Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.9%

Time: 18.2s

Alternatives: 15

Speedup: 1.3×

Specification

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (cos delta)
    (*
     (sin phi1)
     (sin
      (asin
       (+
        (* (sin phi1) (cos delta))
        (* (* (cos phi1) (sin delta)) (cos theta))))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (cos delta)
    (*
     (sin phi1)
     (sin
      (asin
       (+
        (* (sin phi1) (cos delta))
        (* (* (cos phi1) (sin delta)) (cos theta))))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)}, \color{blue}{\lambda_1}\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in lambda1 around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \color{blue}{\left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}}{\lambda_1}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}}{\lambda_1}\right)}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}, \color{blue}{\lambda_1}\right)\right)\right) \]

8. Simplified 99.9%

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

9. Taylor expanded in lambda1 around 0

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

    1. +-commutative N/A

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

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot {\cos \phi_1}^{2} - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}, \color{blue}{\lambda_1}\right) \]

11. Simplified 99.9%

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos \phi_1 \cdot \left(\cos \phi_1 \cdot \cos delta + \sin \phi_1 \cdot \left(\cos theta \cdot \left(0 - \sin delta\right)\right)\right)} + \lambda_1} \]

12. Taylor expanded in phi1 around 0

    \[\leadsto \mathsf{+.f64}\left(\color{blue}{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos \phi_1 \cdot \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)}}, \lambda_1\right) \]
13. Step-by-step derivation

    1. atan2-lowering-atan2.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\right), \left(\cos \phi_1 \cdot \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta\right), \left(\cos \phi_1 \cdot \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\cos \phi_1 \cdot \sin delta\right), \sin theta\right), \left(\cos \phi_1 \cdot \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(\sin delta \cdot \cos \phi_1\right), \sin theta\right), \left(\cos \phi_1 \cdot \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos \phi_1\right), \sin theta\right), \left(\cos \phi_1 \cdot \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    6. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos \phi_1\right), \sin theta\right), \left(\cos \phi_1 \cdot \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    7. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \sin theta\right), \left(\cos \phi_1 \cdot \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    8. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \left(\cos \phi_1 \cdot \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\cos \phi_1, \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    10. cos-lowering-cos.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(-1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right) + \cos delta \cdot \cos \phi_1\right)\right)\right), \lambda_1\right) \]

    11. +-commutative N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos delta \cdot \cos \phi_1 + -1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right)\right), \lambda_1\right) \]

    12. mul-1-neg N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos delta \cdot \cos \phi_1 + \left(\mathsf{neg}\left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right)\right)\right), \lambda_1\right) \]

    13. unsub-neg N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos delta \cdot \cos \phi_1 - \cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

    14. --lowering--.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{\_.f64}\left(\left(\cos delta \cdot \cos \phi_1\right), \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right)\right), \lambda_1\right) \]

14. Simplified 99.9%

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

15. Final simplification 99.9%

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

Alternative 2: 94.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot {\cos \phi_1}^{2} - \cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), ((cos(delta) * (cos(phi1) ** 2.0d0)) - (cos(phi1) * (sin(delta) * sin(phi1)))))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), ((Math.cos(delta) * Math.pow(Math.cos(phi1), 2.0)) - (Math.cos(phi1) * (Math.sin(delta) * Math.sin(phi1)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[delta], $MachinePrecision] * N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)}, \color{blue}{\lambda_1}\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in theta around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \color{blue}{\left(\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}\right), \lambda_1\right) \]
7. Step-by-step derivation

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\left(\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right)\right), \left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\left(\cos delta \cdot \left(1 - \sin \phi_1 \cdot \sin \phi_1\right)\right), \left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   3. 1-sub-sin N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\left(\cos delta \cdot \left(\cos \phi_1 \cdot \cos \phi_1\right)\right), \left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\left(\cos delta \cdot {\cos \phi_1}^{2}\right), \left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos delta, \left({\cos \phi_1}^{2}\right)\right), \left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \left({\cos \phi_1}^{2}\right)\right), \left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   7. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\cos \phi_1, 2\right)\right), \left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right), \left(\cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right), \mathsf{*.f64}\left(\cos \phi_1, \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   10. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \phi_1 \cdot \sin delta\right)\right)\right)\right), \lambda_1\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\sin \phi_1, \sin delta\right)\right)\right)\right), \lambda_1\right) \]

   13. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \sin delta\right)\right)\right)\right), \lambda_1\right) \]

   14. sin-lowering-sin.f64 96.2%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(delta\right)\right)\right)\right)\right), \lambda_1\right) \]

8. Simplified 96.2%

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

9. Final simplification 96.2%

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

Alternative 3: 94.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)}, \color{blue}{\lambda_1}\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in lambda1 around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \color{blue}{\left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}}{\lambda_1}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}}{\lambda_1}\right)}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}, \color{blue}{\lambda_1}\right)\right)\right) \]

8. Simplified 99.9%

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

9. Taylor expanded in lambda1 around 0

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

    1. +-commutative N/A

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

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot {\cos \phi_1}^{2} - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}, \color{blue}{\lambda_1}\right) \]

11. Simplified 99.9%

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos \phi_1 \cdot \left(\cos \phi_1 \cdot \cos delta + \sin \phi_1 \cdot \left(\cos theta \cdot \left(0 - \sin delta\right)\right)\right)} + \lambda_1} \]

12. Taylor expanded in theta around 0

    \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\left(-1 \cdot \left(\sin delta \cdot \sin \phi_1\right) + \cos delta \cdot \cos \phi_1\right)}\right)\right), \lambda_1\right) \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos delta \cdot \cos \phi_1 + -1 \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos delta \cdot \cos \phi_1 + \left(\mathsf{neg}\left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right)\right), \lambda_1\right) \]

