Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.9%

Time: 22.9s

Alternatives: 14

Speedup: 1.4×

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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Derivation

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. 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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate--r+ N/A

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

4. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. sqr-sin-a N/A

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

   3. unpow2 N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. distribute-rgt1-in N/A

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

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

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

6. Applied egg-rr 99.8%

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

   1. +-commutative N/A

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

   2. associate-+r+ N/A

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

   3. metadata-eval N/A

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

   4. sqr-cos-a N/A

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

   5. pow2 N/A

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

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

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

   7. cos-lowering-cos.f64 99.9%

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

8. Applied egg-rr 99.9%

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. fmm-def N/A

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

   5. *-commutative N/A

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

   6. fmm-def N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. associate-*l* N/A

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

   11. distribute-lft-out-- N/A

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

10. Applied egg-rr 99.9%

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

Alternative 2: 99.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta \cdot \left(0.5 + \frac{\cos \left(\phi_1 \cdot 2\right)}{2}\right) - \frac{\sin \left(\phi_1 \cdot 2\right)}{2} \cdot \left(\sin delta \cdot \cos theta\right)} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (* (cos delta) (+ 0.5 (/ (cos (* phi1 2.0)) 2.0)))
    (* (/ (sin (* phi1 2.0)) 2.0) (* (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) * (0.5 + (cos((phi1 * 2.0)) / 2.0))) - ((sin((phi1 * 2.0)) / 2.0) * (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) * (0.5d0 + (cos((phi1 * 2.0d0)) / 2.0d0))) - ((sin((phi1 * 2.0d0)) / 2.0d0) * (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) * (0.5 + (Math.cos((phi1 * 2.0)) / 2.0))) - ((Math.sin((phi1 * 2.0)) / 2.0) * (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) * (0.5 + (math.cos((phi1 * 2.0)) / 2.0))) - ((math.sin((phi1 * 2.0)) / 2.0) * (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(Float64(cos(delta) * Float64(0.5 + Float64(cos(Float64(phi1 * 2.0)) / 2.0))) - Float64(Float64(sin(Float64(phi1 * 2.0)) / 2.0) * Float64(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) * (0.5 + (cos((phi1 * 2.0)) / 2.0))) - ((sin((phi1 * 2.0)) / 2.0) * (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[(N[Cos[delta], $MachinePrecision] * N[(0.5 + N[(N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[Sin[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. 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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate--r+ N/A

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

4. Applied egg-rr 99.8%

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

   1. cancel-sign-sub-inv N/A

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

   2. sqr-sin-a N/A

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

   3. unpow2 N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. unpow2 N/A

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

   6. sqr-sin-a N/A

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

   7. sub-neg N/A

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

   8. distribute-lft-in N/A

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

   9. associate--r+ N/A

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

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

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

   11. --lowering--.f64 N/A

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

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

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

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

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

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

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

   15. *-commutative N/A

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

6. Applied egg-rr 99.8%

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

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta \cdot \left(0.5 + \frac{\cos \left(2 \cdot \phi_1\right)}{2}\right) - \frac{0 + \sin \left(2 \cdot \phi_1\right)}{2} \cdot \left(\sin delta \cdot \cos theta\right)} + \lambda_1} \]

9. Final simplification 99.8%

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

Alternative 3: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta \cdot \left(\left(-0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)\right) + 1\right) - \cos \phi_1 \cdot \left(\sin delta \cdot \sin \phi_1\right)} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (-
    (* (cos delta) (+ (+ -0.5 (* 0.5 (cos (* phi1 2.0)))) 1.0))
    (* (cos phi1) (* (sin delta) (sin phi1)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), ((cos(delta) * ((-0.5 + (0.5 * cos((phi1 * 2.0)))) + 1.0)) - (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(((sin(theta) * sin(delta)) * cos(phi1)), ((cos(delta) * (((-0.5d0) + (0.5d0 * cos((phi1 * 2.0d0)))) + 1.0d0)) - (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.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), ((Math.cos(delta) * ((-0.5 + (0.5 * Math.cos((phi1 * 2.0)))) + 1.0)) - (Math.cos(phi1) * (Math.sin(delta) * Math.sin(phi1)))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), ((math.cos(delta) * ((-0.5 + (0.5 * math.cos((phi1 * 2.0)))) + 1.0)) - (math.cos(phi1) * (math.sin(delta) * math.sin(phi1)))))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(Float64(cos(delta) * Float64(Float64(-0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0)))) + 1.0)) - Float64(cos(phi1) * Float64(sin(delta) * sin(phi1))))))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), ((cos(delta) * ((-0.5 + (0.5 * cos((phi1 * 2.0)))) + 1.0)) - (cos(phi1) * (sin(delta) * sin(phi1)))));
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[(N[Cos[delta], $MachinePrecision] * N[(N[(-0.5 + N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $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 99.7%

