Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 17.9s

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: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \left(\phi_1 \cdot -2\right)\\ \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\left(\left({\left(\mathsf{fma}\left(0.5, t\_1, -0.5\right)\right)}^{2} + -1\right) \cdot \frac{1}{\mathsf{fma}\left(0.5, t\_1, -1.5\right)}\right) \cdot \cos delta - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)} + \lambda_1 \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (cos (* phi1 -2.0))))
   (+
    (atan2
     (* (cos phi1) (* (sin delta) (sin theta)))
     (-
      (*
       (* (+ (pow (fma 0.5 t_1 -0.5) 2.0) -1.0) (/ 1.0 (fma 0.5 t_1 -1.5)))
       (cos delta))
      (* (cos phi1) (* (cos theta) (* (sin delta) (sin phi1))))))
    lambda1)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos((phi1 * -2.0));
	return atan2((cos(phi1) * (sin(delta) * sin(theta))), ((((pow(fma(0.5, t_1, -0.5), 2.0) + -1.0) * (1.0 / fma(0.5, t_1, -1.5))) * cos(delta)) - (cos(phi1) * (cos(theta) * (sin(delta) * sin(phi1)))))) + lambda1;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(Float64(phi1 * -2.0))
	return Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), Float64(Float64(Float64(Float64((fma(0.5, t_1, -0.5) ^ 2.0) + -1.0) * Float64(1.0 / fma(0.5, t_1, -1.5))) * cos(delta)) - Float64(cos(phi1) * Float64(cos(theta) * Float64(sin(delta) * sin(phi1)))))) + lambda1)
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Power[N[(0.5 * t$95$1 + -0.5), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(0.5 * t$95$1 + -1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

4. Applied egg-rr 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 99.8

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

   11. lift-cos.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. cos-2 N/A

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

   14. cos-sum N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-+.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

9. Simplified 99.8%

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

    1. lift-*.f64 N/A

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

    2. lift-cos.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. flip-+ N/A

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

    5. div-inv N/A

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

    6. lower-*.f64 N/A

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

    7. metadata-eval N/A

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

    8. sub-neg N/A

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

    9. metadata-eval N/A

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

    10. lower-+.f64 N/A

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

    11. pow2 N/A

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

    12. lower-pow.f64 N/A

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

    13. lower-/.f64 N/A

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

11. Applied egg-rr 99.9%

    \[\leadsto \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\color{blue}{\left(\left({\left(\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), -0.5\right)\right)}^{2} + -1\right) \cdot \frac{1}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), -1.5\right)}\right)} \cdot \cos delta - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin \phi_1 \cdot \sin delta\right)\right)} + \lambda_1 \]

12. Final simplification 99.9%

    \[\leadsto \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\left(\left({\left(\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), -0.5\right)\right)}^{2} + -1\right) \cdot \frac{1}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot -2\right), -1.5\right)}\right) \cdot \cos delta - \cos \phi_1 \cdot \left(\cos theta \cdot \left(\sin delta \cdot \sin \phi_1\right)\right)} + \lambda_1 \]
13. Add Preprocessing

Alternative 2: 83.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot t\_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\\ t_3 := \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;t\_2 \leq -500000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(-0.5, delta \cdot delta, 1\right)}\\ \mathbf{elif}\;t\_2 \leq -0.1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \mathbf{elif}\;t\_2 \leq 3.14159265:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot theta\right)}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta)))
        (t_2
         (+
          lambda1
          (atan2
           (* (cos phi1) t_1)
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin
               (+
                (* (cos delta) (sin phi1))
                (* (cos theta) (* (cos phi1) (sin delta)))))))))))
        (t_3 (atan2 t_1 (cos delta))))
   (if (<= t_2 -500000.0)
     (+ lambda1 (atan2 t_1 (fma -0.5 (* delta delta) 1.0)))
     (if (<= t_2 -0.1)
       t_3
       (if (<= t_2 1e-7)
         (+ lambda1 (atan2 t_1 1.0))
         (if (<= t_2 3.14159265)
           t_3
           (+
            lambda1
            (atan2
             (* (cos phi1) (* (sin delta) theta))
             (- 1.0 (* phi1 phi1))))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double t_2 = lambda1 + atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (cos(phi1) * sin(delta)))))))));
	double t_3 = atan2(t_1, cos(delta));
	double tmp;
	if (t_2 <= -500000.0) {
		tmp = lambda1 + atan2(t_1, fma(-0.5, (delta * delta), 1.0));
	} else if (t_2 <= -0.1) {
		tmp = t_3;
	} else if (t_2 <= 1e-7) {
		tmp = lambda1 + atan2(t_1, 1.0);
	} else if (t_2 <= 3.14159265) {
		tmp = t_3;
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (sin(delta) * theta)), (1.0 - (phi1 * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	t_2 = Float64(lambda1 + atan(Float64(cos(phi1) * t_1), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(cos(phi1) * sin(delta))))))))))
	t_3 = atan(t_1, cos(delta))
	tmp = 0.0
	if (t_2 <= -500000.0)
		tmp = Float64(lambda1 + atan(t_1, fma(-0.5, Float64(delta * delta), 1.0)));
	elseif (t_2 <= -0.1)
		tmp = t_3;
	elseif (t_2 <= 1e-7)
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	elseif (t_2 <= 3.14159265)
		tmp = t_3;
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * theta)), Float64(1.0 - Float64(phi1 * phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -500000.0], N[(lambda1 + N[ArcTan[t$95$1 / N[(-0.5 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], t$95$3, If[LessEqual[t$95$2, 1e-7], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 3.14159265], t$95$3, N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -5e5

