Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.9%

Time: 16.9s

Alternatives: 16

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. sin-asin N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. associate--r+ N/A

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-sub0 N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. lift-+.f64 N/A

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

   8. count-2 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. associate--r- N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

   15. associate-+l+ N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lift-cos.f64 N/A

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

   20. lift-*.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * (-N[Cos[phi1], $MachinePrecision])), $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[Cos[delta], $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
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. sin-asin N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. associate--r+ N/A

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-sub0 N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. lift-+.f64 N/A

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

   8. count-2 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. associate--r- N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

   15. associate-+l+ N/A

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

   16. metadata-eval N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[phi1], $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
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. sin-asin N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate--r+ N/A

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

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

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

   7. metadata-eval N/A

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

   8. lift-*.f64 N/A

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

   9. count-2 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-fma.f64 N/A

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

   13. lift-neg.f64 N/A

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

   14. distribute-rgt1-in N/A

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 94.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[((-N[Sin[delta], $MachinePrecision]) * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $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
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. sin-asin N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. associate--r+ N/A

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. neg-sub0 N/A

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

   4. lift-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. metadata-eval N/A

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

   7. lift-+.f64 N/A

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

   8. count-2 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. lift-*.f64 N/A

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

   12. associate--r- N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

   15. associate-+l+ N/A

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lift-cos.f64 N/A

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

   20. lift-*.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in theta around 0

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

    1. mul-1-neg N/A

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

    2. distribute-rgt-neg-in N/A

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

    3. *-commutative N/A

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

    4. unpow2 N/A

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

    5. associate-*l* N/A

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

    6. distribute-lft-out N/A

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

    7. lower-*.f64 N/A

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

    8. lower-cos.f64 N/A

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

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

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

    10. lower-fma.f64 N/A

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

    11. lower-neg.f64 N/A

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

    12. lower-sin.f64 N/A

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

    13. lower-sin.f64 N/A

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

    14. lower-*.f64 N/A

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

    15. lower-cos.f64 N/A

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

    16. lower-cos.f64 95.3

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

11. Applied rewrites 95.3%

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

12. Final simplification 95.3%

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

Alternative 5: 91.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in delta around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-sin.f64 85.3

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

5. Applied rewrites 85.3%

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

6. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-sin.f64 95.2

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

8. Applied rewrites 95.2%

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

10. Applied rewrites 93.0%

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

11. Final simplification 93.0%

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

Alternative 6: 92.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 92.9

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

5. Applied rewrites 92.9%

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

6. Final simplification 92.9%

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

Alternative 7: 91.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2 (* (* (cos phi1) (sin theta)) (sin delta)) (cos delta))
          lambda1)))
   (if (<= delta -2.4e+19)
     t_1
     (if (<= delta 1.9e-9)
       (+
        (atan2 (* (* (sin delta) (sin theta)) (cos phi1)) (pow (cos phi1) 2.0))
        lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -2.4e+19) {
		tmp = t_1;
	} else if (delta <= 1.9e-9) {
		tmp = atan2(((sin(delta) * sin(theta)) * cos(phi1)), pow(cos(phi1), 2.0)) + lambda1;
	} 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 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1
    if (delta <= (-2.4d+19)) then
        tmp = t_1
    else if (delta <= 1.9d-9) then
        tmp = atan2(((sin(delta) * sin(theta)) * cos(phi1)), (cos(phi1) ** 2.0d0)) + lambda1
    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 = Math.atan2(((Math.cos(phi1) * Math.sin(theta)) * Math.sin(delta)), Math.cos(delta)) + lambda1;
	double tmp;
	if (delta <= -2.4e+19) {
		tmp = t_1;
	} else if (delta <= 1.9e-9) {
		tmp = Math.atan2(((Math.sin(delta) * Math.sin(theta)) * Math.cos(phi1)), Math.pow(Math.cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2(((math.cos(phi1) * math.sin(theta)) * math.sin(delta)), math.cos(delta)) + lambda1
	tmp = 0
	if delta <= -2.4e+19:
		tmp = t_1
	elif delta <= 1.9e-9:
		tmp = math.atan2(((math.sin(delta) * math.sin(theta)) * math.cos(phi1)), math.pow(math.cos(phi1), 2.0)) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -2.4e+19)
		tmp = t_1;
	elseif (delta <= 1.9e-9)
		tmp = Float64(atan(Float64(Float64(sin(delta) * sin(theta)) * cos(phi1)), (cos(phi1) ^ 2.0)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	tmp = 0.0;
	if (delta <= -2.4e+19)
		tmp = t_1;
	elseif (delta <= 1.9e-9)
		tmp = atan2(((sin(delta) * sin(theta)) * cos(phi1)), (cos(phi1) ^ 2.0)) + lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -2.4e+19], t$95$1, If[LessEqual[delta, 1.9e-9], N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.4e19 or 1.90000000000000006e-9 < 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