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

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate--r+ N/A

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

4. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. sqr-sin-a N/A

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

   3. unpow2 N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. distribute-rgt1-in N/A

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

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

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in theta around 0

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

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

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

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

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

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

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

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

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

   5. sin-lowering-sin.f64 93.7%

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

9. Simplified 93.7%

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

10. Final simplification 93.7%

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

Alternative 4: 92.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(0 - \sin \phi_1, \sin \phi_1, \cos delta\right)} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (fma (- 0.0 (sin phi1)) (sin phi1) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), fma((0.0 - sin(phi1)), sin(phi1), cos(delta)));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), fma(Float64(0.0 - sin(phi1)), sin(phi1), cos(delta))))
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[(0.0 - N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[Cos[delta], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 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 91.0%

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

5. Simplified 91.0%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. unpow2 N/A

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

   4. distribute-lft-neg-in N/A

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

   5. fma-define N/A

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

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

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

   7. neg-sub0 N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{fma.f64}\left(\left(0 - \sin \phi_1\right), \sin \color{blue}{\phi_1}, \cos delta\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \sin \phi_1\right), \sin \color{blue}{\phi_1}, \cos delta\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\phi_1\right)\right), \sin \phi_1, \cos delta\right)\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(\phi_1\right), \cos delta\right)\right)\right) \]

   11. cos-lowering-cos.f64 91.0%

       \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\phi_1\right)\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(delta\right)\right)\right)\right) \]

7. Applied egg-rr 91.0%

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

Alternative 5: 91.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -1760000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (* (sin theta) (sin delta)) (cos phi1))))
   (if (<= delta -1760000000.0)
     (+ lambda1 (atan2 t_1 (- (cos delta) (* phi1 phi1))))
     (if (<= delta 1.35e-40)
       (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0)))
       (+
        lambda1
        (atan2 (* (sin delta) (* (sin theta) (cos phi1))) (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	double tmp;
	if (delta <= -1760000000.0) {
		tmp = lambda1 + atan2(t_1, (cos(delta) - (phi1 * phi1)));
	} else if (delta <= 1.35e-40) {
		tmp = lambda1 + atan2(t_1, pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	}
	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) :: tmp
    t_1 = (sin(theta) * sin(delta)) * cos(phi1)
    if (delta <= (-1760000000.0d0)) then
        tmp = lambda1 + atan2(t_1, (cos(delta) - (phi1 * phi1)))
    else if (delta <= 1.35d-40) then
        tmp = lambda1 + atan2(t_1, (cos(phi1) ** 2.0d0))
    else
        tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta))
    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.sin(theta) * Math.sin(delta)) * Math.cos(phi1);
	double tmp;
	if (delta <= -1760000000.0) {
		tmp = lambda1 + Math.atan2(t_1, (Math.cos(delta) - (phi1 * phi1)));
	} else if (delta <= 1.35e-40) {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = (math.sin(theta) * math.sin(delta)) * math.cos(phi1)
	tmp = 0
	if delta <= -1760000000.0:
		tmp = lambda1 + math.atan2(t_1, (math.cos(delta) - (phi1 * phi1)))
	elif delta <= 1.35e-40:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * (math.sin(theta) * math.cos(phi1))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(Float64(sin(theta) * sin(delta)) * cos(phi1))
	tmp = 0.0
	if (delta <= -1760000000.0)
		tmp = Float64(lambda1 + atan(t_1, Float64(cos(delta) - Float64(phi1 * phi1))));
	elseif (delta <= 1.35e-40)
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	tmp = 0.0;
	if (delta <= -1760000000.0)
		tmp = lambda1 + atan2(t_1, (cos(delta) - (phi1 * phi1)));
	elseif (delta <= 1.35e-40)
		tmp = lambda1 + atan2(t_1, (cos(phi1) ^ 2.0));
	else
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -1760000000.0], N[(lambda1 + N[ArcTan[t$95$1 / N[(N[Cos[delta], $MachinePrecision] - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 1.35e-40], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -1.76e9

