Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.7%

Time: 20.6s

Alternatives: 13

Speedup: 1.2×

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: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sin-asin N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

   1. associate--r+ N/A

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

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

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. associate-*l* N/A

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

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

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

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

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

   16. sqr-sin-a N/A

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

6. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   15. cos-lowering-cos.f64 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), ((cos(delta) - (sin(delta) * (sin(phi1) * (cos(phi1) * cos(theta))))) - (cos(delta) * (0.5d0 - (0.5d0 * cos((phi1 * 2.0d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sin-asin N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

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

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

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

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 99.8%

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

   1. associate--r+ N/A

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

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

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

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

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

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

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

   5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. associate-*l* N/A

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

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

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

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

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

   16. sqr-sin-a N/A

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sin-asin N/A

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

   2. flip3-+ N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\sin \phi_1 \cdot \frac{{\left(\sin \phi_1 \cdot \cos delta\right)}^{3} + {\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}^{3}}{\color{blue}{\left(\sin \phi_1 \cdot \cos delta\right) \cdot \left(\sin \phi_1 \cdot \cos delta\right) + \left(\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) - \left(\sin \phi_1 \cdot \cos delta\right) \cdot \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}}\right)\right)\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(theta\right), \mathsf{sin.f64}\left(delta\right)\right), \mathsf{cos.f64}\left(\phi_1\right)\right), \mathsf{\_.f64}\left(\mathsf{cos.f64}\left(delta\right), \left(\sin \phi_1 \cdot \frac{1}{\color{blue}{\frac{\left(\sin \phi_1 \cdot \cos delta\right) \cdot \left(\sin \phi_1 \cdot \cos delta\right) + \left(\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) \cdot \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right) - \left(\sin \phi_1 \cdot \cos delta\right) \cdot \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)\right)}{{\left(\sin \phi_1 \cdot \cos delta\right)}^{3} + {\left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}^{3}}}}\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. +-commutative N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 5: 95.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in theta around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. sin-lowering-sin.f64 94.6%

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

5. Simplified 94.6%

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

6. Final simplification 94.6%

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

Alternative 6: 92.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in delta around 0

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

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

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

   2. sin-lowering-sin.f64 93.5%

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

5. Simplified 93.5%

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

6. Final simplification 93.5%

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

Alternative 7: 92.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;delta \leq -0.7:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;delta \leq 0.000175:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin theta) (sin delta))))
        (t_2 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= delta -0.7)
     t_2
     (if (<= delta 0.000175)
       (+ lambda1 (atan2 t_1 (+ 0.5 (* 0.5 (cos (* phi1 2.0))))))
       t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(theta) * sin(delta));
	double t_2 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (delta <= -0.7) {
		tmp = t_2;
	} else if (delta <= 0.000175) {
		tmp = lambda1 + atan2(t_1, (0.5 + (0.5 * cos((phi1 * 2.0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(phi1) * (sin(theta) * sin(delta))
    t_2 = lambda1 + atan2(t_1, cos(delta))
    if (delta <= (-0.7d0)) then
        tmp = t_2
    else if (delta <= 0.000175d0) then
        tmp = lambda1 + atan2(t_1, (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta));
	double t_2 = lambda1 + Math.atan2(t_1, Math.cos(delta));
	double tmp;
	if (delta <= -0.7) {
		tmp = t_2;
	} else if (delta <= 0.000175) {
		tmp = lambda1 + Math.atan2(t_1, (0.5 + (0.5 * Math.cos((phi1 * 2.0)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(theta) * math.sin(delta))
	t_2 = lambda1 + math.atan2(t_1, math.cos(delta))
	tmp = 0
	if delta <= -0.7:
		tmp = t_2
	elif delta <= 0.000175:
		tmp = lambda1 + math.atan2(t_1, (0.5 + (0.5 * math.cos((phi1 * 2.0)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(theta) * sin(delta)))
	t_2 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (delta <= -0.7)
		tmp = t_2;
	elseif (delta <= 0.000175)
		tmp = Float64(lambda1 + atan(t_1, Float64(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(theta) * sin(delta));
	t_2 = lambda1 + atan2(t_1, cos(delta));
	tmp = 0.0;
	if (delta <= -0.7)
		tmp = t_2;
	elseif (delta <= 0.000175)
		tmp = lambda1 + atan2(t_1, (0.5 + (0.5 * cos((phi1 * 2.0)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -0.7], t$95$2, If[LessEqual[delta, 0.000175], N[(lambda1 + N[ArcTan[t$95$1 / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if delta < -0.69999999999999996 or 1.74999999999999998e-4 < delta