    3. unsub-neg N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos delta \cdot \cos \phi_1 - \sin delta \cdot \sin \phi_1\right)\right)\right), \lambda_1\right) \]

    4. --lowering--.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{\_.f64}\left(\left(\cos delta \cdot \cos \phi_1\right), \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\cos delta, \cos \phi_1\right), \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

    6. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \cos \phi_1\right), \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

    7. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\sin delta \cdot \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\sin delta, \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

    9. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin \phi_1\right)\right)\right)\right), \lambda_1\right) \]

    10. sin-lowering-sin.f64 96.2%

        \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(\phi_1\right)\right)\right)\right)\right), \lambda_1\right) \]

14. Simplified 96.2%

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

15. Final simplification 96.2%

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

Alternative 4: 92.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - {\sin \phi_1}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (sin(phi1) ** 2.0d0)))
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), (Math.cos(delta) - Math.pow(Math.sin(phi1), 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in delta around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \color{blue}{\left({\sin \phi_1}^{2}\right)}\right)\right)\right) \]
4. Step-by-step derivation

   1. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\sin \phi_1, \color{blue}{2}\right)\right)\right)\right) \]

   2. sin-lowering-sin.f64 93.6%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{pow.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), 2\right)\right)\right)\right) \]

5. Simplified 93.6%

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

6. Final simplification 93.6%

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

Alternative 5: 92.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;delta \leq -1.95 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 \cdot \left(1 + \frac{\tan^{-1}_* \frac{t\_1}{{\cos \phi_1}^{2}}}{\lambda_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin delta) (sin theta))))
        (t_2 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= delta -1.95e-6)
     t_2
     (if (<= delta 1.25e-5)
       (* lambda1 (+ 1.0 (/ (atan2 t_1 (pow (cos phi1) 2.0)) lambda1)))
       t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(delta) * sin(theta));
	double t_2 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (delta <= -1.95e-6) {
		tmp = t_2;
	} else if (delta <= 1.25e-5) {
		tmp = lambda1 * (1.0 + (atan2(t_1, pow(cos(phi1), 2.0)) / lambda1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(phi1) * (sin(delta) * sin(theta))
    t_2 = lambda1 + atan2(t_1, cos(delta))
    if (delta <= (-1.95d-6)) then
        tmp = t_2
    else if (delta <= 1.25d-5) then
        tmp = lambda1 * (1.0d0 + (atan2(t_1, (cos(phi1) ** 2.0d0)) / lambda1))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta));
	double t_2 = lambda1 + Math.atan2(t_1, Math.cos(delta));
	double tmp;
	if (delta <= -1.95e-6) {
		tmp = t_2;
	} else if (delta <= 1.25e-5) {
		tmp = lambda1 * (1.0 + (Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0)) / lambda1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(delta) * math.sin(theta))
	t_2 = lambda1 + math.atan2(t_1, math.cos(delta))
	tmp = 0
	if delta <= -1.95e-6:
		tmp = t_2
	elif delta <= 1.25e-5:
		tmp = lambda1 * (1.0 + (math.atan2(t_1, math.pow(math.cos(phi1), 2.0)) / lambda1))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(delta) * sin(theta)))
	t_2 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (delta <= -1.95e-6)
		tmp = t_2;
	elseif (delta <= 1.25e-5)
		tmp = Float64(lambda1 * Float64(1.0 + Float64(atan(t_1, (cos(phi1) ^ 2.0)) / lambda1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(delta) * sin(theta));
	t_2 = lambda1 + atan2(t_1, cos(delta));
	tmp = 0.0;
	if (delta <= -1.95e-6)
		tmp = t_2;
	elseif (delta <= 1.25e-5)
		tmp = lambda1 * (1.0 + (atan2(t_1, (cos(phi1) ^ 2.0)) / lambda1));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -1.95e-6], t$95$2, If[LessEqual[delta, 1.25e-5], N[(lambda1 * N[(1.0 + N[(N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.95e-6 or 1.25000000000000006e-5 < delta

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\cos delta}\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 86.4%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   5. Simplified 86.4%

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

   if -1.95e-6 < delta < 1.25000000000000006e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)}, \color{blue}{\lambda_1}\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in lambda1 around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\lambda_1, \color{blue}{\left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}}{\lambda_1}\right)}\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}}{\lambda_1}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}, \color{blue}{\lambda_1}\right)\right)\right) \]

   8. Simplified 99.9%

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

   9. Taylor expanded in delta around 0

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \color{blue}{\left({\cos \phi_1}^{2}\right)}\right), \lambda_1\right)\right)\right) \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{pow.f64}\left(\cos \phi_1, 2\right)\right), \lambda_1\right)\right)\right) \]

       2. cos-lowering-cos.f64 99.4%

          \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right), \lambda_1\right)\right)\right) \]

