambda2) * cos(lambda1)) + (lambda1 * sin(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[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(lambda1 * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.9%

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

   1. sin-diff 98.9%

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

4. Applied egg-rr 98.9%

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

   1. cos-diff 99.7%

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

   2. +-commutative 99.7%

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

   3. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in lambda1 around 0 99.2%

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

8. Final simplification 99.2%

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

Alternative 4: 98.8% 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 98.9%

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

   1. sin-diff 98.9%

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

4. Applied egg-rr 98.9%

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

Alternative 5: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq -0.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\cos \phi_1 + t\_0}\\ \mathbf{elif}\;\cos \phi_1 \leq 0.997:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_0 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2))))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) -0.8)
     (+ lambda1 (atan2 (* lambda1 (cos phi2)) (+ (cos phi1) t_0)))
     (if (<= (cos phi1) 0.997)
       (+ lambda1 (atan2 t_1 (+ (cos phi1) (cos (- lambda2 lambda1)))))
       (+ lambda1 (atan2 t_1 (+ t_0 1.0)))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= -0.8) {
		tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + t_0));
	} else if (cos(phi1) <= 0.997) {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + cos((lambda2 - lambda1))));
	} else {
		tmp = lambda1 + atan2(t_1, (t_0 + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi2) * cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi1) <= (-0.8d0)) then
        tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + t_0))
    else if (cos(phi1) <= 0.997d0) then
        tmp = lambda1 + atan2(t_1, (cos(phi1) + cos((lambda2 - lambda1))))
    else
        tmp = lambda1 + atan2(t_1, (t_0 + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi1) <= -0.8) {
		tmp = lambda1 + Math.atan2((lambda1 * Math.cos(phi2)), (Math.cos(phi1) + t_0));
	} else if (Math.cos(phi1) <= 0.997) {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(phi1) + Math.cos((lambda2 - lambda1))));
	} else {
		tmp = lambda1 + Math.atan2(t_1, (t_0 + 1.0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi1) <= -0.8:
		tmp = lambda1 + math.atan2((lambda1 * math.cos(phi2)), (math.cos(phi1) + t_0))
	elif math.cos(phi1) <= 0.997:
		tmp = lambda1 + math.atan2(t_1, (math.cos(phi1) + math.cos((lambda2 - lambda1))))
	else:
		tmp = lambda1 + math.atan2(t_1, (t_0 + 1.0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= -0.8)
		tmp = Float64(lambda1 + atan(Float64(lambda1 * cos(phi2)), Float64(cos(phi1) + t_0)));
	elseif (cos(phi1) <= 0.997)
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(t_0 + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi1) <= -0.8)
		tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + t_0));
	elseif (cos(phi1) <= 0.997)
		tmp = lambda1 + atan2(t_1, (cos(phi1) + cos((lambda2 - lambda1))));
	else
		tmp = lambda1 + atan2(t_1, (t_0 + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], -0.8], N[(lambda1 + N[ArcTan[N[(lambda1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.997], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 phi1) < -0.80000000000000004

   1. Initial program 96.3%

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

   3. Taylor expanded in lambda1 around 0 95.4%

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

      1. +-commutative 95.4%

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

      2. sin-neg 95.4%

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

      3. unsub-neg 95.4%

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

      4. cos-neg 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in lambda2 around 0 71.3%

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

      1. *-commutative 71.3%

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

   8. Simplified 71.3%

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

   if -0.80000000000000004 < (cos.f64 phi1) < 0.996999999999999997

   1. Initial program 99.0%

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

   3. Taylor expanded in phi2 around 0 74.3%

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

      1. +-commutative 74.3%

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

      2. sub-neg 74.3%

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

      3. remove-double-neg 74.3%

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

      4. mul-1-neg 74.3%

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

      5. distribute-neg-in 74.3%

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

      6. +-commutative 74.3%

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

      7. cos-neg 74.3%

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

      8. mul-1-neg 74.3%

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

      9. unsub-neg 74.3%

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

   5. Simplified 74.3%

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

   if 0.996999999999999997 < (cos.f64 phi1)