   1. Initial program 100.0%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   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 100.0

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

   5. Simplified 100.0%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 100.0

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

   8. Simplified 100.0%

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

   9. Taylor expanded in delta around 0

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

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 99.8

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

   11. Simplified 99.8%

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

   if -5e5 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -0.10000000000000001 or 9.9999999999999995e-8 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 3.14159265000000021

   1. Initial program 99.3%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   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 68.7

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

   5. Simplified 68.7%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 59.2

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

   8. Simplified 59.2%

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

   9. Taylor expanded in lambda1 around 0

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

       1. lower-atan2.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-cos.f64 59.2

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

   11. Simplified 59.2%

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

   if -0.10000000000000001 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 9.9999999999999995e-8

   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 87.4

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

   5. Simplified 87.4%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 86.1

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

   8. Simplified 86.1%

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

   9. Taylor expanded in delta around 0

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

   11. Simplified 83.4%

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

   if 3.14159265000000021 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))

   1. Initial program 100.0%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. lower-fma.f64 N/A

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

   5. Simplified 92.7%

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

   6. Taylor expanded in delta around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-cos.f64 89.1

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

   8. Simplified 89.1%

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

   9. Taylor expanded in phi1 around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 97.3

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

   11. Simplified 97.3%

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

   12. Taylor expanded in theta around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 97.3

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

   14. Simplified 97.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq -500000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\mathsf{fma}\left(-0.5, delta \cdot delta, 1\right)}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq -0.1:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq 3.14159265:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot theta\right)}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ t_2 := \tan^{-1}_* \frac{\cos \phi_1 \cdot t\_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\\ \mathbf{if}\;t\_2 \leq -1.736:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(-0.5, delta \cdot delta, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 1.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot theta\right)}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta)))
        (t_2
         (atan2
          (* (cos phi1) t_1)
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (cos delta) (sin phi1))
               (* (cos theta) (* (cos phi1) (sin delta)))))))))))
   (if (<= t_2 -1.736)
     (+ lambda1 (atan2 t_1 (fma -0.5 (* delta delta) 1.0)))
     (if (<= t_2 1.5)
       (+ lambda1 (atan2 t_1 1.0))
       (+
        lambda1
        (atan2 (* (cos phi1) (* (sin delta) theta)) (- 1.0 (* phi1 phi1))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double t_2 = atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (cos(phi1) * sin(delta)))))))));
	double tmp;
	if (t_2 <= -1.736) {
		tmp = lambda1 + atan2(t_1, fma(-0.5, (delta * delta), 1.0));
	} else if (t_2 <= 1.5) {
		tmp = lambda1 + atan2(t_1, 1.0);
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (sin(delta) * theta)), (1.0 - (phi1 * phi1)));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	t_2 = atan(Float64(cos(phi1) * t_1), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(cos(phi1) * sin(delta)))))))))
	tmp = 0.0
	if (t_2 <= -1.736)
		tmp = Float64(lambda1 + atan(t_1, fma(-0.5, Float64(delta * delta), 1.0)));
	elseif (t_2 <= 1.5)
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * theta)), Float64(1.0 - Float64(phi1 * phi1))));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -1.736], N[(lambda1 + N[ArcTan[t$95$1 / N[(-0.5 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.5], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -1.73599999999999999

   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 89.6

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

   5. Simplified 89.6%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 83.1

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

   8. Simplified 83.1%

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

   9. Taylor expanded in delta around 0

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

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 72.5

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

   11. Simplified 72.5%

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

   if -1.73599999999999999 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 1.5

