Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.8%

Time: 17.3s

Alternatives: 14

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. count-2 N/A

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

   6. sqr-cos-a N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in delta around inf

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. metadata-eval N/A

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

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

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

    7. lower-fma.f64 N/A

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

    8. cos-neg N/A

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

    9. lower-cos.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 N/A

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

    12. lower-cos.f64 N/A

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

    13. *-commutative N/A

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

    14. associate-*l* N/A

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

    15. *-commutative N/A

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

    16. lower-*.f64 N/A

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

11. Applied rewrites 99.9%

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

12. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.9%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. lift-+.f64 N/A

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

   8. +-lft-identity N/A

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

   9. metadata-eval N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 99.9

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 4: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 99.9

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

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lower-fma.f64 N/A

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

10. Applied rewrites 99.9%

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

11. Final simplification 99.9%

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

Alternative 5: 94.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.9%

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

9. Taylor expanded in theta around 0

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

    1. metadata-eval N/A

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

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

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

    3. sin-neg N/A

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

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

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

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

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

    6. metadata-eval N/A

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

    7. lower-*.f64 N/A

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

    8. lower-sin.f64 N/A

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

    9. *-commutative N/A

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

    10. lower-*.f64 92.1

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

11. Applied rewrites 92.1%

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

12. Final simplification 92.1%

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

Alternative 6: 92.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in delta around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \color{blue}{{\sin \phi_1}^{2}}} \]
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 90.1

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

5. Applied rewrites 90.1%

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

6. Final simplification 90.1%

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

Alternative 7: 91.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -6 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (sin delta) (* (sin theta) (cos phi1))) (cos delta)))))
   (if (<= delta -6e-6)
     t_1
     (if (<= delta 9e-7)
       (+
        lambda1
        (atan2
         (* (cos phi1) (* (sin theta) (sin delta)))
         (pow (cos phi1) 2.0)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	double tmp;
	if (delta <= -6e-6) {
		tmp = t_1;
	} else if (delta <= 9e-7) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), pow(cos(phi1), 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta))
    if (delta <= (-6d-6)) then
        tmp = t_1
    else if (delta <= 9d-7) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(phi1) ** 2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1))), Math.cos(delta));
	double tmp;
	if (delta <= -6e-6) {
		tmp = t_1;
	} else if (delta <= 9e-7) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(delta) * (math.sin(theta) * math.cos(phi1))), math.cos(delta))
	tmp = 0
	if delta <= -6e-6:
		tmp = t_1
	elif delta <= 9e-7:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), cos(delta)))
	tmp = 0.0
	if (delta <= -6e-6)
		tmp = t_1;
	elseif (delta <= 9e-7)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), (cos(phi1) ^ 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	tmp = 0.0;
	if (delta <= -6e-6)
		tmp = t_1;
	elseif (delta <= 9e-7)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(phi1) ^ 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -6e-6], t$95$1, If[LessEqual[delta, 9e-7], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -6.0000000000000002e-6 or 8.99999999999999959e-7 < delta

   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 81.3

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

   5. Applied rewrites 81.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if -6.0000000000000002e-6 < delta < 8.99999999999999959e-7

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around 0

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

      1. lower--.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -6 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. 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 87.2

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

5. Applied rewrites 87.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 87.2

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

7. Applied rewrites 87.2%

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

8. Final simplification 87.2%

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

Alternative 9: 85.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. 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 87.2

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

5. Applied rewrites 87.2%

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

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 85.7

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

8. Applied rewrites 85.7%

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

9. Final simplification 85.7%

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

Alternative 10: 79.8% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -1100000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 3.1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\sin theta \cdot \left(\mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right) + \left(delta \cdot delta\right) \cdot \left(\left(delta \cdot delta\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -1100000.0)
   (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
   (if (<= delta 3.1)
     (+
      lambda1
      (atan2
       (*
        delta
        (*
         (sin theta)
         (+
          (fma -0.16666666666666666 (* delta delta) 1.0)
          (* (* delta delta) (* (* delta delta) 0.008333333333333333)))))
       (cos delta)))
     (+
      lambda1
      (atan2
       (*
        (sin delta)
        (fma theta (* -0.16666666666666666 (* theta theta)) theta))
       (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -1100000.0) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else if (delta <= 3.1) {
		tmp = lambda1 + atan2((delta * (sin(theta) * (fma(-0.16666666666666666, (delta * delta), 1.0) + ((delta * delta) * ((delta * delta) * 0.008333333333333333))))), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(delta) * fma(theta, (-0.16666666666666666 * (theta * theta)), theta)), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -1100000.0)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	elseif (delta <= 3.1)
		tmp = Float64(lambda1 + atan(Float64(delta * Float64(sin(theta) * Float64(fma(-0.16666666666666666, Float64(delta * delta), 1.0) + Float64(Float64(delta * delta) * Float64(Float64(delta * delta) * 0.008333333333333333))))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * fma(theta, Float64(-0.16666666666666666 * Float64(theta * theta)), theta)), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -1100000.0], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 3.1], N[(lambda1 + N[ArcTan[N[(delta * N[(N[Sin[theta], $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision] + N[(N[(delta * delta), $MachinePrecision] * N[(N[(delta * delta), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(theta * N[(-0.16666666666666666 * N[(theta * theta), $MachinePrecision]), $MachinePrecision] + theta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -1.1e6