   1. Initial program 99.5%

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

   3. Taylor expanded in phi1 around 0 98.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq -0.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\cos \phi_1 \leq 0.997:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq -0.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\cos \phi_1 \leq 0.997:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) -0.8)
     (+
      lambda1
      (atan2
       (* lambda1 (cos phi2))
       (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
     (if (<= (cos phi1) 0.997)
       (+ lambda1 (atan2 t_0 (+ (cos phi1) (cos (- lambda2 lambda1)))))
       (+ lambda1 (atan2 t_0 (+ 1.0 (* (cos phi2) (cos lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= -0.8) {
		tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	} else if (cos(phi1) <= 0.997) {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda2 - lambda1))));
	} else {
		tmp = lambda1 + atan2(t_0, (1.0 + (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 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi1) <= (-0.8d0)) then
        tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
    else if (cos(phi1) <= 0.997d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda2 - lambda1))))
    else
        tmp = lambda1 + atan2(t_0, (1.0d0 + (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.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi1) <= -0.8) {
		tmp = lambda1 + Math.atan2((lambda1 * Math.cos(phi2)), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
	} else if (Math.cos(phi1) <= 0.997) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + Math.cos((lambda2 - lambda1))));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (1.0 + (Math.cos(phi2) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi1) <= -0.8:
		tmp = lambda1 + math.atan2((lambda1 * math.cos(phi2)), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))
	elif math.cos(phi1) <= 0.997:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + math.cos((lambda2 - lambda1))))
	else:
		tmp = lambda1 + math.atan2(t_0, (1.0 + (math.cos(phi2) * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= -0.8)
		tmp = Float64(lambda1 + atan(Float64(lambda1 * cos(phi2)), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	elseif (cos(phi1) <= 0.997)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(1.0 + Float64(cos(phi2) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi1) <= -0.8)
		tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	elseif (cos(phi1) <= 0.997)
		tmp = lambda1 + atan2(t_0, (cos(phi1) + cos((lambda2 - lambda1))));
	else
		tmp = lambda1 + atan2(t_0, (1.0 + (cos(phi2) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], -0.8], N[(lambda1 + N[ArcTan[N[(lambda1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.997], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 phi1) < -0.80000000000000004

   1. Initial program 96.3%

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

   3. Taylor expanded in lambda1 around 0 95.4%

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

      1. +-commutative 95.4%

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

      2. sin-neg 95.4%

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

      3. unsub-neg 95.4%

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

      4. cos-neg 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in lambda2 around 0 71.3%

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

      1. *-commutative 71.3%

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

   8. Simplified 71.3%

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

   if -0.80000000000000004 < (cos.f64 phi1) < 0.996999999999999997

   1. Initial program 99.0%

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

   3. Taylor expanded in phi2 around 0 74.3%

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

      1. +-commutative 74.3%

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

      2. sub-neg 74.3%

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

      3. remove-double-neg 74.3%

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

      4. mul-1-neg 74.3%

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

      5. distribute-neg-in 74.3%

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

      6. +-commutative 74.3%

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

      7. cos-neg 74.3%

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

      8. mul-1-neg 74.3%

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

      9. unsub-neg 74.3%

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

   5. Simplified 74.3%

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

   if 0.996999999999999997 < (cos.f64 phi1)

   1. Initial program 99.5%

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

      1. add-cbrt-cube 99.4%

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

      2. pow3 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-define 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in phi1 around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in lambda1 around 0 97.5%