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

      1. lower-cos.f64 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if -2.4e19 < delta < 1.90000000000000006e-9

   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

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   7. Applied rewrites 99.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), 0.5, 0.5\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2 (* (* (cos phi1) (sin theta)) (sin delta)) (cos delta))
          lambda1)))
   (if (<= delta -2.4e+19)
     t_1
     (if (<= delta 1.9e-9)
       (+
        (atan2
         (* (* (sin delta) (sin theta)) (cos phi1))
         (fma (cos (+ phi1 phi1)) 0.5 0.5))
        lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -2.4e+19) {
		tmp = t_1;
	} else if (delta <= 1.9e-9) {
		tmp = atan2(((sin(delta) * sin(theta)) * cos(phi1)), fma(cos((phi1 + phi1)), 0.5, 0.5)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -2.4e+19)
		tmp = t_1;
	elseif (delta <= 1.9e-9)
		tmp = Float64(atan(Float64(Float64(sin(delta) * sin(theta)) * cos(phi1)), fma(cos(Float64(phi1 + phi1)), 0.5, 0.5)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -2.4e+19], t$95$1, If[LessEqual[delta, 1.9e-9], N[(N[ArcTan[N[(N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.4e19 or 1.90000000000000006e-9 < 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

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

      1. lower-cos.f64 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if -2.4e19 < delta < 1.90000000000000006e-9

   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

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.4

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

   7. Applied rewrites 99.5%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin delta \cdot \sin theta\right) \cdot \cos \phi_1}{\mathsf{fma}\left(\cos \left(\phi_1 + \phi_1\right), 0.5, 0.5\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot delta\right) \cdot \sin theta}{1 - {\sin \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2 (* (* (cos phi1) (sin theta)) (sin delta)) (cos delta))
          lambda1)))
   (if (<= delta -2.4e+19)
     t_1
     (if (<= delta 1.9e-9)
       (+
        (atan2
         (* (* (cos phi1) delta) (sin theta))
         (- 1.0 (pow (sin phi1) 2.0)))
        lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -2.4e+19) {
		tmp = t_1;
	} else if (delta <= 1.9e-9) {
		tmp = atan2(((cos(phi1) * delta) * sin(theta)), (1.0 - pow(sin(phi1), 2.0))) + lambda1;
	} 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 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1
    if (delta <= (-2.4d+19)) then
        tmp = t_1
    else if (delta <= 1.9d-9) then
        tmp = atan2(((cos(phi1) * delta) * sin(theta)), (1.0d0 - (sin(phi1) ** 2.0d0))) + lambda1
    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 = Math.atan2(((Math.cos(phi1) * Math.sin(theta)) * Math.sin(delta)), Math.cos(delta)) + lambda1;
	double tmp;
	if (delta <= -2.4e+19) {
		tmp = t_1;
	} else if (delta <= 1.9e-9) {
		tmp = Math.atan2(((Math.cos(phi1) * delta) * Math.sin(theta)), (1.0 - Math.pow(Math.sin(phi1), 2.0))) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2(((math.cos(phi1) * math.sin(theta)) * math.sin(delta)), math.cos(delta)) + lambda1
	tmp = 0
	if delta <= -2.4e+19:
		tmp = t_1
	elif delta <= 1.9e-9:
		tmp = math.atan2(((math.cos(phi1) * delta) * math.sin(theta)), (1.0 - math.pow(math.sin(phi1), 2.0))) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -2.4e+19)
		tmp = t_1;
	elseif (delta <= 1.9e-9)
		tmp = Float64(atan(Float64(Float64(cos(phi1) * delta) * sin(theta)), Float64(1.0 - (sin(phi1) ^ 2.0))) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	tmp = 0.0;
	if (delta <= -2.4e+19)
		tmp = t_1;
	elseif (delta <= 1.9e-9)
		tmp = atan2(((cos(phi1) * delta) * sin(theta)), (1.0 - (sin(phi1) ^ 2.0))) + lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -2.4e+19], t$95$1, If[LessEqual[delta, 1.9e-9], N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * delta), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.4e19 or 1.90000000000000006e-9 < 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