   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 87.4%

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

   5. Simplified 87.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 87.6%

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

   8. Simplified 87.6%

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

   if -1.76e9 < delta < 1.35e-40

   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 delta around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 99.2%

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

   5. Simplified 99.2%

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. pow2 N/A

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

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

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

      5. cos-lowering-cos.f64 99.4%

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

   7. Applied egg-rr 99.4%

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

   if 1.35e-40 < 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. Step-by-step derivation

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

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

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

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1760000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{0.5 \cdot \cos \left(\phi_1 \cdot 2\right) + \left(\cos delta - 0.5\right)} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (sin theta) (* (sin delta) (cos phi1)))
   (+ (* 0.5 (cos (* phi1 2.0))) (- (cos delta) 0.5)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), ((0.5 * cos((phi1 * 2.0))) + (cos(delta) - 0.5)));
}

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))), ((0.5d0 * cos((phi1 * 2.0d0))) + (cos(delta) - 0.5d0)))
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))), ((0.5 * Math.cos((phi1 * 2.0))) + (Math.cos(delta) - 0.5)));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), ((0.5 * math.cos((phi1 * 2.0))) + (math.cos(delta) - 0.5)))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(Float64(0.5 * cos(Float64(phi1 * 2.0))) + Float64(cos(delta) - 0.5))))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), ((0.5 * cos((phi1 * 2.0))) + (cos(delta) - 0.5)));
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 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 91.0%

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

5. Simplified 91.0%

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

   1. +-commutative N/A

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

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

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

7. Applied egg-rr 91.0%

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

8. Final simplification 91.0%

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

Alternative 7: 92.3% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (+ (cos delta) (/ (+ (cos (* phi1 2.0)) -1.0) 2.0)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) + ((cos((phi1 * 2.0)) + -1.0) / 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(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) + ((cos((phi1 * 2.0d0)) + (-1.0d0)) / 2.0d0)))
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.cos((phi1 * 2.0)) + -1.0) / 2.0)));
}

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.cos((phi1 * 2.0)) + -1.0) / 2.0)))

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) + Float64(Float64(cos(Float64(phi1 * 2.0)) + -1.0) / 2.0))))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) + ((cos((phi1 * 2.0)) + -1.0) / 2.0)));
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[(N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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 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 91.0%