   11. Simplified 99.4%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.95 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 \cdot \left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}}{\lambda_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;delta \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;delta \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin delta) (sin theta))))
        (t_2 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= delta -6.2e-5)
     t_2
     (if (<= delta 2.05e-7)
       (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0)))
       t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(delta) * sin(theta));
	double t_2 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (delta <= -6.2e-5) {
		tmp = t_2;
	} else if (delta <= 2.05e-7) {
		tmp = lambda1 + atan2(t_1, pow(cos(phi1), 2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(phi1) * (sin(delta) * sin(theta))
    t_2 = lambda1 + atan2(t_1, cos(delta))
    if (delta <= (-6.2d-5)) then
        tmp = t_2
    else if (delta <= 2.05d-7) then
        tmp = lambda1 + atan2(t_1, (cos(phi1) ** 2.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta));
	double t_2 = lambda1 + Math.atan2(t_1, Math.cos(delta));
	double tmp;
	if (delta <= -6.2e-5) {
		tmp = t_2;
	} else if (delta <= 2.05e-7) {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(delta) * math.sin(theta))
	t_2 = lambda1 + math.atan2(t_1, math.cos(delta))
	tmp = 0
	if delta <= -6.2e-5:
		tmp = t_2
	elif delta <= 2.05e-7:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(delta) * sin(theta)))
	t_2 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (delta <= -6.2e-5)
		tmp = t_2;
	elseif (delta <= 2.05e-7)
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(delta) * sin(theta));
	t_2 = lambda1 + atan2(t_1, cos(delta));
	tmp = 0.0;
	if (delta <= -6.2e-5)
		tmp = t_2;
	elseif (delta <= 2.05e-7)
		tmp = lambda1 + atan2(t_1, (cos(phi1) ^ 2.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -6.2e-5], t$95$2, If[LessEqual[delta, 2.05e-7], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if delta < -6.20000000000000027e-5 or 2.05e-7 < delta

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\cos delta}\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 86.4%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   5. Simplified 86.4%

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

   if -6.20000000000000027e-5 < delta < 2.05e-7

   1. Initial program 99.8%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)}, \color{blue}{\lambda_1}\right) \]

   5. Simplified 99.8%

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

   6. Taylor expanded in delta around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \color{blue}{\left(1 - {\sin \phi_1}^{2}\right)}\right), \lambda_1\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \left(1 - \sin \phi_1 \cdot \sin \phi_1\right)\right), \lambda_1\right) \]

      2. 1-sub-sin N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \left(\cos \phi_1 \cdot \cos \phi_1\right)\right), \lambda_1\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \left({\cos \phi_1}^{2}\right)\right), \lambda_1\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{pow.f64}\left(\cos \phi_1, 2\right)\right), \lambda_1\right) \]

      5. cos-lowering-cos.f64 99.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), 2\right)\right), \lambda_1\right) \]

   8. Simplified 99.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -6.2 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 2.05 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;delta \leq -1.36 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;delta \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin delta) (sin theta))))
        (t_2 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= delta -1.36e-6)
     t_2
     (if (<= delta 1.6e-5)
       (+ lambda1 (atan2 t_1 (+ 0.5 (* 0.5 (cos (* phi1 2.0))))))
       t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(delta) * sin(theta));
	double t_2 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (delta <= -1.36e-6) {
		tmp = t_2;
	} else if (delta <= 1.6e-5) {
		tmp = lambda1 + atan2(t_1, (0.5 + (0.5 * cos((phi1 * 2.0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(phi1) * (sin(delta) * sin(theta))
    t_2 = lambda1 + atan2(t_1, cos(delta))
    if (delta <= (-1.36d-6)) then
        tmp = t_2
    else if (delta <= 1.6d-5) then
        tmp = lambda1 + atan2(t_1, (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta));
	double t_2 = lambda1 + Math.atan2(t_1, Math.cos(delta));
	double tmp;
	if (delta <= -1.36e-6) {
		tmp = t_2;
	} else if (delta <= 1.6e-5) {
		tmp = lambda1 + Math.atan2(t_1, (0.5 + (0.5 * Math.cos((phi1 * 2.0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(delta) * math.sin(theta))
	t_2 = lambda1 + math.atan2(t_1, math.cos(delta))
	tmp = 0
	if delta <= -1.36e-6:
		tmp = t_2
	elif delta <= 1.6e-5:
		tmp = lambda1 + math.atan2(t_1, (0.5 + (0.5 * math.cos((phi1 * 2.0)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(delta) * sin(theta)))
	t_2 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (delta <= -1.36e-6)
		tmp = t_2;
	elseif (delta <= 1.6e-5)
		tmp = Float64(lambda1 + atan(t_1, Float64(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(delta) * sin(theta));
	t_2 = lambda1 + atan2(t_1, cos(delta));
	tmp = 0.0;
	if (delta <= -1.36e-6)
		tmp = t_2;
	elseif (delta <= 1.6e-5)
		tmp = lambda1 + atan2(t_1, (0.5 + (0.5 * cos((phi1 * 2.0)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -1.36e-6], t$95$2, If[LessEqual[delta, 1.6e-5], N[(lambda1 + N[ArcTan[t$95$1 / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.3599999999999999e-6 or 1.59999999999999993e-5 < delta

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\cos delta}\right)\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 86.4%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

   5. Simplified 86.4%

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

   if -1.3599999999999999e-6 < delta < 1.59999999999999993e-5

   1. Initial program 99.8%

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

      1. sin-asin N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta + \color{blue}{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta}\right)\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta - \sin \phi_1 \cdot \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \color{blue}{\sin \phi_1 \cdot \cos delta}\right)\right)\right)\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta - \left(\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1 + \color{blue}{\left(\sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}\right)\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\left(\cos delta - \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1\right) - \color{blue}{\left(\sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}\right)\right)\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\left(\cos delta - \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1\right) - \sin \phi_1 \cdot \color{blue}{\left(\cos delta \cdot \sin \phi_1\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\left(\cos delta - \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1\right) - \sin \phi_1 \cdot \left(\sin \phi_1 \cdot \color{blue}{\cos delta}\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\left(\cos delta - \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \sin \phi_1\right), \color{blue}{\left(\sin \phi_1 \cdot \left(\sin \phi_1 \cdot \cos delta\right)\right)}\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\cos delta - \left(\sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right) \cdot \sin \phi_1\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \phi_1\right)\right) \cdot \cos delta}} \]