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

      1. lower-cos.f64 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 85.0

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

   7. Applied rewrites 85.0%

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

   if -2.4e19 < delta < 1.90000000000000006e-9

   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

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in delta around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot delta\right) \cdot \sin theta}{1 - {\sin \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.6

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

5. Applied rewrites 87.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 87.6

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

7. Applied rewrites 87.6%

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

8. Final simplification 87.6%

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

Alternative 11: 86.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in phi1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 85.3

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

8. Applied rewrites 85.3%

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

9. Final simplification 85.3%

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

Alternative 12: 80.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot \sin delta\right) \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 2.2 \cdot 10^{+24}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(delta \cdot delta, -0.16666666666666666, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -2.4e+19)
   (+
    (atan2
     (* (* (fma (* theta theta) -0.16666666666666666 1.0) (sin delta)) theta)
     (cos delta))
    lambda1)
   (if (<= delta 2.2e+24)
     (+
      (atan2
       (* (* (fma (* delta delta) -0.16666666666666666 1.0) (sin theta)) delta)
       (cos delta))
      lambda1)
     (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -2.4e+19) {
		tmp = atan2(((fma((theta * theta), -0.16666666666666666, 1.0) * sin(delta)) * theta), cos(delta)) + lambda1;
	} else if (delta <= 2.2e+24) {
		tmp = atan2(((fma((delta * delta), -0.16666666666666666, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1;
	} else {
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -2.4e+19)
		tmp = Float64(atan(Float64(Float64(fma(Float64(theta * theta), -0.16666666666666666, 1.0) * sin(delta)) * theta), cos(delta)) + lambda1);
	elseif (delta <= 2.2e+24)
		tmp = Float64(atan(Float64(Float64(fma(Float64(delta * delta), -0.16666666666666666, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1);
	else
		tmp = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -2.4e+19], N[(N[ArcTan[N[(N[(N[(N[(theta * theta), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], If[LessEqual[delta, 2.2e+24], N[(N[ArcTan[N[(N[(N[(N[(delta * delta), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -2.4e19

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 89.4

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

   5. Applied rewrites 89.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 85.4

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

   8. Applied rewrites 85.4%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 70.0%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot \sin delta\right) \cdot \color{blue}{theta}}{\cos delta} \]

   if -2.4e19 < delta < 2.20000000000000002e24

   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

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

      1. lower-cos.f64 90.6

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

   5. Applied rewrites 90.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 89.3

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

   8. Applied rewrites 89.3%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 89.4%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\mathsf{fma}\left(delta \cdot delta, -0.16666666666666666, 1\right) \cdot \sin theta\right) \cdot \color{blue}{delta}}{\cos delta} \]

   if 2.20000000000000002e24 < 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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 75.3