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

      1. cos-neg 97.5%

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

      2. *-commutative 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq -0.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\cos \phi_1 \leq 0.997:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq -0.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\cos \phi_1 \leq 0.997:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \lambda_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) -0.8)
     (+
      lambda1
      (atan2
       (* lambda1 (cos phi2))
       (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
     (if (<= (cos phi1) 0.997)
       (+ lambda1 (atan2 t_0 (+ (cos lambda2) (cos phi1))))
       (+ lambda1 (atan2 t_0 (+ 1.0 (* (cos phi2) (cos lambda2)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= -0.8) {
		tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	} else if (cos(phi1) <= 0.997) {
		tmp = lambda1 + atan2(t_0, (cos(lambda2) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, (1.0 + (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 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi1) <= (-0.8d0)) then
        tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
    else if (cos(phi1) <= 0.997d0) then
        tmp = lambda1 + atan2(t_0, (cos(lambda2) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, (1.0d0 + (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.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi1) <= -0.8) {
		tmp = lambda1 + Math.atan2((lambda1 * Math.cos(phi2)), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
	} else if (Math.cos(phi1) <= 0.997) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(lambda2) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (1.0 + (Math.cos(phi2) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi1) <= -0.8:
		tmp = lambda1 + math.atan2((lambda1 * math.cos(phi2)), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))
	elif math.cos(phi1) <= 0.997:
		tmp = lambda1 + math.atan2(t_0, (math.cos(lambda2) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, (1.0 + (math.cos(phi2) * math.cos(lambda2))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= -0.8)
		tmp = Float64(lambda1 + atan(Float64(lambda1 * cos(phi2)), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	elseif (cos(phi1) <= 0.997)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(lambda2) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(1.0 + Float64(cos(phi2) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi1) <= -0.8)
		tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	elseif (cos(phi1) <= 0.997)
		tmp = lambda1 + atan2(t_0, (cos(lambda2) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, (1.0 + (cos(phi2) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], -0.8], N[(lambda1 + N[ArcTan[N[(lambda1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.997], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(1.0 + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 phi1) < -0.80000000000000004

   1. Initial program 96.3%

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

   3. Taylor expanded in lambda1 around 0 95.4%

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

      1. +-commutative 95.4%

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

      2. sin-neg 95.4%

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

      3. unsub-neg 95.4%

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

      4. cos-neg 95.4%

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

   5. Simplified 95.4%

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

   6. Taylor expanded in lambda2 around 0 71.3%

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

      1. *-commutative 71.3%

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

   8. Simplified 71.3%

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

   if -0.80000000000000004 < (cos.f64 phi1) < 0.996999999999999997

   1. Initial program 99.0%

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

   3. Taylor expanded in phi2 around 0 74.3%

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

      1. +-commutative 74.3%

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

      2. sub-neg 74.3%

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

      3. remove-double-neg 74.3%

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

      4. mul-1-neg 74.3%

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

      5. distribute-neg-in 74.3%

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

      6. +-commutative 74.3%

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

      7. cos-neg 74.3%

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

      8. mul-1-neg 74.3%

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

      9. unsub-neg 74.3%

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

   5. Simplified 74.3%

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

   6. Taylor expanded in lambda1 around 0 74.3%

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

   if 0.996999999999999997 < (cos.f64 phi1)

   1. Initial program 99.5%

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

      1. add-cbrt-cube 99.4%

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

      2. pow3 99.4%

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

      3. +-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. fma-define 99.4%

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in phi1 around 0 98.5%

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

      1. +-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in lambda1 around 0 97.5%

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

      1. cos-neg 97.5%

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

      2. *-commutative 97.5%

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

   10. Simplified 97.5%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq -0.8:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{elif}\;\cos \phi_1 \leq 0.997:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \phi_2 \cdot \cos \lambda_2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.997:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.997)
     (+ lambda1 (atan2 t_0 (+ (cos phi1) (* (cos phi2) (cos lambda1)))))
     (+
      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) <= 0.997) {
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos(lambda1))));
	} 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) <= 0.997d0) then
        tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos(lambda1))))
    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) <= 0.997) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(phi1) + (Math.cos(phi2) * Math.cos(lambda1))));
	} 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) <= 0.997:
		tmp = lambda1 + math.atan2(t_0, (math.cos(phi1) + (math.cos(phi2) * math.cos(lambda1))))
	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) <= 0.997)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(phi1) + Float64(cos(phi2) * cos(lambda1)))));
	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) <= 0.997)
		tmp = lambda1 + atan2(t_0, (cos(phi1) + (cos(phi2) * cos(lambda1))));
	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], 0.997], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $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) < 0.996999999999999997