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 92.0

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

   5. Simplified 92.0%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in delta around 0

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

   11. Simplified 86.5%

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

   if 1.5 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. lower-fma.f64 N/A

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

   5. Simplified 58.9%

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

   6. Taylor expanded in delta around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-cos.f64 51.5

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

   8. Simplified 51.5%

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

   9. Taylor expanded in phi1 around 0

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

       1. mul-1-neg N/A

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

       2. unsub-neg N/A

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

       3. lower--.f64 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f64 74.0

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

   11. Simplified 74.0%

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

   12. Taylor expanded in theta around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 73.3

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

   14. Simplified 73.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq -1.736:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\mathsf{fma}\left(-0.5, delta \cdot delta, 1\right)}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq 1.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot theta\right)}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ t_2 := \tan^{-1}_* \frac{\cos \phi_1 \cdot t\_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\\ \mathbf{if}\;t\_2 \leq -1.736:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(-0.5, delta \cdot delta, 1\right)}\\ \mathbf{elif}\;t\_2 \leq 1.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta)))
        (t_2
         (atan2
          (* (cos phi1) t_1)
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (cos delta) (sin phi1))
               (* (cos theta) (* (cos phi1) (sin delta)))))))))))
   (if (<= t_2 -1.736)
     (+ lambda1 (atan2 t_1 (fma -0.5 (* delta delta) 1.0)))
     (if (<= t_2 1.5)
       (+ lambda1 (atan2 t_1 1.0))
       (+
        lambda1
        (atan2 (* delta (sin theta)) (fma delta (* delta -0.5) 1.0)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double t_2 = atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (cos(phi1) * sin(delta)))))))));
	double tmp;
	if (t_2 <= -1.736) {
		tmp = lambda1 + atan2(t_1, fma(-0.5, (delta * delta), 1.0));
	} else if (t_2 <= 1.5) {
		tmp = lambda1 + atan2(t_1, 1.0);
	} else {
		tmp = lambda1 + atan2((delta * sin(theta)), fma(delta, (delta * -0.5), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	t_2 = atan(Float64(cos(phi1) * t_1), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(cos(phi1) * sin(delta)))))))))
	tmp = 0.0
	if (t_2 <= -1.736)
		tmp = Float64(lambda1 + atan(t_1, fma(-0.5, Float64(delta * delta), 1.0)));
	elseif (t_2 <= 1.5)
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	else
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), fma(delta, Float64(delta * -0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -1.736], N[(lambda1 + N[ArcTan[t$95$1 / N[(-0.5 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.5], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -1.73599999999999999

   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 89.6

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

   5. Simplified 89.6%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 83.1

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

   8. Simplified 83.1%

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

   9. Taylor expanded in delta around 0

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

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. lower-fma.f64 N/A

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

       3. unpow2 N/A

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

       4. lower-*.f64 72.5

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

   11. Simplified 72.5%

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

   if -1.73599999999999999 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 1.5

   1. Initial program 99.8%

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

   3. Taylor expanded in 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 92.0

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

   5. Simplified 92.0%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 91.1

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

   8. Simplified 91.1%

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

   9. Taylor expanded in delta around 0

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

   11. Simplified 86.5%

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

   if 1.5 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 84.3

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

   5. Simplified 84.3%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 74.0

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

   8. Simplified 74.0%

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

   9. Taylor expanded in delta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 57.6

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

   11. Simplified 57.6%

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

   12. Taylor expanded in delta around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{1 + \frac{-1}{2} \cdot {delta}^{2}}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \frac{-1}{2}} + 1} \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 70.1