   5. Taylor expanded in delta around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{2} \cdot \cos \left(2 \cdot \phi_1\right)\right)}\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \color{blue}{\cos \left(2 \cdot \phi_1\right)}\right)\right)\right)\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\left(2 \cdot \phi_1\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 99.4%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{1}{2}, \mathsf{cos.f64}\left(\mathsf{*.f64}\left(2, \phi_1\right)\right)\right)\right)\right)\right) \]

   7. Simplified 99.4%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{0.5 + 0.5 \cdot \cos \left(2 \cdot \phi_1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.36 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.6 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.0% accurate, 2.6× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+ lambda1 (atan2 (* (cos phi1) (* (sin delta) (sin theta))) (cos delta))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\cos delta}\right)\right) \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 89.3%

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{cos.f64}\left(delta\right)\right)\right) \]

5. Simplified 89.3%

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

6. Final simplification 89.3%

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

Alternative 9: 86.7% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (*
  lambda1
  (+ 1.0 (/ (atan2 (* (sin delta) (sin theta)) (cos delta)) lambda1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 * (1.0 + (atan2((sin(delta) * sin(theta)), cos(delta)) / lambda1));
}

Fortran

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

Java

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

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 * (1.0 + (math.atan2((math.sin(delta) * math.sin(theta)), math.cos(delta)) / lambda1))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 * Float64(1.0 + Float64(atan(Float64(sin(delta) * sin(theta)), cos(delta)) / lambda1)))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 * (1.0 + (atan2((sin(delta) * sin(theta)), cos(delta)) / lambda1));
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 * N[(1.0 + N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \left(\cos delta \cdot \sin \phi_1 + \cos \phi_1 \cdot \left(\cos theta \cdot \sin delta\right)\right)}, \color{blue}{\lambda_1}\right) \]

5. Simplified 99.8%

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

6. Taylor expanded in lambda1 around inf

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \color{blue}{\left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}}{\lambda_1}\right)}\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}}{\lambda_1}\right)}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta \cdot \left(1 - {\sin \phi_1}^{2}\right) - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)}, \color{blue}{\lambda_1}\right)\right)\right) \]

8. Simplified 99.9%

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

9. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \color{blue}{\cos delta}\right), \lambda_1\right)\right)\right) \]
10. Step-by-step derivation

    1. cos-lowering-cos.f64 89.3%

       \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{cos.f64}\left(delta\right)\right), \lambda_1\right)\right)\right) \]

11. Simplified 89.3%

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

12. Taylor expanded in phi1 around 0

    \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right), \lambda_1\right)\right)\right) \]
13. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right), \lambda_1\right)\right)\right) \]

    2. sin-lowering-sin.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{cos.f64}\left(delta\right)\right), \lambda_1\right)\right)\right) \]

    3. sin-lowering-sin.f64 88.1%

       \[\leadsto \mathsf{*.f64}\left(\lambda_1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{cos.f64}\left(delta\right)\right), \lambda_1\right)\right)\right) \]

14. Simplified 88.1%

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

Alternative 10: 75.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.1 \cdot 10^{+49}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \left(1 + \phi_1 \cdot \left(\phi_1 \cdot -0.5\right)\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -5.1e+49)
   lambda1
   (if (<= phi1 1.0)
     (+
      lambda1
      (atan2
       (* (* (sin delta) (sin theta)) (+ 1.0 (* phi1 (* phi1 -0.5))))
       (- 1.0 (* phi1 phi1))))
     lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -5.1e+49) {
		tmp = lambda1;
	} else if (phi1 <= 1.0) {
		tmp = lambda1 + atan2(((sin(delta) * sin(theta)) * (1.0 + (phi1 * (phi1 * -0.5)))), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-5.1d+49)) then
        tmp = lambda1
    else if (phi1 <= 1.0d0) then
        tmp = lambda1 + atan2(((sin(delta) * sin(theta)) * (1.0d0 + (phi1 * (phi1 * (-0.5d0))))), (1.0d0 - (phi1 * phi1)))
    else
        tmp = lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -5.1e+49) {
		tmp = lambda1;
	} else if (phi1 <= 1.0) {
		tmp = lambda1 + Math.atan2(((Math.sin(delta) * Math.sin(theta)) * (1.0 + (phi1 * (phi1 * -0.5)))), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -5.1e+49:
		tmp = lambda1
	elif phi1 <= 1.0:
		tmp = lambda1 + math.atan2(((math.sin(delta) * math.sin(theta)) * (1.0 + (phi1 * (phi1 * -0.5)))), (1.0 - (phi1 * phi1)))
	else:
		tmp = lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -5.1e+49)
		tmp = lambda1;
	elseif (phi1 <= 1.0)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(delta) * sin(theta)) * Float64(1.0 + Float64(phi1 * Float64(phi1 * -0.5)))), Float64(1.0 - Float64(phi1 * phi1))));
	else
		tmp = lambda1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -5.1e+49)
		tmp = lambda1;
	elseif (phi1 <= 1.0)
		tmp = lambda1 + atan2(((sin(delta) * sin(theta)) * (1.0 + (phi1 * (phi1 * -0.5)))), (1.0 - (phi1 * phi1)));
	else
		tmp = lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -5.1e+49], lambda1, If[LessEqual[phi1, 1.0], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(phi1 * N[(phi1 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], lambda1]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.09999999999999956e49 or 1 < phi1