   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 lambda2 around 0 80.0%

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

   if 0.996999999999999997 < (cos.f64 phi1)

   1. Initial program 99.5%

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

   3. Taylor expanded in phi1 around 0 98.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.997:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \lambda_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

   1. cos-neg 98.9%

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

   2. cos-neg 98.9%

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

   3. cos-neg 98.9%

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

   4. +-commutative 98.9%

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

   5. cos-neg 98.9%

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

   6. fma-define 98.9%

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

3. Simplified 98.9%

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

Alternative 10: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.98:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos phi2) 0.98)
   (+
    lambda1
    (atan2
     (* lambda1 (cos phi2))
     (+ (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
   (+
    lambda1
    (atan2
     (* (cos phi2) (sin (- lambda1 lambda2)))
     (+ (cos lambda2) (cos phi1))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos(phi2) <= 0.98) {
		tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + cos(phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (cos(phi2) <= 0.98d0) then
        tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + cos(phi1)))
    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) <= 0.98) {
		tmp = lambda1 + Math.atan2((lambda1 * Math.cos(phi2)), (Math.cos(phi1) + (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (Math.cos(lambda2) + Math.cos(phi1)));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.cos(phi2) <= 0.98:
		tmp = lambda1 + math.atan2((lambda1 * math.cos(phi2)), (math.cos(phi1) + (math.cos(phi2) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (math.cos(lambda2) + math.cos(phi1)))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(phi2) <= 0.98)
		tmp = Float64(lambda1 + atan(Float64(lambda1 * cos(phi2)), Float64(cos(phi1) + Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(cos(lambda2) + cos(phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (cos(phi2) <= 0.98)
		tmp = lambda1 + atan2((lambda1 * cos(phi2)), (cos(phi1) + (cos(phi2) * cos((lambda1 - lambda2)))));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (cos(lambda2) + cos(phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[phi2], $MachinePrecision], 0.98], N[(lambda1 + N[ArcTan[N[(lambda1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[lambda2], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi2) < 0.97999999999999998

   1. Initial program 98.8%

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

   3. Taylor expanded in lambda1 around 0 98.0%

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

      1. +-commutative 98.0%

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

      2. sin-neg 98.0%

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

      3. unsub-neg 98.0%

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

      4. cos-neg 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in lambda2 around 0 62.7%

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

      1. *-commutative 62.7%

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

   8. Simplified 62.7%

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

   if 0.97999999999999998 < (cos.f64 phi2)