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

5. Simplified 91.0%

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

   1. unpow2 N/A

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

   2. sin-mult N/A

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

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

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

   4. count-2 N/A

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

   5. cos-diff N/A

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

   6. cos-sin-sum N/A

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

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

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

   8. count-2 N/A

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

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

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

   10. count-2 N/A

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

   11. *-lowering-*.f64 91.0%

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

7. Applied egg-rr 91.0%

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

8. Final simplification 91.0%

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

Alternative 8: 91.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -820000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + \frac{\cos \left(\phi_1 \cdot 2\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -820000000.0)
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (- (cos delta) (* phi1 phi1))))
   (if (<= delta 1.35e-40)
     (+
      lambda1
      (atan2
       (* (cos phi1) (* (sin theta) delta))
       (+ 0.5 (/ (cos (* phi1 2.0)) 2.0))))
     (+
      lambda1
      (atan2 (* (sin delta) (* (sin theta) (cos phi1))) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -820000000.0) {
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (phi1 * phi1)));
	} else if (delta <= 1.35e-40) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (cos((phi1 * 2.0)) / 2.0)));
	} else {
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	}
	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 (delta <= (-820000000.0d0)) then
        tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (phi1 * phi1)))
    else if (delta <= 1.35d-40) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5d0 + (cos((phi1 * 2.0d0)) / 2.0d0)))
    else
        tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -820000000.0) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (Math.cos(delta) - (phi1 * phi1)));
	} else if (delta <= 1.35e-40) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (0.5 + (Math.cos((phi1 * 2.0)) / 2.0)));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -820000000.0:
		tmp = lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (math.cos(delta) - (phi1 * phi1)))
	elif delta <= 1.35e-40:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (0.5 + (math.cos((phi1 * 2.0)) / 2.0)))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * (math.sin(theta) * math.cos(phi1))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -820000000.0)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(cos(delta) - Float64(phi1 * phi1))));
	elseif (delta <= 1.35e-40)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(0.5 + Float64(cos(Float64(phi1 * 2.0)) / 2.0))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -820000000.0)
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (cos(delta) - (phi1 * phi1)));
	elseif (delta <= 1.35e-40)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (cos((phi1 * 2.0)) / 2.0)));
	else
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -820000000.0], 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[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 1.35e-40], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -8.2e8

   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 87.4%

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

   5. Simplified 87.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 87.6%

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

   8. Simplified 87.6%

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

   if -8.2e8 < delta < 1.35e-40

   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 delta around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in delta around 0

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

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

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

      2. sin-lowering-sin.f64 99.2%

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

   8. Simplified 99.2%

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

      1. +-commutative N/A

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

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

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

   10. Applied egg-rr 99.3%

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

   if 1.35e-40 < 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. Step-by-step derivation

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

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

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

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -820000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 1.35 \cdot 10^{-40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + \frac{\cos \left(\phi_1 \cdot 2\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

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

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

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

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

3. Simplified 99.7%

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

5. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 87.7%

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

7. Simplified 87.7%

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

8. Final simplification 87.7%

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

Alternative 10: 81.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{if}\;delta \leq -1.42 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + \frac{\cos \left(\phi_1 \cdot 2\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2
           (* (* (sin theta) (sin delta)) (cos phi1))
           (- 1.0 (* phi1 phi1))))))
   (if (<= delta -1.42e+17)
     t_1
     (if (<= delta 3e+109)
       (+
        lambda1
        (atan2
         (* (cos phi1) (* (sin theta) delta))
         (+ 0.5 (/ (cos (* phi1 2.0)) 2.0))))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (1.0 - (phi1 * phi1)));
	double tmp;
	if (delta <= -1.42e+17) {
		tmp = t_1;
	} else if (delta <= 3e+109) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (cos((phi1 * 2.0)) / 2.0)));
	} else {
		tmp = t_1;
	}
	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) :: tmp
    t_1 = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (1.0d0 - (phi1 * phi1)))
    if (delta <= (-1.42d+17)) then
        tmp = t_1
    else if (delta <= 3d+109) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5d0 + (cos((phi1 * 2.0d0)) / 2.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (1.0 - (phi1 * phi1)));
	double tmp;
	if (delta <= -1.42e+17) {
		tmp = t_1;
	} else if (delta <= 3e+109) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (0.5 + (Math.cos((phi1 * 2.0)) / 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (1.0 - (phi1 * phi1)))
	tmp = 0
	if delta <= -1.42e+17:
		tmp = t_1
	elif delta <= 3e+109:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (0.5 + (math.cos((phi1 * 2.0)) / 2.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(1.0 - Float64(phi1 * phi1))))
	tmp = 0.0
	if (delta <= -1.42e+17)
		tmp = t_1;
	elseif (delta <= 3e+109)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(0.5 + Float64(cos(Float64(phi1 * 2.0)) / 2.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (1.0 - (phi1 * phi1)));
	tmp = 0.0;
	if (delta <= -1.42e+17)
		tmp = t_1;
	elseif (delta <= 3e+109)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (cos((phi1 * 2.0)) / 2.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -1.42e+17], t$95$1, If[LessEqual[delta, 3e+109], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.42e17 or 3.00000000000000015e109 < 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 delta around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 67.0%