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

   14. Simplified 70.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq -1.736:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\mathsf{fma}\left(-0.5, delta \cdot delta, 1\right)}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq 1.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)}\\ t_2 := delta \cdot \sin theta\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-30}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.18:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, delta \cdot delta, -0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (atan2
          (* (cos phi1) (* (sin delta) (sin theta)))
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (cos delta) (sin phi1))
               (* (cos theta) (* (cos phi1) (sin delta))))))))))
        (t_2 (* delta (sin theta)))
        (t_3 (+ lambda1 (atan2 t_2 (fma delta (* delta -0.5) 1.0)))))
   (if (<= t_1 -2e-30)
     t_3
     (if (<= t_1 0.18)
       (+
        lambda1
        (atan2
         t_2
         (fma
          (* delta delta)
          (fma 0.041666666666666664 (* delta delta) -0.5)
          1.0)))
       t_3))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (cos(phi1) * sin(delta)))))))));
	double t_2 = delta * sin(theta);
	double t_3 = lambda1 + atan2(t_2, fma(delta, (delta * -0.5), 1.0));
	double tmp;
	if (t_1 <= -2e-30) {
		tmp = t_3;
	} else if (t_1 <= 0.18) {
		tmp = lambda1 + atan2(t_2, fma((delta * delta), fma(0.041666666666666664, (delta * delta), -0.5), 1.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(cos(phi1) * sin(delta)))))))))
	t_2 = Float64(delta * sin(theta))
	t_3 = Float64(lambda1 + atan(t_2, fma(delta, Float64(delta * -0.5), 1.0)))
	tmp = 0.0
	if (t_1 <= -2e-30)
		tmp = t_3;
	elseif (t_1 <= 0.18)
		tmp = Float64(lambda1 + atan(t_2, fma(Float64(delta * delta), fma(0.041666666666666664, Float64(delta * delta), -0.5), 1.0)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[t$95$2 / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-30], t$95$3, If[LessEqual[t$95$1, 0.18], N[(lambda1 + N[ArcTan[t$95$2 / N[(N[(delta * delta), $MachinePrecision] * N[(0.041666666666666664 * N[(delta * delta), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -2e-30 or 0.17999999999999999 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   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 83.2

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

   5. Simplified 83.2%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 76.5

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

   8. Simplified 76.5%

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

   9. Taylor expanded in delta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 56.4

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

   11. Simplified 56.4%

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

   12. Taylor expanded in delta around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{1 + \frac{-1}{2} \cdot {delta}^{2}}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \frac{-1}{2}} + 1} \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 63.4

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

   14. Simplified 63.4%

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

   if -2e-30 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 0.17999999999999999

   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 95.8

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

   5. Simplified 95.8%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 95.4

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

   8. Simplified 95.4%

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

   9. Taylor expanded in delta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 90.5

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

   11. Simplified 90.5%

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

   12. Taylor expanded in delta around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{1 + {delta}^{2} \cdot \left(\frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}\right)}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \left(\frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\mathsf{fma}\left({delta}^{2}, \frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}, 1\right)}} \]

       3. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}, 1\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}, 1\right)} \]

       5. sub-neg N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{\frac{1}{24} \cdot {delta}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)} \]

       6. metadata-eval N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \frac{1}{24} \cdot {delta}^{2} + \color{blue}{\frac{-1}{2}}, 1\right)} \]

       7. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{\mathsf{fma}\left(\frac{1}{24}, {delta}^{2}, \frac{-1}{2}\right)}, 1\right)} \]

       8. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{1}{24}, \color{blue}{delta \cdot delta}, \frac{-1}{2}\right), 1\right)} \]

       9. lower-*.f64 93.6

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, \color{blue}{delta \cdot delta}, -0.5\right), 1\right)} \]

   14. Simplified 93.6%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, delta \cdot delta, -0.5\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq -2 \cdot 10^{-30}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq 0.18:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, delta \cdot delta, -0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot t\_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq 1.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta))))
   (if (<=
        (atan2
         (* (cos phi1) t_1)
         (-
          (cos delta)
          (*
           (sin phi1)
           (sin
            (asin
             (+
              (* (cos delta) (sin phi1))
              (* (cos theta) (* (cos phi1) (sin delta)))))))))
        1.5)
     (+ lambda1 (atan2 t_1 1.0))
     (+
      lambda1
      (atan2 (* delta (sin theta)) (fma delta (* delta -0.5) 1.0))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double tmp;
	if (atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (cos(phi1) * sin(delta))))))))) <= 1.5) {
		tmp = lambda1 + atan2(t_1, 1.0);
	} else {
		tmp = lambda1 + atan2((delta * sin(theta)), fma(delta, (delta * -0.5), 1.0));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	tmp = 0.0
	if (atan(Float64(cos(phi1) * t_1), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(cos(phi1) * sin(delta))))))))) <= 1.5)
		tmp = Float64(lambda1 + atan(t_1, 1.0));
	else
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), fma(delta, Float64(delta * -0.5), 1.0)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.5], N[(lambda1 + N[ArcTan[t$95$1 / 1.0], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 1.5