   1. Initial program 99.7%

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

   3. Taylor expanded in lambda1 around inf

      \[\leadsto \color{blue}{\lambda_1} \]
   4. Step-by-step derivation

   5. Simplified 77.2%

      \[\leadsto \color{blue}{\lambda_1} \]

   if -5.09999999999999956e49 < phi1 < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) - \cos theta \cdot \sin delta\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) + \color{blue}{\left(\mathsf{neg}\left(\cos theta \cdot \sin delta\right)\right)}\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\left(\mathsf{neg}\left(\phi_1 \cdot \cos delta\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\cos theta \cdot \sin delta}\right)\right)\right)\right)\right)\right) \]

      3. distribute-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\mathsf{neg}\left(\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \left(\mathsf{neg}\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta - \color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\cos delta, \color{blue}{\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\color{blue}{\phi_1} \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \left(\cos theta \cdot \sin delta + \color{blue}{\phi_1 \cdot \cos delta}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\cos theta \cdot \sin delta\right), \color{blue}{\left(\phi_1 \cdot \cos delta\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\sin delta \cdot \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos theta\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\cos delta}\right)\right)\right)\right)\right)\right) \]

      16. cos-lowering-cos.f64 97.0%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \mathsf{cos.f64}\left(delta\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 97.0%

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

   6. Taylor expanded in delta around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(1 - {\phi_1}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left({\phi_1}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \left(\phi_1 \cdot \color{blue}{\phi_1}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 74.9%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \color{blue}{\phi_1}\right)\right)\right)\right) \]

   8. Simplified 74.9%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \left(\sin delta \cdot \sin theta\right)\right) + \sin delta \cdot \sin theta\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\sin delta \cdot \sin theta\right) + \sin delta \cdot \sin theta\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       2. distribute-lft1-in N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(\frac{-1}{2} \cdot {\phi_1}^{2} + 1\right) \cdot \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       3. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right), \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{2} \cdot {\phi_1}^{2}\right)\right), \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({\phi_1}^{2} \cdot \frac{-1}{2}\right)\right), \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\left(\phi_1 \cdot \phi_1\right) \cdot \frac{-1}{2}\right)\right), \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       8. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\phi_1 \cdot \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right), \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\phi_1 \cdot \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right), \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \left(\frac{-1}{2} \cdot \phi_1\right)\right)\right), \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right), \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right), \left(\sin delta \cdot \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\sin delta, \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       14. sin-lowering-sin.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       15. sin-lowering-sin.f64 74.8%

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

   11. Simplified 74.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(1 + \phi_1 \cdot \left(\phi_1 \cdot -0.5\right)\right) \cdot \left(\sin delta \cdot \sin theta\right)}}{1 - \phi_1 \cdot \phi_1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.1 \cdot 10^{+49}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \left(1 + \phi_1 \cdot \left(\phi_1 \cdot -0.5\right)\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.1:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 1.55:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -1.1)
   lambda1
   (if (<= phi1 1.55)
     (+ lambda1 (atan2 (* (sin delta) (sin theta)) (- 1.0 (* phi1 phi1))))
     lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -1.1) {
		tmp = lambda1;
	} else if (phi1 <= 1.55) {
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-1.1d0)) then
        tmp = lambda1
    else if (phi1 <= 1.55d0) then
        tmp = lambda1 + atan2((sin(delta) * sin(theta)), (1.0d0 - (phi1 * phi1)))
    else
        tmp = lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -1.1) {
		tmp = lambda1;
	} else if (phi1 <= 1.55) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -1.1:
		tmp = lambda1
	elif phi1 <= 1.55:
		tmp = lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), (1.0 - (phi1 * phi1)))
	else:
		tmp = lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -1.1)
		tmp = lambda1;
	elseif (phi1 <= 1.55)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), Float64(1.0 - Float64(phi1 * phi1))));
	else
		tmp = lambda1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -1.1)
		tmp = lambda1;
	elseif (phi1 <= 1.55)
		tmp = lambda1 + atan2((sin(delta) * sin(theta)), (1.0 - (phi1 * phi1)));
	else
		tmp = lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -1.1], lambda1, If[LessEqual[phi1, 1.55], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], lambda1]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.1000000000000001 or 1.55000000000000004 < phi1

   1. Initial program 99.6%

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

   3. Taylor expanded in lambda1 around inf

      \[\leadsto \color{blue}{\lambda_1} \]
   4. Step-by-step derivation

   5. Simplified 74.8%

      \[\leadsto \color{blue}{\lambda_1} \]

   if -1.1000000000000001 < phi1 < 1.55000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) - \cos theta \cdot \sin delta\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) + \color{blue}{\left(\mathsf{neg}\left(\cos theta \cdot \sin delta\right)\right)}\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\left(\mathsf{neg}\left(\phi_1 \cdot \cos delta\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\cos theta \cdot \sin delta}\right)\right)\right)\right)\right)\right) \]

      3. distribute-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\mathsf{neg}\left(\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \left(\mathsf{neg}\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta - \color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\cos delta, \color{blue}{\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\color{blue}{\phi_1} \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \left(\cos theta \cdot \sin delta + \color{blue}{\phi_1 \cdot \cos delta}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\cos theta \cdot \sin delta\right), \color{blue}{\left(\phi_1 \cdot \cos delta\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\sin delta \cdot \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos theta\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\cos delta}\right)\right)\right)\right)\right)\right) \]

      16. cos-lowering-cos.f64 99.7%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \mathsf{cos.f64}\left(delta\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.7%

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

   6. Taylor expanded in delta around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(1 - {\phi_1}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left({\phi_1}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \left(\phi_1 \cdot \color{blue}{\phi_1}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 76.3%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \color{blue}{\phi_1}\right)\right)\right)\right) \]

   8. Simplified 76.3%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       2. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       3. sin-lowering-sin.f64 76.3%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