   1. Initial program 99.0%

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

   3. Taylor expanded in phi2 around 0 97.1%

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

      1. +-commutative 97.1%

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

      2. sub-neg 97.1%

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

      3. remove-double-neg 97.1%

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

      4. mul-1-neg 97.1%

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

      5. distribute-neg-in 97.1%

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

      6. +-commutative 97.1%

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

      7. cos-neg 97.1%

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

      8. mul-1-neg 97.1%

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

      9. unsub-neg 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in lambda1 around 0 96.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_2 \leq 0.98:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\lambda_1 \cdot \cos \phi_2}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_2 + \cos \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.9999999995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{\cos \lambda_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_0}{1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= (cos phi1) 0.9999999995)
     (+ lambda1 (atan2 t_0 (+ (cos lambda1) (cos phi1))))
     (+ lambda1 (atan2 t_0 (+ 1.0 (cos (- lambda2 lambda1))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (cos(phi1) <= 0.9999999995) {
		tmp = lambda1 + atan2(t_0, (cos(lambda1) + cos(phi1)));
	} else {
		tmp = lambda1 + atan2(t_0, (1.0 + cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi2) * sin((lambda1 - lambda2))
    if (cos(phi1) <= 0.9999999995d0) then
        tmp = lambda1 + atan2(t_0, (cos(lambda1) + cos(phi1)))
    else
        tmp = lambda1 + atan2(t_0, (1.0d0 + cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (Math.cos(phi1) <= 0.9999999995) {
		tmp = lambda1 + Math.atan2(t_0, (Math.cos(lambda1) + Math.cos(phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_0, (1.0 + Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if math.cos(phi1) <= 0.9999999995:
		tmp = lambda1 + math.atan2(t_0, (math.cos(lambda1) + math.cos(phi1)))
	else:
		tmp = lambda1 + math.atan2(t_0, (1.0 + math.cos((lambda2 - lambda1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (cos(phi1) <= 0.9999999995)
		tmp = Float64(lambda1 + atan(t_0, Float64(cos(lambda1) + cos(phi1))));
	else
		tmp = Float64(lambda1 + atan(t_0, Float64(1.0 + cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (cos(phi1) <= 0.9999999995)
		tmp = lambda1 + atan2(t_0, (cos(lambda1) + cos(phi1)));
	else
		tmp = lambda1 + atan2(t_0, (1.0 + cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.9999999995], N[(lambda1 + N[ArcTan[t$95$0 / N[(N[Cos[lambda1], $MachinePrecision] + N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$0 / N[(1.0 + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.99999999949999996

   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 phi2 around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. sub-neg 69.3%

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

      3. remove-double-neg 69.3%

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

      4. mul-1-neg 69.3%

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

      5. distribute-neg-in 69.3%

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

      6. +-commutative 69.3%

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

      7. cos-neg 69.3%

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

      8. mul-1-neg 69.3%

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

      9. unsub-neg 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in lambda2 around 0 60.1%

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

      1. cos-neg 60.1%

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

   8. Simplified 60.1%

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

   if 0.99999999949999996 < (cos.f64 phi1)

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0 78.0%

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

      1. +-commutative 78.0%

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

      2. sub-neg 78.0%

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

      3. remove-double-neg 78.0%

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

      4. mul-1-neg 78.0%

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

      5. distribute-neg-in 78.0%

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

      6. +-commutative 78.0%

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

      7. cos-neg 78.0%

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

      8. mul-1-neg 78.0%

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

      9. unsub-neg 78.0%

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

   5. Simplified 78.0%

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

   6. Taylor expanded in phi1 around 0 78.0%

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

      1. +-commutative 78.0%

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

   8. Simplified 78.0%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.9999999995:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \lambda_1 + \cos \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 98.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

Alternative 13: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in lambda1 around 0 98.2%