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 88.7%

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

   5. Simplified 88.7%

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

   if -0.69999999999999996 < delta < 1.74999999999999998e-4

   1. Initial program 99.8%

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

      1. sin-asin N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

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

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

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

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 99.8%

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

      1. associate--r+ N/A

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

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

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. associate-*l* N/A

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

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

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

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

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

      16. sqr-sin-a N/A

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in delta around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 99.7%

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

   9. Simplified 99.7%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.7:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.000175:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 90.8%

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

5. Simplified 90.8%

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

6. Final simplification 90.8%

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

Alternative 9: 85.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 \cdot \left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos delta}}{\lambda_1}\right)\\ \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(1 + \phi_1 \cdot \left(\phi_1 \cdot -0.5\right)\right) \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (*
          lambda1
          (+
           1.0
           (/
            (atan2 (* (cos phi1) (* (sin theta) delta)) (cos delta))
            lambda1)))))
   (if (<= phi1 -3.5e+20)
     t_1
     (if (<= phi1 1.05e-32)
       (+
        lambda1
        (atan2
         (* (+ 1.0 (* phi1 (* phi1 -0.5))) (* (sin theta) (sin delta)))
         (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 * (1.0 + (atan2((cos(phi1) * (sin(theta) * delta)), cos(delta)) / lambda1));
	double tmp;
	if (phi1 <= -3.5e+20) {
		tmp = t_1;
	} else if (phi1 <= 1.05e-32) {
		tmp = lambda1 + atan2(((1.0 + (phi1 * (phi1 * -0.5))) * (sin(theta) * sin(delta))), cos(delta));
	} 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 * (1.0d0 + (atan2((cos(phi1) * (sin(theta) * delta)), cos(delta)) / lambda1))
    if (phi1 <= (-3.5d+20)) then
        tmp = t_1
    else if (phi1 <= 1.05d-32) then
        tmp = lambda1 + atan2(((1.0d0 + (phi1 * (phi1 * (-0.5d0)))) * (sin(theta) * sin(delta))), cos(delta))
    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 * (1.0 + (Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), Math.cos(delta)) / lambda1));
	double tmp;
	if (phi1 <= -3.5e+20) {
		tmp = t_1;
	} else if (phi1 <= 1.05e-32) {
		tmp = lambda1 + Math.atan2(((1.0 + (phi1 * (phi1 * -0.5))) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 * (1.0 + (math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), math.cos(delta)) / lambda1))
	tmp = 0
	if phi1 <= -3.5e+20:
		tmp = t_1
	elif phi1 <= 1.05e-32:
		tmp = lambda1 + math.atan2(((1.0 + (phi1 * (phi1 * -0.5))) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 * Float64(1.0 + Float64(atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), cos(delta)) / lambda1)))
	tmp = 0.0
	if (phi1 <= -3.5e+20)
		tmp = t_1;
	elseif (phi1 <= 1.05e-32)
		tmp = Float64(lambda1 + atan(Float64(Float64(1.0 + Float64(phi1 * Float64(phi1 * -0.5))) * Float64(sin(theta) * sin(delta))), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 * (1.0 + (atan2((cos(phi1) * (sin(theta) * delta)), cos(delta)) / lambda1));
	tmp = 0.0;
	if (phi1 <= -3.5e+20)
		tmp = t_1;
	elseif (phi1 <= 1.05e-32)
		tmp = lambda1 + atan2(((1.0 + (phi1 * (phi1 * -0.5))) * (sin(theta) * sin(delta))), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 * N[(1.0 + N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.5e+20], t$95$1, If[LessEqual[phi1, 1.05e-32], N[(lambda1 + N[ArcTan[N[(N[(1.0 + N[(phi1 * N[(phi1 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.5e20 or 1.05e-32 < phi1

   1. Initial program 99.8%

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

   3. Taylor expanded in lambda1 around inf

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

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

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

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

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

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

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

   5. Simplified 99.7%

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

   6. Taylor expanded in delta around 0

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

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

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

      2. sin-lowering-sin.f64 83.4%

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

   8. Simplified 83.4%

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

   9. Taylor expanded in phi1 around 0

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

       1. cos-lowering-cos.f64 76.9%

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

   11. Simplified 76.9%

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

   if -3.5e20 < phi1 < 1.05e-32

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. sin-lowering-sin.f64 97.8%