   11. Simplified 76.3%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1 - \phi_1 \cdot \phi_1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 73.7% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0006:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 0.94:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \left(1 + \left(delta \cdot delta\right) \cdot \left(-0.16666666666666666 + \left(delta \cdot delta\right) \cdot \left(0.008333333333333333 + \left(delta \cdot delta\right) \cdot -0.0001984126984126984\right)\right)\right)\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -0.0006)
   lambda1
   (if (<= phi1 0.94)
     (+
      lambda1
      (atan2
       (*
        (sin theta)
        (*
         delta
         (+
          1.0
          (*
           (* delta delta)
           (+
            -0.16666666666666666
            (*
             (* delta delta)
             (+
              0.008333333333333333
              (* (* delta delta) -0.0001984126984126984))))))))
       (- 1.0 (* phi1 phi1))))
     lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -0.0006) {
		tmp = lambda1;
	} else if (phi1 <= 0.94) {
		tmp = lambda1 + atan2((sin(theta) * (delta * (1.0 + ((delta * delta) * (-0.16666666666666666 + ((delta * delta) * (0.008333333333333333 + ((delta * delta) * -0.0001984126984126984)))))))), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-0.0006d0)) then
        tmp = lambda1
    else if (phi1 <= 0.94d0) then
        tmp = lambda1 + atan2((sin(theta) * (delta * (1.0d0 + ((delta * delta) * ((-0.16666666666666666d0) + ((delta * delta) * (0.008333333333333333d0 + ((delta * delta) * (-0.0001984126984126984d0))))))))), (1.0d0 - (phi1 * phi1)))
    else
        tmp = lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -0.0006) {
		tmp = lambda1;
	} else if (phi1 <= 0.94) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (delta * (1.0 + ((delta * delta) * (-0.16666666666666666 + ((delta * delta) * (0.008333333333333333 + ((delta * delta) * -0.0001984126984126984)))))))), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -0.0006:
		tmp = lambda1
	elif phi1 <= 0.94:
		tmp = lambda1 + math.atan2((math.sin(theta) * (delta * (1.0 + ((delta * delta) * (-0.16666666666666666 + ((delta * delta) * (0.008333333333333333 + ((delta * delta) * -0.0001984126984126984)))))))), (1.0 - (phi1 * phi1)))
	else:
		tmp = lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -0.0006)
		tmp = lambda1;
	elseif (phi1 <= 0.94)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(delta * Float64(1.0 + Float64(Float64(delta * delta) * Float64(-0.16666666666666666 + Float64(Float64(delta * delta) * Float64(0.008333333333333333 + Float64(Float64(delta * delta) * -0.0001984126984126984)))))))), Float64(1.0 - Float64(phi1 * phi1))));
	else
		tmp = lambda1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -0.0006)
		tmp = lambda1;
	elseif (phi1 <= 0.94)
		tmp = lambda1 + atan2((sin(theta) * (delta * (1.0 + ((delta * delta) * (-0.16666666666666666 + ((delta * delta) * (0.008333333333333333 + ((delta * delta) * -0.0001984126984126984)))))))), (1.0 - (phi1 * phi1)));
	else
		tmp = lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -0.0006], lambda1, If[LessEqual[phi1, 0.94], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(1.0 + N[(N[(delta * delta), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(delta * delta), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(delta * delta), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], lambda1]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.99999999999999947e-4 or 0.93999999999999995 < phi1

   1. Initial program 99.6%

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

   3. Taylor expanded in lambda1 around inf

      \[\leadsto \color{blue}{\lambda_1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\lambda_1} \]

   if -5.99999999999999947e-4 < phi1 < 0.93999999999999995

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) - \cos theta \cdot \sin delta\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) + \color{blue}{\left(\mathsf{neg}\left(\cos theta \cdot \sin delta\right)\right)}\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\left(\mathsf{neg}\left(\phi_1 \cdot \cos delta\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\cos theta \cdot \sin delta}\right)\right)\right)\right)\right)\right) \]

      3. distribute-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\mathsf{neg}\left(\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \left(\mathsf{neg}\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta - \color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\cos delta, \color{blue}{\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\color{blue}{\phi_1} \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \left(\cos theta \cdot \sin delta + \color{blue}{\phi_1 \cdot \cos delta}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\cos theta \cdot \sin delta\right), \color{blue}{\left(\phi_1 \cdot \cos delta\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\sin delta \cdot \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos theta\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\cos delta}\right)\right)\right)\right)\right)\right) \]

      16. cos-lowering-cos.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \mathsf{cos.f64}\left(delta\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in delta around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(1 - {\phi_1}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left({\phi_1}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \left(\phi_1 \cdot \color{blue}{\phi_1}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 76.5%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \color{blue}{\phi_1}\right)\right)\right)\right) \]

   8. Simplified 76.5%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       2. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       3. sin-lowering-sin.f64 76.5%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

   11. Simplified 76.5%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1 - \phi_1 \cdot \phi_1} \]

   12. Taylor expanded in delta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(delta \cdot \left(1 + {delta}^{2} \cdot \left({delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right) - \frac{1}{6}\right)\right)\right)}, \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \left(1 + {delta}^{2} \cdot \left({delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right) - \frac{1}{6}\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \left({delta}^{2} \cdot \left({delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({delta}^{2}\right), \left({delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(delta \cdot delta\right), \left({delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \left({delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right) - \frac{1}{6}\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       6. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \left({delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       7. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \left({delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       8. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \left(\frac{-1}{6} + {delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{-1}{6}, \left({delta}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({delta}^{2}\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(delta \cdot delta\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \left(\frac{1}{120} + \frac{-1}{5040} \cdot {delta}^{2}\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       13. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{1}{120}, \left(\frac{-1}{5040} \cdot {delta}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({delta}^{2} \cdot \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({delta}^{2}\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       16. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(delta \cdot delta\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       17. *-lowering-*.f64 75.6%