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

   1. cos-neg 98.2%

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

5. Simplified 98.2%

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

Alternative 14: 82.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq 0.041:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{t\_0 + 1}\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{+236}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2))))
        (t_1 (* (cos phi2) (sin (- lambda1 lambda2)))))
   (if (<= phi1 0.041)
     (+ lambda1 (atan2 t_1 (+ t_0 1.0)))
     (if (<= phi1 4e+236)
       (+
        lambda1
        (atan2 (* (cos phi2) (- lambda1 lambda2)) (+ (cos phi1) t_0)))
       (+ lambda1 (atan2 t_1 (+ (cos phi1) (cos (- lambda2 lambda1)))))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos((lambda1 - lambda2));
	double t_1 = cos(phi2) * sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 0.041) {
		tmp = lambda1 + atan2(t_1, (t_0 + 1.0));
	} else if (phi1 <= 4e+236) {
		tmp = lambda1 + atan2((cos(phi2) * (lambda1 - lambda2)), (cos(phi1) + t_0));
	} else {
		tmp = lambda1 + atan2(t_1, (cos(phi1) + cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi2) * cos((lambda1 - lambda2))
    t_1 = cos(phi2) * sin((lambda1 - lambda2))
    if (phi1 <= 0.041d0) then
        tmp = lambda1 + atan2(t_1, (t_0 + 1.0d0))
    else if (phi1 <= 4d+236) then
        tmp = lambda1 + atan2((cos(phi2) * (lambda1 - lambda2)), (cos(phi1) + t_0))
    else
        tmp = lambda1 + atan2(t_1, (cos(phi1) + cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.cos((lambda1 - lambda2));
	double t_1 = Math.cos(phi2) * Math.sin((lambda1 - lambda2));
	double tmp;
	if (phi1 <= 0.041) {
		tmp = lambda1 + Math.atan2(t_1, (t_0 + 1.0));
	} else if (phi1 <= 4e+236) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * (lambda1 - lambda2)), (Math.cos(phi1) + t_0));
	} else {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(phi1) + Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.cos((lambda1 - lambda2))
	t_1 = math.cos(phi2) * math.sin((lambda1 - lambda2))
	tmp = 0
	if phi1 <= 0.041:
		tmp = lambda1 + math.atan2(t_1, (t_0 + 1.0))
	elif phi1 <= 4e+236:
		tmp = lambda1 + math.atan2((math.cos(phi2) * (lambda1 - lambda2)), (math.cos(phi1) + t_0))
	else:
		tmp = lambda1 + math.atan2(t_1, (math.cos(phi1) + math.cos((lambda2 - lambda1))))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))
	t_1 = Float64(cos(phi2) * sin(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= 0.041)
		tmp = Float64(lambda1 + atan(t_1, Float64(t_0 + 1.0)));
	elseif (phi1 <= 4e+236)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * Float64(lambda1 - lambda2)), Float64(cos(phi1) + t_0)));
	else
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(phi1) + cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * cos((lambda1 - lambda2));
	t_1 = cos(phi2) * sin((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= 0.041)
		tmp = lambda1 + atan2(t_1, (t_0 + 1.0));
	elseif (phi1 <= 4e+236)
		tmp = lambda1 + atan2((cos(phi2) * (lambda1 - lambda2)), (cos(phi1) + t_0));
	else
		tmp = lambda1 + atan2(t_1, (cos(phi1) + cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 0.041], N[(lambda1 + N[ArcTan[t$95$1 / N[(t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4e+236], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] + N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < 0.0410000000000000017

   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. Taylor expanded in phi1 around 0 84.3%

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

   if 0.0410000000000000017 < phi1 < 4.00000000000000021e236

   1. Initial program 98.8%

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

   3. Taylor expanded in lambda1 around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. sin-neg 98.9%

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

      3. unsub-neg 98.9%

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

      4. cos-neg 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in lambda2 around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. associate-*r* 82.6%

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

      3. neg-mul-1 82.6%

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

      4. distribute-rgt-out 82.6%

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

      5. sub-neg 82.6%

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

   8. Simplified 82.6%

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

   if 4.00000000000000021e236 < phi1

   1. Initial program 94.6%

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

   3. Taylor expanded in phi2 around 0 80.8%

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

      1. +-commutative 80.8%

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

      2. sub-neg 80.8%

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

      3. remove-double-neg 80.8%

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

      4. mul-1-neg 80.8%

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

      5. distribute-neg-in 80.8%

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

      6. +-commutative 80.8%

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

      7. cos-neg 80.8%

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

      8. mul-1-neg 80.8%

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

      9. unsub-neg 80.8%

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

   5. Simplified 80.8%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 0.041:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + 1}\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{+236}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\cos \phi_1 \leq 0.55:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= (cos phi1) 0.55)
     (+ lambda1 (atan2 (* (cos phi2) (- lambda1 lambda2)) (+ (cos phi1) t_0)))
     (+
      lambda1
      (atan2 (* (cos phi2) (sin (- lambda1 lambda2))) (+ 1.0 t_0))))))

C

double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (cos(phi1) <= 0.55) {
		tmp = lambda1 + atan2((cos(phi2) * (lambda1 - lambda2)), (cos(phi1) + t_0));
	} else {
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (1.0 + t_0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (cos(phi1) <= 0.55d0) then
        tmp = lambda1 + atan2((cos(phi2) * (lambda1 - lambda2)), (cos(phi1) + t_0))
    else
        tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (1.0d0 + t_0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (Math.cos(phi1) <= 0.55) {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * (lambda1 - lambda2)), (Math.cos(phi1) + t_0));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi2) * Math.sin((lambda1 - lambda2))), (1.0 + t_0));
	}
	return tmp;
}

