Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.9%

Time: 19.5s

Alternatives: 14

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.8%

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

8. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. sqr-cos-a N/A

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

   3. pow2 N/A

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

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

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

   5. cos-lowering-cos.f64 99.9%

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

10. Applied egg-rr 99.9%

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

Alternative 2: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2(((sin(theta) * cos(phi1)) * sin(delta)), ((cos(delta) * (0.5 + (0.5 * cos((phi1 * 2.0))))) - (sin(phi1) * (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) * cos(phi1)) * sin(delta)), ((cos(delta) * (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0))))) - (sin(phi1) * (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.cos(phi1)) * Math.sin(delta)), ((Math.cos(delta) * (0.5 + (0.5 * Math.cos((phi1 * 2.0))))) - (Math.sin(phi1) * (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.cos(phi1)) * math.sin(delta)), ((math.cos(delta) * (0.5 + (0.5 * math.cos((phi1 * 2.0))))) - (math.sin(phi1) * (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) * cos(phi1)) * sin(delta)), Float64(Float64(cos(delta) * Float64(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))) - Float64(sin(phi1) * Float64(cos(phi1) * Float64(sin(delta) * cos(theta)))))))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   8. sin-lowering-sin.f64 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.8%

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

   1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

   6. sin-lowering-sin.f64 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin theta) (sin delta)))
   (-
    (* (cos delta) (+ 0.5 (* 0.5 (cos (* phi1 2.0)))))
    (* (sin phi1) (* (cos phi1) (* (sin delta) (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) * (0.5 + (0.5 * cos((phi1 * 2.0))))) - (sin(phi1) * (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((cos(phi1) * (sin(theta) * sin(delta))), ((cos(delta) * (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0))))) - (sin(phi1) * (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.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), ((Math.cos(delta) * (0.5 + (0.5 * Math.cos((phi1 * 2.0))))) - (Math.sin(phi1) * (Math.cos(phi1) * (Math.sin(delta) * 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) * (0.5 + (0.5 * math.cos((phi1 * 2.0))))) - (math.sin(phi1) * (math.cos(phi1) * (math.sin(delta) * math.cos(theta))))))

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(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))) - Float64(sin(phi1) * Float64(cos(phi1) * Float64(sin(delta) * cos(theta)))))))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 5: 94.6% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin theta) (sin delta)))
   (-
    (* (cos delta) (+ 0.5 (* 0.5 (cos (* phi1 2.0)))))
    (* (sin phi1) (* (cos phi1) (sin 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) * (0.5 + (0.5 * cos((phi1 * 2.0))))) - (sin(phi1) * (cos(phi1) * sin(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) * (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0))))) - (sin(phi1) * (cos(phi1) * sin(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) * (0.5 + (0.5 * Math.cos((phi1 * 2.0))))) - (Math.sin(phi1) * (Math.cos(phi1) * Math.sin(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) * (0.5 + (0.5 * math.cos((phi1 * 2.0))))) - (math.sin(phi1) * (math.cos(phi1) * math.sin(delta)))))

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(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))) - Float64(sin(phi1) * Float64(cos(phi1) * sin(delta))))))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in theta around 0

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

   1. sin-lowering-sin.f64 94.2%

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

9. Simplified 94.2%

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

10. Final simplification 94.2%

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

Alternative 6: 91.6% accurate, 1.6× speedup

Math

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

FPCore

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

Fortran

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

Java

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

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(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))) - Float64(sin(delta) * sin(phi1)))))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in phi1 around 0

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

9. Simplified 91.9%

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

10. Taylor expanded in theta around 0

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

    1. sin-lowering-sin.f64 91.6%

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

12. Simplified 91.6%

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

13. Final simplification 91.6%

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

Alternative 7: 91.9% 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 99.7%

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

3. Taylor expanded in delta around 0

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

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

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

   2. sin-lowering-sin.f64 91.1%

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

5. Simplified 91.1%

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

6. Final simplification 91.1%

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

Alternative 8: 91.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;delta \leq -0.056:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;delta \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin theta) (sin delta))))
        (t_2 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= delta -0.056)
     t_2
     (if (<= delta 3.3e-5) (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0))) t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(theta) * sin(delta));
	double t_2 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (delta <= -0.056) {
		tmp = t_2;
	} else if (delta <= 3.3e-5) {
		tmp = lambda1 + atan2(t_1, pow(cos(phi1), 2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(phi1) * (sin(theta) * sin(delta))
    t_2 = lambda1 + atan2(t_1, cos(delta))
    if (delta <= (-0.056d0)) then
        tmp = t_2
    else if (delta <= 3.3d-5) then
        tmp = lambda1 + atan2(t_1, (cos(phi1) ** 2.0d0))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta));
	double t_2 = lambda1 + Math.atan2(t_1, Math.cos(delta));
	double tmp;
	if (delta <= -0.056) {
		tmp = t_2;
	} else if (delta <= 3.3e-5) {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(theta) * math.sin(delta))
	t_2 = lambda1 + math.atan2(t_1, math.cos(delta))
	tmp = 0
	if delta <= -0.056:
		tmp = t_2
	elif delta <= 3.3e-5:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(theta) * sin(delta)))
	t_2 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (delta <= -0.056)
		tmp = t_2;
	elseif (delta <= 3.3e-5)
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(theta) * sin(delta));
	t_2 = lambda1 + atan2(t_1, cos(delta));
	tmp = 0.0;
	if (delta <= -0.056)
		tmp = t_2;
	elseif (delta <= 3.3e-5)
		tmp = lambda1 + atan2(t_1, (cos(phi1) ^ 2.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[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, -0.056], t$95$2, If[LessEqual[delta, 3.3e-5], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if delta < -0.0560000000000000012 or 3.3000000000000003e-5 < delta