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

   8. Simplified 67.0%

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

   if -1.42e17 < delta < 3.00000000000000015e109

   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 delta around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 93.1%

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

   5. Simplified 93.1%

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

   6. Taylor expanded in delta around 0

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

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

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

      2. sin-lowering-sin.f64 92.9%

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

   8. Simplified 92.9%

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

      1. +-commutative N/A

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

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

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

   10. Applied egg-rr 93.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.42 \cdot 10^{+17}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;delta \leq 3 \cdot 10^{+109}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + \frac{\cos \left(\phi_1 \cdot 2\right)}{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\\ \mathbf{if}\;delta \leq -1.32 \cdot 10^{+17}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (* (sin theta) (sin delta)) (cos phi1))))
   (if (<= delta -1.32e+17)
     (+ lambda1 (atan2 t_1 (- 1.0 (* phi1 phi1))))
     (+ lambda1 (atan2 t_1 1.0)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	double tmp;
	if (delta <= -1.32e+17) {
		tmp = lambda1 + atan2(t_1, (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1 + atan2(t_1, 1.0);
	}
	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) :: tmp
    t_1 = (sin(theta) * sin(delta)) * cos(phi1)
    if (delta <= (-1.32d+17)) then
        tmp = lambda1 + atan2(t_1, (1.0d0 - (phi1 * phi1)))
    else
        tmp = lambda1 + atan2(t_1, 1.0d0)
    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.sin(theta) * Math.sin(delta)) * Math.cos(phi1);
	double tmp;
	if (delta <= -1.32e+17) {
		tmp = lambda1 + Math.atan2(t_1, (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1 + Math.atan2(t_1, 1.0);
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = (math.sin(theta) * math.sin(delta)) * math.cos(phi1)
	tmp = 0
	if delta <= -1.32e+17:
		tmp = lambda1 + math.atan2(t_1, (1.0 - (phi1 * phi1)))
	else:
		tmp = lambda1 + math.atan2(t_1, 1.0)
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(Float64(sin(theta) * sin(delta)) * cos(phi1))
	tmp = 0.0
	if (delta <= -1.32e+17)
		tmp = Float64(lambda1 + atan(t_1, Float64(1.0 - Float64(phi1 * phi1))));
	else
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	tmp = 0.0;
	if (delta <= -1.32e+17)
		tmp = lambda1 + atan2(t_1, (1.0 - (phi1 * phi1)));
	else
		tmp = lambda1 + atan2(t_1, 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -1.32e+17], N[(lambda1 + N[ArcTan[t$95$1 / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.32e17

   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

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

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

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

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

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

      3. sin-lowering-sin.f64 64.4%

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

   5. Simplified 64.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 71.7%

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

   8. Simplified 71.7%

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

   if -1.32e17 < 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 delta around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 83.7%

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

   5. Simplified 83.7%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 79.8%

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

Alternative 12: 77.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

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(((sin(theta) * sin(delta)) * cos(phi1)), 1.0d0)
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)), 1.0);
}

Python

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

Julia

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

MATLAB

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

Derivation

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 delta around 0

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

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

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

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

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

   3. sin-lowering-sin.f64 78.9%

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

5. Simplified 78.9%

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

6. Taylor expanded in phi1 around 0

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

8. Simplified 75.9%

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

Alternative 13: 73.7% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

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(theta) * delta)), 1.0d0)
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(theta) * delta)), 1.0);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 delta around 0

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

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

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

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

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

   3. sin-lowering-sin.f64 78.9%

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

5. Simplified 78.9%

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

6. Taylor expanded in delta around 0

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

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

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

   2. sin-lowering-sin.f64 76.5%

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

8. Simplified 76.5%

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

9. Taylor expanded in phi1 around 0

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

11. Simplified 73.7%

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

12. Final simplification 73.7%

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

Alternative 14: 69.8% 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 99.7%

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

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

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

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

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

3. Simplified 99.7%

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

5. Taylor expanded in lambda1 around inf

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

7. Simplified 69.2%

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

Reproduce

herbie shell --seed 2024164 
(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))))))))))