   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 91.6

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

   5. Simplified 91.6%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 89.7

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

   8. Simplified 89.7%

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

   9. Taylor expanded in delta around 0

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

   11. Simplified 82.2%

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

   if 1.5 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 84.3

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

   5. Simplified 84.3%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 74.0

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

   8. Simplified 74.0%

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

   9. Taylor expanded in delta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 57.6

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

   11. Simplified 57.6%

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

   12. Taylor expanded in delta around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{1 + \frac{-1}{2} \cdot {delta}^{2}}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \frac{-1}{2}} + 1} \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 70.1

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

   14. Simplified 70.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)\right)} \leq 1.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

4. Applied egg-rr 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 99.8

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

   11. lift-cos.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. cos-2 N/A

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

   14. cos-sum N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-+.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

9. Simplified 99.8%

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

    1. lift-*.f64 N/A

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

    2. lift-cos.f64 N/A

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

    3. associate-+l+ N/A

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

    4. *-commutative N/A

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

    5. metadata-eval N/A

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

    6. lower-fma.f64 99.8

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

11. Applied egg-rr 99.8%

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

12. Final simplification 99.8%

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

Alternative 8: 94.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 95.0

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

7. Simplified 95.0%

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

8. Final simplification 95.0%

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

Alternative 9: 94.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. sin-asin N/A

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

4. Applied egg-rr 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 99.8

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

   11. lift-cos.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. cos-2 N/A

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

   14. cos-sum N/A

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

   15. lower-cos.f64 N/A

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

   16. lower-+.f64 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in theta around 0

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

   1. associate--r+ N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. distribute-rgt1-in N/A

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

   6. lower-fma.f64 N/A

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

9. Simplified 95.0%

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

10. Final simplification 95.0%

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

Alternative 10: 92.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in delta around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{{\sin \phi_1}^{2}}} \]
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 92.3

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

5. Simplified 92.3%

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

6. Final simplification 92.3%

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

Alternative 11: 88.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 90.7

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

5. Simplified 90.7%

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

6. Final simplification 90.7%

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

Alternative 12: 86.8% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 90.7

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

5. Simplified 90.7%

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

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. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 87.7

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

8. Simplified 87.7%

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

Alternative 13: 78.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq 2.9 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta 2.9e-12)
   (+ lambda1 (atan2 (* delta (sin theta)) (fma delta (* delta -0.5) 1.0)))
   (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= 2.9e-12) {
		tmp = lambda1 + atan2((delta * sin(theta)), fma(delta, (delta * -0.5), 1.0));
	} else {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= 2.9e-12)
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), fma(delta, Float64(delta * -0.5), 1.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, 2.9e-12], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < 2.9000000000000002e-12

   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 92.7

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

   5. Simplified 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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 89.9

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

   8. Simplified 89.9%

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

   9. Taylor expanded in delta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 82.4

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

   11. Simplified 82.4%

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

   12. Taylor expanded in delta around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{1 + \frac{-1}{2} \cdot {delta}^{2}}} \]
   13. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

       2. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \frac{-1}{2}} + 1} \]

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 84.3

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

   14. Simplified 84.3%

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

   if 2.9000000000000002e-12 < 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 85.1

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

   5. Simplified 85.1%

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

   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. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 81.6

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

   8. Simplified 81.6%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 70.4

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

   11. Simplified 70.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq 2.9 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.4% accurate, 6.1× speedup

Math

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

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((delta * sin(theta)), fma(delta, (delta * -0.5), 1.0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 90.7

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

5. Simplified 90.7%

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

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. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 87.7

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

8. Simplified 87.7%

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

9. Taylor expanded in delta around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sin.f64 76.6

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

11. Simplified 76.6%

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

12. Taylor expanded in delta around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{1 + \frac{-1}{2} \cdot {delta}^{2}}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\frac{-1}{2} \cdot {delta}^{2} + 1}} \]

    2. *-commutative N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \frac{-1}{2}} + 1} \]

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-*.f64 77.8

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

14. Simplified 77.8%

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

Alternative 15: 73.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 90.7

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

5. Simplified 90.7%

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

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. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 87.7

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

8. Simplified 87.7%

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

9. Taylor expanded in delta around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sin.f64 76.6

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

11. Simplified 76.6%

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

12. Taylor expanded in delta around 0

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

14. Simplified 75.9%

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

Alternative 16: 67.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 90.7

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

5. Simplified 90.7%

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

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. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 87.7

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

8. Simplified 87.7%

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

9. Taylor expanded in delta around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sin.f64 76.6

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

11. Simplified 76.6%

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

12. Taylor expanded in theta around 0

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

    1. lower-*.f64 69.6

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

14. Simplified 69.6%

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

Reproduce

herbie shell --seed 2024210 
(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))))))))))