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

   8. Simplified 97.8%

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

   9. Taylor expanded in phi1 around 0

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

       1. cos-lowering-cos.f64 98.9%

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

   11. Simplified 98.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{+20}:\\ \;\;\;\;\lambda_1 \cdot \left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos delta}}{\lambda_1}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(1 + \phi_1 \cdot \left(\phi_1 \cdot -0.5\right)\right) \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 \cdot \left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos delta}}{\lambda_1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 \cdot \left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos delta}}{\lambda_1}\right)\\ \mathbf{if}\;\phi_1 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (*
          lambda1
          (+
           1.0
           (/
            (atan2 (* (cos phi1) (* (sin theta) delta)) (cos delta))
            lambda1)))))
   (if (<= phi1 -1.0)
     t_1
     (if (<= phi1 1.05e-32)
       (+
        lambda1
        (atan2 (* (sin theta) (sin delta)) (- (cos delta) (* phi1 phi1))))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 * (1.0 + (atan2((cos(phi1) * (sin(theta) * delta)), cos(delta)) / lambda1));
	double tmp;
	if (phi1 <= -1.0) {
		tmp = t_1;
	} else if (phi1 <= 1.05e-32) {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), (cos(delta) - (phi1 * phi1)));
	} 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 * (1.0d0 + (atan2((cos(phi1) * (sin(theta) * delta)), cos(delta)) / lambda1))
    if (phi1 <= (-1.0d0)) then
        tmp = t_1
    else if (phi1 <= 1.05d-32) then
        tmp = lambda1 + atan2((sin(theta) * sin(delta)), (cos(delta) - (phi1 * phi1)))
    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 * (1.0 + (Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), Math.cos(delta)) / lambda1));
	double tmp;
	if (phi1 <= -1.0) {
		tmp = t_1;
	} else if (phi1 <= 1.05e-32) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * Math.sin(delta)), (Math.cos(delta) - (phi1 * phi1)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 * (1.0 + (math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), math.cos(delta)) / lambda1))
	tmp = 0
	if phi1 <= -1.0:
		tmp = t_1
	elif phi1 <= 1.05e-32:
		tmp = lambda1 + math.atan2((math.sin(theta) * math.sin(delta)), (math.cos(delta) - (phi1 * phi1)))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 * Float64(1.0 + Float64(atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), cos(delta)) / lambda1)))
	tmp = 0.0
	if (phi1 <= -1.0)
		tmp = t_1;
	elseif (phi1 <= 1.05e-32)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), Float64(cos(delta) - Float64(phi1 * phi1))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 * (1.0 + (atan2((cos(phi1) * (sin(theta) * delta)), cos(delta)) / lambda1));
	tmp = 0.0;
	if (phi1 <= -1.0)
		tmp = t_1;
	elseif (phi1 <= 1.05e-32)
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), (cos(delta) - (phi1 * phi1)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 * N[(1.0 + N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.0], t$95$1, If[LessEqual[phi1, 1.05e-32], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1 or 1.05e-32 < phi1

   1. Initial program 99.8%

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

   3. Taylor expanded in lambda1 around inf

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

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

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

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

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

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

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

   5. Simplified 99.7%

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

   6. Taylor expanded in delta around 0

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

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

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

      2. sin-lowering-sin.f64 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in phi1 around 0

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

       1. cos-lowering-cos.f64 76.7%

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

   11. Simplified 76.7%

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

   if -1 < phi1 < 1.05e-32

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in delta around 0

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

   8. Simplified 99.1%

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

   9. Taylor expanded in phi1 around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 98.9%

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

   11. Simplified 98.9%

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

   12. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 98.9%

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

   14. Simplified 98.9%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1:\\ \;\;\;\;\lambda_1 \cdot \left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos delta}}{\lambda_1}\right)\\ \mathbf{elif}\;\phi_1 \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 \cdot \left(1 + \frac{\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{\cos delta}}{\lambda_1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.98:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -0.98)
   lambda1
   (if (<= phi1 2.2e-26)
     (+
      lambda1
      (atan2 (* (sin theta) (sin delta)) (- (cos delta) (* phi1 phi1))))
     lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -0.98) {
		tmp = lambda1;
	} else if (phi1 <= 2.2e-26) {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), (cos(delta) - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-0.98d0)) then
        tmp = lambda1
    else if (phi1 <= 2.2d-26) then
        tmp = lambda1 + atan2((sin(theta) * sin(delta)), (cos(delta) - (phi1 * phi1)))
    else
        tmp = lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -0.98) {
		tmp = lambda1;
	} else if (phi1 <= 2.2e-26) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * Math.sin(delta)), (Math.cos(delta) - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -0.98:
		tmp = lambda1
	elif phi1 <= 2.2e-26:
		tmp = lambda1 + math.atan2((math.sin(theta) * math.sin(delta)), (math.cos(delta) - (phi1 * phi1)))
	else:
		tmp = lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -0.98)
		tmp = lambda1;
	elseif (phi1 <= 2.2e-26)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), Float64(cos(delta) - Float64(phi1 * phi1))));
	else
		tmp = lambda1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -0.98)
		tmp = lambda1;
	elseif (phi1 <= 2.2e-26)
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), (cos(delta) - (phi1 * phi1)));
	else
		tmp = lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -0.98], lambda1, If[LessEqual[phi1, 2.2e-26], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], lambda1]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.97999999999999998 or 2.2000000000000001e-26 < phi1