           \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, delta\right), \frac{-1}{5040}\right)\right)\right)\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

   14. Simplified 75.6%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(delta \cdot \left(1 + \left(delta \cdot delta\right) \cdot \left(-0.16666666666666666 + \left(delta \cdot delta\right) \cdot \left(0.008333333333333333 + \left(delta \cdot delta\right) \cdot -0.0001984126984126984\right)\right)\right)\right)} \cdot \sin theta}{1 - \phi_1 \cdot \phi_1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0006:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 0.94:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \left(1 + \left(delta \cdot delta\right) \cdot \left(-0.16666666666666666 + \left(delta \cdot delta\right) \cdot \left(0.008333333333333333 + \left(delta \cdot delta\right) \cdot -0.0001984126984126984\right)\right)\right)\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.8% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00074:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 1.55:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \left(1 + \left(delta \cdot delta\right) \cdot -0.16666666666666666\right)\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -0.00074)
   lambda1
   (if (<= phi1 1.55)
     (+
      lambda1
      (atan2
       (*
        (sin theta)
        (* delta (+ 1.0 (* (* delta delta) -0.16666666666666666))))
       (- 1.0 (* phi1 phi1))))
     lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -0.00074) {
		tmp = lambda1;
	} else if (phi1 <= 1.55) {
		tmp = lambda1 + atan2((sin(theta) * (delta * (1.0 + ((delta * delta) * -0.16666666666666666)))), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-0.00074d0)) then
        tmp = lambda1
    else if (phi1 <= 1.55d0) then
        tmp = lambda1 + atan2((sin(theta) * (delta * (1.0d0 + ((delta * delta) * (-0.16666666666666666d0))))), (1.0d0 - (phi1 * phi1)))
    else
        tmp = lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -0.00074) {
		tmp = lambda1;
	} else if (phi1 <= 1.55) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (delta * (1.0 + ((delta * delta) * -0.16666666666666666)))), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -0.00074:
		tmp = lambda1
	elif phi1 <= 1.55:
		tmp = lambda1 + math.atan2((math.sin(theta) * (delta * (1.0 + ((delta * delta) * -0.16666666666666666)))), (1.0 - (phi1 * phi1)))
	else:
		tmp = lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -0.00074)
		tmp = lambda1;
	elseif (phi1 <= 1.55)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(delta * Float64(1.0 + Float64(Float64(delta * delta) * -0.16666666666666666)))), Float64(1.0 - Float64(phi1 * phi1))));
	else
		tmp = lambda1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -0.00074)
		tmp = lambda1;
	elseif (phi1 <= 1.55)
		tmp = lambda1 + atan2((sin(theta) * (delta * (1.0 + ((delta * delta) * -0.16666666666666666)))), (1.0 - (phi1 * phi1)));
	else
		tmp = lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -0.00074], lambda1, If[LessEqual[phi1, 1.55], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(1.0 + N[(N[(delta * delta), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], lambda1]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.3999999999999999e-4 or 1.55000000000000004 < phi1

   1. Initial program 99.6%

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

   3. Taylor expanded in lambda1 around inf

      \[\leadsto \color{blue}{\lambda_1} \]
   4. Step-by-step derivation

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\lambda_1} \]

   if -7.3999999999999999e-4 < phi1 < 1.55000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) - \cos theta \cdot \sin delta\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) + \color{blue}{\left(\mathsf{neg}\left(\cos theta \cdot \sin delta\right)\right)}\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\left(\mathsf{neg}\left(\phi_1 \cdot \cos delta\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\cos theta \cdot \sin delta}\right)\right)\right)\right)\right)\right) \]

      3. distribute-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\mathsf{neg}\left(\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \left(\mathsf{neg}\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta - \color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\cos delta, \color{blue}{\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\color{blue}{\phi_1} \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \left(\cos theta \cdot \sin delta + \color{blue}{\phi_1 \cdot \cos delta}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\cos theta \cdot \sin delta\right), \color{blue}{\left(\phi_1 \cdot \cos delta\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\sin delta \cdot \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos theta\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\cos delta}\right)\right)\right)\right)\right)\right) \]

      16. cos-lowering-cos.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \mathsf{cos.f64}\left(delta\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 99.9%

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

   6. Taylor expanded in delta around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(1 - {\phi_1}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left({\phi_1}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \left(\phi_1 \cdot \color{blue}{\phi_1}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 76.5%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \color{blue}{\phi_1}\right)\right)\right)\right) \]

   8. Simplified 76.5%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       2. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       3. sin-lowering-sin.f64 76.5%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

   11. Simplified 76.5%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1 - \phi_1 \cdot \phi_1} \]

   12. Taylor expanded in delta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(delta \cdot \left(1 + \frac{-1}{6} \cdot {delta}^{2}\right)\right)}, \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \left(1 + \frac{-1}{6} \cdot {delta}^{2}\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \left(\frac{-1}{6} \cdot {delta}^{2}\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left({delta}^{2}\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \left(delta \cdot delta\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       5. *-lowering-*.f64 75.4%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(delta, delta\right)\right)\right)\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