Python

def code(lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if math.cos(phi1) <= 0.55:
		tmp = lambda1 + math.atan2((math.cos(phi2) * (lambda1 - lambda2)), (math.cos(phi1) + t_0))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi2) * math.sin((lambda1 - lambda2))), (1.0 + t_0))
	return tmp

Julia

function code(lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (cos(phi1) <= 0.55)
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * Float64(lambda1 - lambda2)), Float64(cos(phi1) + t_0)));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi2) * sin(Float64(lambda1 - lambda2))), Float64(1.0 + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (cos(phi1) <= 0.55)
		tmp = lambda1 + atan2((cos(phi2) * (lambda1 - lambda2)), (cos(phi1) + t_0));
	else
		tmp = lambda1 + atan2((cos(phi2) * sin((lambda1 - lambda2))), (1.0 + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Cos[phi1], $MachinePrecision], 0.55], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi2], $MachinePrecision] * N[Sin[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 phi1) < 0.55000000000000004

   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 68.4%

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

      1. +-commutative 68.4%

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

      2. sub-neg 68.4%

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

      3. remove-double-neg 68.4%

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

      4. mul-1-neg 68.4%

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

      5. distribute-neg-in 68.4%

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

      6. +-commutative 68.4%

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

      7. cos-neg 68.4%

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

      8. mul-1-neg 68.4%

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

      9. unsub-neg 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in lambda2 around 0 57.6%

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

      1. mul-1-neg 57.6%

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

      2. unsub-neg 57.6%

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

   8. Simplified 57.6%

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

   9. Taylor expanded in lambda1 around 0 57.3%

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

       1. associate-*r* 57.3%

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

       2. neg-mul-1 57.3%

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

       3. distribute-rgt-out 57.3%

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

       4. +-commutative 57.3%

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

       5. sub-neg 57.3%

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

   11. Simplified 57.3%

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

   if 0.55000000000000004 < (cos.f64 phi1)

   1. Initial program 99.5%

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

   3. Taylor expanded in phi2 around 0 76.5%

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

      1. +-commutative 76.5%

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

      2. sub-neg 76.5%

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

      3. remove-double-neg 76.5%

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

      4. mul-1-neg 76.5%

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

      5. distribute-neg-in 76.5%

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

      6. +-commutative 76.5%

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

      7. cos-neg 76.5%

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

      8. mul-1-neg 76.5%

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

      9. unsub-neg 76.5%

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

   5. Simplified 76.5%

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

   6. Taylor expanded in phi1 around 0 74.3%

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

      1. +-commutative 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \phi_1 \leq 0.55:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \left(\lambda_1 - \lambda_2\right)}{\cos \phi_1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_2 \cdot \sin \left(\lambda_1 - \lambda_2\right)}{1 + \cos \left(\lambda_2 - \lambda_1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 77.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in phi2 around 0 73.7%

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

   1. +-commutative 73.7%

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

   2. sub-neg 73.7%

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

   3. remove-double-neg 73.7%

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

   4. mul-1-neg 73.7%

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

   5. distribute-neg-in 73.7%

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

   6. +-commutative 73.7%

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

   7. cos-neg 73.7%

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

   8. mul-1-neg 73.7%

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

   9. unsub-neg 73.7%

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

5. Simplified 73.7%

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

6. Taylor expanded in lambda1 around 0 73.3%

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

Alternative 17: 67.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in phi2 around 0 73.7%

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

   1. +-commutative 73.7%

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

   2. sub-neg 73.7%

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

   3. remove-double-neg 73.7%

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

   4. mul-1-neg 73.7%

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

   5. distribute-neg-in 73.7%

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

   6. +-commutative 73.7%

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

   7. cos-neg 73.7%

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

   8. mul-1-neg 73.7%

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

   9. unsub-neg 73.7%

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

5. Simplified 73.7%

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

6. Taylor expanded in phi1 around 0 66.2%

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

   1. +-commutative 66.2%

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

8. Simplified 66.2%

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

9. Final simplification 66.2%

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

Alternative 18: 67.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in phi2 around 0 73.7%