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 83.7%

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

   5. Simplified 83.7%

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

   if -0.0560000000000000012 < delta < 3.3000000000000003e-5

   1. Initial program 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in delta around 0

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

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

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

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

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

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

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

      4. *-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 99.6%

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

   7. Simplified 99.6%

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

      1. *-commutative N/A

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

      2. sqr-cos-a N/A

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

      3. pow2 N/A

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

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

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

      5. cos-lowering-cos.f64 99.7%

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

   9. Applied egg-rr 99.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.056:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 3.3 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 0.00032:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \cos \phi_1\right) \cdot \left(delta \cdot \left(1 + -0.16666666666666666 \cdot \left(delta \cdot delta\right)\right)\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))))
   (if (<= delta -2.85e-5)
     t_1
     (if (<= delta 0.00032)
       (+
        lambda1
        (atan2
         (*
          (* (sin theta) (cos phi1))
          (* delta (+ 1.0 (* -0.16666666666666666 (* delta delta)))))
         (+ 0.5 (* 0.5 (cos (* phi1 2.0))))))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	double tmp;
	if (delta <= -2.85e-5) {
		tmp = t_1;
	} else if (delta <= 0.00032) {
		tmp = lambda1 + atan2(((sin(theta) * cos(phi1)) * (delta * (1.0 + (-0.16666666666666666 * (delta * delta))))), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    if (delta <= (-2.85d-5)) then
        tmp = t_1
    else if (delta <= 0.00032d0) then
        tmp = lambda1 + atan2(((sin(theta) * cos(phi1)) * (delta * (1.0d0 + ((-0.16666666666666666d0) * (delta * delta))))), (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	double tmp;
	if (delta <= -2.85e-5) {
		tmp = t_1;
	} else if (delta <= 0.00032) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * Math.cos(phi1)) * (delta * (1.0 + (-0.16666666666666666 * (delta * delta))))), (0.5 + (0.5 * Math.cos((phi1 * 2.0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	tmp = 0
	if delta <= -2.85e-5:
		tmp = t_1
	elif delta <= 0.00032:
		tmp = lambda1 + math.atan2(((math.sin(theta) * math.cos(phi1)) * (delta * (1.0 + (-0.16666666666666666 * (delta * delta))))), (0.5 + (0.5 * math.cos((phi1 * 2.0)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)))
	tmp = 0.0
	if (delta <= -2.85e-5)
		tmp = t_1;
	elseif (delta <= 0.00032)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * cos(phi1)) * Float64(delta * Float64(1.0 + Float64(-0.16666666666666666 * Float64(delta * delta))))), Float64(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	tmp = 0.0;
	if (delta <= -2.85e-5)
		tmp = t_1;
	elseif (delta <= 0.00032)
		tmp = lambda1 + atan2(((sin(theta) * cos(phi1)) * (delta * (1.0 + (-0.16666666666666666 * (delta * delta))))), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -2.85e-5], t$95$1, If[LessEqual[delta, 0.00032], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[(delta * N[(1.0 + N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.8500000000000002e-5 or 3.20000000000000026e-4 < delta

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 83.7%

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

   5. Simplified 83.7%

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

   if -2.8500000000000002e-5 < delta < 3.20000000000000026e-4

   1. Initial program 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in delta around 0

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

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

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

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

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

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

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

      4. *-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 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-rgt-out N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 99.6%