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

   8. Applied rewrites 75.3%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 64.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.4 \cdot 10^{+19}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot \sin delta\right) \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 2.2 \cdot 10^{+24}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(delta \cdot delta, -0.16666666666666666, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -2.5 \cdot 10^{+53}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(delta \cdot delta, -0.16666666666666666, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 1.52 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= theta -2.5e+53)
   (+
    (atan2
     (* (* (fma (* delta delta) -0.16666666666666666 1.0) (sin theta)) delta)
     (cos delta))
    lambda1)
   (if (<= theta 1.52e-18)
     (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)
     (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= -2.5e+53) {
		tmp = atan2(((fma((delta * delta), -0.16666666666666666, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1;
	} else if (theta <= 1.52e-18) {
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	} else {
		tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (theta <= -2.5e+53)
		tmp = Float64(atan(Float64(Float64(fma(Float64(delta * delta), -0.16666666666666666, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1);
	elseif (theta <= 1.52e-18)
		tmp = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1);
	else
		tmp = Float64(atan(Float64(delta * sin(theta)), cos(delta)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[theta, -2.5e+53], N[(N[ArcTan[N[(N[(N[(N[(delta * delta), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], If[LessEqual[theta, 1.52e-18], N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if theta < -2.5000000000000002e53

   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

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

      1. lower-cos.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 79.3

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 62.0%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\mathsf{fma}\left(delta \cdot delta, -0.16666666666666666, 1\right) \cdot \sin theta\right) \cdot \color{blue}{delta}}{\cos delta} \]

   if -2.5000000000000002e53 < theta < 1.52e-18

   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

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

      1. lower-cos.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 89.9

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 89.6%

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

   if 1.52e-18 < theta

   1. Initial program 99.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 80.3

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

   8. Applied rewrites 80.3%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 69.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -2.5 \cdot 10^{+53}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(delta \cdot delta, -0.16666666666666666, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 1.52 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{if}\;theta \leq -2.4 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 1.52 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1)))
   (if (<= theta -2.4e+50)
     t_1
     (if (<= theta 1.52e-18)
       (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	double tmp;
	if (theta <= -2.4e+50) {
		tmp = t_1;
	} else if (theta <= 1.52e-18) {
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	} 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 = atan2((delta * sin(theta)), cos(delta)) + lambda1
    if (theta <= (-2.4d+50)) then
        tmp = t_1
    else if (theta <= 1.52d-18) then
        tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1
    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 = Math.atan2((delta * Math.sin(theta)), Math.cos(delta)) + lambda1;
	double tmp;
	if (theta <= -2.4e+50) {
		tmp = t_1;
	} else if (theta <= 1.52e-18) {
		tmp = Math.atan2((Math.sin(delta) * theta), Math.cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2((delta * math.sin(theta)), math.cos(delta)) + lambda1
	tmp = 0
	if theta <= -2.4e+50:
		tmp = t_1
	elif theta <= 1.52e-18:
		tmp = math.atan2((math.sin(delta) * theta), math.cos(delta)) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(delta * sin(theta)), cos(delta)) + lambda1)
	tmp = 0.0
	if (theta <= -2.4e+50)
		tmp = t_1;
	elseif (theta <= 1.52e-18)
		tmp = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	tmp = 0.0;
	if (theta <= -2.4e+50)
		tmp = t_1;
	elseif (theta <= 1.52e-18)
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[theta, -2.4e+50], t$95$1, If[LessEqual[theta, 1.52e-18], N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -2.4000000000000002e50 or 1.52e-18 < theta

   1. Initial program 99.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 81.8

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

   5. Applied rewrites 81.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 79.7

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

   8. Applied rewrites 79.7%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 65.8%

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

   if -2.4000000000000002e50 < theta < 1.52e-18

   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

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

      1. lower-cos.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 90.2

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

   8. Applied rewrites 90.2%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 90.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -2.4 \cdot 10^{+50}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 1.52 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in phi1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 85.3

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

8. Applied rewrites 85.3%

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

9. Taylor expanded in theta around 0

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

11. Applied rewrites 72.0%

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

12. Final simplification 72.0%

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

Alternative 16: 67.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.6

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

5. Applied rewrites 87.6%

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

6. Taylor expanded in phi1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 85.3

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

8. Applied rewrites 85.3%

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

9. Taylor expanded in theta around 0

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

11. Applied rewrites 72.0%

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

12. Taylor expanded in delta around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot theta}{\cos delta} \]
13. Step-by-step derivation

14. Applied rewrites 67.1%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{theta \cdot delta}{\cos delta} \]

15. Final simplification 67.1%

    \[\leadsto \tan^{-1}_* \frac{delta \cdot theta}{\cos delta} + \lambda_1 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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))))))))))