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

   1. +-commutative 73.7%

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

   2. sub-neg 73.7%

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

   3. remove-double-neg 73.7%

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

   4. mul-1-neg 73.7%

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

   5. distribute-neg-in 73.7%

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

   6. +-commutative 73.7%

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

   7. cos-neg 73.7%

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

   8. mul-1-neg 73.7%

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

   9. unsub-neg 73.7%

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

5. Simplified 73.7%

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

6. Taylor expanded in phi1 around 0 66.2%

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

   1. +-commutative 66.2%

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

8. Simplified 66.2%

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

9. Taylor expanded in lambda1 around 0 65.8%

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

Alternative 19: 66.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in phi2 around 0 73.7%

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

   1. +-commutative 73.7%

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

   2. sub-neg 73.7%

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

   3. remove-double-neg 73.7%

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

   4. mul-1-neg 73.7%

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

   5. distribute-neg-in 73.7%

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

   6. +-commutative 73.7%

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

   7. cos-neg 73.7%

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

   8. mul-1-neg 73.7%

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

   9. unsub-neg 73.7%

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

5. Simplified 73.7%

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

6. Taylor expanded in phi1 around 0 66.2%

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

   1. +-commutative 66.2%

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

8. Simplified 66.2%

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

9. Taylor expanded in phi2 around 0 64.6%

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

10. Final simplification 64.6%

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

Alternative 20: 66.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in phi2 around 0 73.7%

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

   1. +-commutative 73.7%

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

   2. sub-neg 73.7%

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

   3. remove-double-neg 73.7%

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

   4. mul-1-neg 73.7%

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

   5. distribute-neg-in 73.7%

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

   6. +-commutative 73.7%

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

   7. cos-neg 73.7%

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

   8. mul-1-neg 73.7%

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

   9. unsub-neg 73.7%

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

5. Simplified 73.7%

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

6. Taylor expanded in phi1 around 0 66.2%

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

   1. +-commutative 66.2%

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

8. Simplified 66.2%

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

9. Taylor expanded in phi2 around 0 64.6%

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

10. Taylor expanded in lambda1 around 0 64.3%

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

    1. +-commutative 64.3%

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

12. Simplified 64.3%

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

Alternative 21: 61.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in phi2 around 0 73.7%

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

   1. +-commutative 73.7%

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

   2. sub-neg 73.7%

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

   3. remove-double-neg 73.7%

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

   4. mul-1-neg 73.7%

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

   5. distribute-neg-in 73.7%

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

   6. +-commutative 73.7%

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

   7. cos-neg 73.7%

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

   8. mul-1-neg 73.7%

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

   9. unsub-neg 73.7%

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

5. Simplified 73.7%

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

6. Taylor expanded in phi1 around 0 66.2%

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

   1. +-commutative 66.2%

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

8. Simplified 66.2%

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

9. Taylor expanded in phi2 around 0 64.6%

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

10. Taylor expanded in lambda2 around 0 61.3%

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

    1. cos-neg 61.3%

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

    2. +-commutative 61.3%

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

12. Simplified 61.3%

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

Alternative 22: 54.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in phi2 around 0 73.7%

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

   1. +-commutative 73.7%

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

   2. sub-neg 73.7%

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

   3. remove-double-neg 73.7%

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

   4. mul-1-neg 73.7%

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

   5. distribute-neg-in 73.7%

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

   6. +-commutative 73.7%

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

   7. cos-neg 73.7%

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

   8. mul-1-neg 73.7%

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

   9. unsub-neg 73.7%

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

5. Simplified 73.7%

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

6. Taylor expanded in phi1 around 0 66.2%

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

   1. +-commutative 66.2%

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

8. Simplified 66.2%

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

9. Taylor expanded in phi2 around 0 64.6%

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

10. Taylor expanded in lambda2 around 0 54.6%

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

11. Final simplification 54.6%

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

Reproduce

herbie shell --seed 2024088 
(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)))))))