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

   10. Simplified 99.6%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \left(delta \cdot \left(1 + -0.16666666666666666 \cdot \left(delta \cdot delta\right)\right)\right)}}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.85 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.00032:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \cos \phi_1\right) \cdot \left(delta \cdot \left(1 + -0.16666666666666666 \cdot \left(delta \cdot delta\right)\right)\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 350000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2
           (* (cos phi1) (* (sin theta) delta))
           (+ 0.5 (* 0.5 (cos (* phi1 2.0))))))))
   (if (<= phi1 -3e+36)
     t_1
     (if (<= phi1 350000.0)
       (+
        lambda1
        (atan2
         (* (* (sin theta) (sin delta)) (+ 1.0 (* -0.5 (* phi1 phi1))))
         (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	double tmp;
	if (phi1 <= -3e+36) {
		tmp = t_1;
	} else if (phi1 <= 350000.0) {
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * (1.0 + (-0.5 * (phi1 * phi1)))), 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 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0)))))
    if (phi1 <= (-3d+36)) then
        tmp = t_1
    else if (phi1 <= 350000.0d0) then
        tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * (1.0d0 + ((-0.5d0) * (phi1 * phi1)))), 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 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (0.5 + (0.5 * Math.cos((phi1 * 2.0)))));
	double tmp;
	if (phi1 <= -3e+36) {
		tmp = t_1;
	} else if (phi1 <= 350000.0) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * (1.0 + (-0.5 * (phi1 * phi1)))), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (0.5 + (0.5 * math.cos((phi1 * 2.0)))))
	tmp = 0
	if phi1 <= -3e+36:
		tmp = t_1
	elif phi1 <= 350000.0:
		tmp = lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * (1.0 + (-0.5 * (phi1 * phi1)))), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))))
	tmp = 0.0
	if (phi1 <= -3e+36)
		tmp = t_1;
	elseif (phi1 <= 350000.0)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * Float64(1.0 + Float64(-0.5 * Float64(phi1 * phi1)))), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	tmp = 0.0;
	if (phi1 <= -3e+36)
		tmp = t_1;
	elseif (phi1 <= 350000.0)
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * (1.0 + (-0.5 * (phi1 * phi1)))), 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[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3e+36], t$95$1, If[LessEqual[phi1, 350000.0], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3e36 or 3.5e5 < phi1

   1. Initial program 99.4%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in delta around 0

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

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

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

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

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

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

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

      4. *-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 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in delta around 0

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

   10. Simplified 72.7%

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

   if -3e36 < phi1 < 3.5e5

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

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

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

      16. cos-lowering-cos.f64 97.2%

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

   5. Simplified 97.2%

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

   6. Taylor expanded in 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. +-commutative N/A

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. sin-lowering-sin.f64 97.2%

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

   8. Simplified 97.2%

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

   9. Taylor expanded in phi1 around 0

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

       1. cos-lowering-cos.f64 98.1%

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

   11. Simplified 98.1%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{+36}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{elif}\;\phi_1 \leq 350000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \left(1 + -0.5 \cdot \left(\phi_1 \cdot \phi_1\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -3200000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -3200000000.0)
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) 1.0))
   (+
    lambda1
    (atan2
     (* (cos phi1) (* (sin theta) delta))
     (+ 0.5 (* 0.5 (cos (* phi1 2.0))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -3200000000.0) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), 1.0);
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (delta <= (-3200000000.0d0)) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), 1.0d0)
    else
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -3200000000.0) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), 1.0);
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (0.5 + (0.5 * Math.cos((phi1 * 2.0)))));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -3200000000.0:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), 1.0)
	else:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (0.5 + (0.5 * math.cos((phi1 * 2.0)))))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -3200000000.0)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), 1.0));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -3200000000.0)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), 1.0);
	else
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -3200000000.0], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -3.2e9

   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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in delta around 0

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

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

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

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

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

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

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

      4. *-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 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in phi1 around 0

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

   10. Simplified 60.7%

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

   if -3.2e9 < delta

   1. Initial program 99.6%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in delta around 0

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

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

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

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

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

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

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

      4. *-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 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in delta around 0

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

   10. Simplified 84.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -3200000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.0% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in delta around 0

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

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

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

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

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

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

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

   4. *-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 79.3%

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

7. Simplified 79.3%

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

8. Taylor expanded in phi1 around 0

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

10. Simplified 75.3%

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

11. Final simplification 75.3%

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

Alternative 13: 70.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq 1.05 \cdot 10^{-72}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta 1.05e-72)
   lambda1
   (+ lambda1 (atan2 (* (sin theta) delta) (- 1.0 (* phi1 phi1))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 1.05e-72) {
		tmp = lambda1;
	} else {
		tmp = lambda1 + atan2((sin(theta) * delta), (1.0 - (phi1 * phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (delta <= 1.05d-72) then
        tmp = lambda1
    else
        tmp = lambda1 + atan2((sin(theta) * delta), (1.0d0 - (phi1 * phi1)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 1.05e-72) {
		tmp = lambda1;
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), (1.0 - (phi1 * phi1)));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= 1.05e-72:
		tmp = lambda1
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), (1.0 - (phi1 * phi1)))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= 1.05e-72)
		tmp = lambda1;
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), Float64(1.0 - Float64(phi1 * phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= 1.05e-72)
		tmp = lambda1;
	else
		tmp = lambda1 + atan2((sin(theta) * delta), (1.0 - (phi1 * phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, 1.05e-72], lambda1, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < 1.05e-72

   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 76.2%

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

   if 1.05e-72 < delta

   1. Initial program 99.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

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

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

      5. unsub-neg N/A

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

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

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

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

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

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

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

      9. +-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

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

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

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

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

      16. cos-lowering-cos.f64 74.0%

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

   5. Simplified 74.0%

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

   6. Taylor expanded in delta around 0

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

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

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 65.0%

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

   8. Simplified 65.0%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 62.3%

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

   11. Simplified 62.3%

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

   12. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 62.8%

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

   14. Simplified 62.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq 1.05 \cdot 10^{-72}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.3% accurate, 1320.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in lambda1 around inf

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

5. Simplified 70.3%

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

Reproduce

herbie shell --seed 2024192 
(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))))))))))