   14. Simplified 75.4%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(delta \cdot \left(1 + -0.16666666666666666 \cdot \left(delta \cdot delta\right)\right)\right)} \cdot \sin theta}{1 - \phi_1 \cdot \phi_1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00074:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 1.55:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \left(1 + \left(delta \cdot delta\right) \cdot -0.16666666666666666\right)\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.6 \cdot 10^{+49}:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 5 \cdot 10^{-76}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -4.6e+49)
   lambda1
   (if (<= phi1 5e-76)
     (+ lambda1 (atan2 (* delta (sin theta)) (- 1.0 (* phi1 phi1))))
     lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -4.6e+49) {
		tmp = lambda1;
	} else if (phi1 <= 5e-76) {
		tmp = lambda1 + atan2((delta * sin(theta)), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-4.6d+49)) then
        tmp = lambda1
    else if (phi1 <= 5d-76) then
        tmp = lambda1 + atan2((delta * sin(theta)), (1.0d0 - (phi1 * phi1)))
    else
        tmp = lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -4.6e+49) {
		tmp = lambda1;
	} else if (phi1 <= 5e-76) {
		tmp = lambda1 + Math.atan2((delta * Math.sin(theta)), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -4.6e+49:
		tmp = lambda1
	elif phi1 <= 5e-76:
		tmp = lambda1 + math.atan2((delta * math.sin(theta)), (1.0 - (phi1 * phi1)))
	else:
		tmp = lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -4.6e+49)
		tmp = lambda1;
	elseif (phi1 <= 5e-76)
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), Float64(1.0 - Float64(phi1 * phi1))));
	else
		tmp = lambda1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -4.6e+49)
		tmp = lambda1;
	elseif (phi1 <= 5e-76)
		tmp = lambda1 + atan2((delta * sin(theta)), (1.0 - (phi1 * phi1)));
	else
		tmp = lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -4.6e+49], lambda1, If[LessEqual[phi1, 5e-76], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], lambda1]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.60000000000000004e49 or 4.9999999999999998e-76 < phi1

   1. Initial program 99.7%

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

   3. Taylor expanded in lambda1 around inf

      \[\leadsto \color{blue}{\lambda_1} \]
   4. Step-by-step derivation

   5. Simplified 76.9%

      \[\leadsto \color{blue}{\lambda_1} \]

   if -4.60000000000000004e49 < phi1 < 4.9999999999999998e-76

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) - \cos theta \cdot \sin delta\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(-1 \cdot \left(\phi_1 \cdot \cos delta\right) + \color{blue}{\left(\mathsf{neg}\left(\cos theta \cdot \sin delta\right)\right)}\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\left(\mathsf{neg}\left(\phi_1 \cdot \cos delta\right)\right) + \left(\mathsf{neg}\left(\color{blue}{\cos theta \cdot \sin delta}\right)\right)\right)\right)\right)\right) \]

      3. distribute-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \phi_1 \cdot \left(\mathsf{neg}\left(\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta + \left(\mathsf{neg}\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \left(\cos delta - \color{blue}{\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\cos delta, \color{blue}{\left(\phi_1 \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)}\right)\right)\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\color{blue}{\phi_1} \cdot \left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(\phi_1 \cdot \cos delta + \cos theta \cdot \sin delta\right)}\right)\right)\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \left(\cos theta \cdot \sin delta + \color{blue}{\phi_1 \cdot \cos delta}\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\cos theta \cdot \sin delta\right), \color{blue}{\left(\phi_1 \cdot \cos delta\right)}\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\left(\sin delta \cdot \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin delta, \cos theta\right), \left(\color{blue}{\phi_1} \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \cos theta\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \left(\phi_1 \cdot \cos delta\right)\right)\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \color{blue}{\cos delta}\right)\right)\right)\right)\right)\right) \]

      16. cos-lowering-cos.f64 96.6%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \mathsf{*.f64}\left(\phi_1, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{cos.f64}\left(theta\right)\right), \mathsf{*.f64}\left(\phi_1, \mathsf{cos.f64}\left(delta\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 96.6%

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

   6. Taylor expanded in delta around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \color{blue}{\left(1 - {\phi_1}^{2}\right)}\right)\right) \]
   7. Step-by-step derivation

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \color{blue}{\left({\phi_1}^{2}\right)}\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \left(\phi_1 \cdot \color{blue}{\phi_1}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 74.8%

         \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \color{blue}{\phi_1}\right)\right)\right)\right) \]

   8. Simplified 74.8%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\sin delta, \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       2. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \sin theta\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       3. sin-lowering-sin.f64 73.8%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(delta\right), \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

   11. Simplified 73.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1 - \phi_1 \cdot \phi_1} \]

   12. Taylor expanded in delta around 0

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(delta \cdot \sin theta\right)}, \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(delta, \sin theta\right), \mathsf{\_.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

       2. sin-lowering-sin.f64 71.8%

          \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(delta, \mathsf{sin.f64}\left(theta\right)\right), \mathsf{\_.f64}\left(1, \mathsf{*.f64}\left(\phi_1, \phi_1\right)\right)\right)\right) \]

   14. Simplified 71.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{delta \cdot \sin theta}}{1 - \phi_1 \cdot \phi_1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 70.5% accurate, 1320.0× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta) :precision binary64 lambda1)

C

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

Fortran

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

Java

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

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return lambda1
end

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := lambda1

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in lambda1 around inf

   \[\leadsto \color{blue}{\lambda_1} \]
4. Step-by-step derivation

5. Simplified 70.9%

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

Reproduce

herbie shell --seed 2024191 
(FPCore (lambda1 phi1 phi2 delta theta)
  :name "Destination given bearing on a great circle"
  :precision binary64
  (+ lambda1 (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (- (cos delta) (* (sin phi1) (sin (asin (+ (* (sin phi1) (cos delta)) (* (* (cos phi1) (sin delta)) (cos theta))))))))))