   1. Initial program 99.8%

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

   3. Taylor expanded in lambda1 around inf

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

   5. Simplified 73.1%

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

   if -0.97999999999999998 < phi1 < 2.2000000000000001e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in delta around 0

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

   8. Simplified 99.1%

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

   9. Taylor expanded in phi1 around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 98.9%

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

   11. Simplified 98.9%

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

   12. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 98.9%

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

   14. Simplified 98.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.98:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.6% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -21000000000000:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(1 + \phi_1 \cdot \left(\phi_1 \cdot -0.5\right)\right) \cdot \left(\sin theta \cdot \sin delta\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -21000000000000.0)
   lambda1
   (if (<= phi1 2.2e-26)
     (+
      lambda1
      (atan2
       (* (+ 1.0 (* phi1 (* phi1 -0.5))) (* (sin theta) (sin delta)))
       (- 1.0 (* phi1 phi1))))
     lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -21000000000000.0) {
		tmp = lambda1;
	} else if (phi1 <= 2.2e-26) {
		tmp = lambda1 + atan2(((1.0 + (phi1 * (phi1 * -0.5))) * (sin(theta) * sin(delta))), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-21000000000000.0d0)) then
        tmp = lambda1
    else if (phi1 <= 2.2d-26) then
        tmp = lambda1 + atan2(((1.0d0 + (phi1 * (phi1 * (-0.5d0)))) * (sin(theta) * sin(delta))), (1.0d0 - (phi1 * phi1)))
    else
        tmp = lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -21000000000000.0) {
		tmp = lambda1;
	} else if (phi1 <= 2.2e-26) {
		tmp = lambda1 + Math.atan2(((1.0 + (phi1 * (phi1 * -0.5))) * (Math.sin(theta) * Math.sin(delta))), (1.0 - (phi1 * phi1)));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -21000000000000.0:
		tmp = lambda1
	elif phi1 <= 2.2e-26:
		tmp = lambda1 + math.atan2(((1.0 + (phi1 * (phi1 * -0.5))) * (math.sin(theta) * math.sin(delta))), (1.0 - (phi1 * phi1)))
	else:
		tmp = lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -21000000000000.0)
		tmp = lambda1;
	elseif (phi1 <= 2.2e-26)
		tmp = Float64(lambda1 + atan(Float64(Float64(1.0 + Float64(phi1 * Float64(phi1 * -0.5))) * Float64(sin(theta) * sin(delta))), Float64(1.0 - Float64(phi1 * phi1))));
	else
		tmp = lambda1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -21000000000000.0)
		tmp = lambda1;
	elseif (phi1 <= 2.2e-26)
		tmp = lambda1 + atan2(((1.0 + (phi1 * (phi1 * -0.5))) * (sin(theta) * sin(delta))), (1.0 - (phi1 * phi1)));
	else
		tmp = lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -21000000000000.0], lambda1, If[LessEqual[phi1, 2.2e-26], N[(lambda1 + N[ArcTan[N[(N[(1.0 + N[(phi1 * N[(phi1 * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], lambda1]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.1e13 or 2.2000000000000001e-26 < phi1

   1. Initial program 99.8%

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

   3. Taylor expanded in lambda1 around inf

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

   5. Simplified 73.4%

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

   if -2.1e13 < phi1 < 2.2000000000000001e-26

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

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

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

      2. +-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. sin-lowering-sin.f64 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in delta around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 84.0%

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

   11. Simplified 84.0%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -21000000000000:\\ \;\;\;\;\lambda_1\\ \mathbf{elif}\;\phi_1 \leq 2.2 \cdot 10^{-26}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(1 + \phi_1 \cdot \left(\phi_1 \cdot -0.5\right)\right) \cdot \left(\sin theta \cdot \sin delta\right)}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.0% accurate, 1320.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in lambda1 around inf

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

5. Simplified 73.4%

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

Reproduce

herbie shell --seed 2024139 
(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))))))))))