   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 80.6

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

   5. Applied rewrites 80.6%

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

   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 78.2

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

   8. Applied rewrites 78.2%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 68.6%

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

   if -1.1e6 < delta < 3.10000000000000009

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 92.9

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

   5. Applied rewrites 92.9%

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

   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 92.7

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

   8. Applied rewrites 92.7%

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

   9. Taylor expanded in delta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \color{blue}{\left(\sin theta + {delta}^{2} \cdot \left(\frac{-1}{6} \cdot \sin theta + \frac{1}{120} \cdot \left({delta}^{2} \cdot \sin theta\right)\right)\right)}}{\cos delta} \]
   10. Step-by-step derivation

   11. Applied rewrites 92.7%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \color{blue}{\left(\sin theta \cdot \left(\mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right) + \left(delta \cdot delta\right) \cdot \left(0.008333333333333333 \cdot \left(delta \cdot delta\right)\right)\right)\right)}}{\cos delta} \]

   if 3.10000000000000009 < delta

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.2

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

   5. Applied rewrites 82.2%

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

   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 79.3

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 74.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1100000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 3.1:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\sin theta \cdot \left(\mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right) + \left(delta \cdot delta\right) \cdot \left(\left(delta \cdot delta\right) \cdot 0.008333333333333333\right)\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -8500000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.078:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\sin theta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -8500000000.0)
   (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
   (if (<= delta 0.078)
     (+
      lambda1
      (atan2
       (* delta (* (sin theta) (fma -0.16666666666666666 (* delta delta) 1.0)))
       (cos delta)))
     (+
      lambda1
      (atan2
       (*
        (sin delta)
        (fma theta (* -0.16666666666666666 (* theta theta)) theta))
       (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -8500000000.0) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else if (delta <= 0.078) {
		tmp = lambda1 + atan2((delta * (sin(theta) * fma(-0.16666666666666666, (delta * delta), 1.0))), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(delta) * fma(theta, (-0.16666666666666666 * (theta * theta)), theta)), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -8500000000.0)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	elseif (delta <= 0.078)
		tmp = Float64(lambda1 + atan(Float64(delta * Float64(sin(theta) * fma(-0.16666666666666666, Float64(delta * delta), 1.0))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * fma(theta, Float64(-0.16666666666666666 * Float64(theta * theta)), theta)), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -8500000000.0], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 0.078], N[(lambda1 + N[ArcTan[N[(delta * N[(N[Sin[theta], $MachinePrecision] * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(theta * N[(-0.16666666666666666 * N[(theta * theta), $MachinePrecision]), $MachinePrecision] + theta), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -8.5e9

   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 80.6

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

   5. Applied rewrites 80.6%

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

   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 78.2

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

   8. Applied rewrites 78.2%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 68.6%

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

   if -8.5e9 < delta < 0.0779999999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 92.9

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

   5. Applied rewrites 92.9%

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

   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 92.7

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

   8. Applied rewrites 92.7%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 92.3%

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

   12. Taylor expanded in delta around 0

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

   14. Applied rewrites 92.7%

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

   if 0.0779999999999999999 < delta

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.2

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

   5. Applied rewrites 82.2%

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

   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 79.3

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 74.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -8500000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.078:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\sin theta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \mathsf{fma}\left(theta, -0.16666666666666666 \cdot \left(theta \cdot theta\right), theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{if}\;delta \leq -8500000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 0.078:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\sin theta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))))
   (if (<= delta -8500000000.0)
     t_1
     (if (<= delta 0.078)
       (+
        lambda1
        (atan2
         (*
          delta
          (* (sin theta) (fma -0.16666666666666666 (* delta delta) 1.0)))
         (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((theta * sin(delta)), cos(delta));
	double tmp;
	if (delta <= -8500000000.0) {
		tmp = t_1;
	} else if (delta <= 0.078) {
		tmp = lambda1 + atan2((delta * (sin(theta) * fma(-0.16666666666666666, (delta * delta), 1.0))), cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)))
	tmp = 0.0
	if (delta <= -8500000000.0)
		tmp = t_1;
	elseif (delta <= 0.078)
		tmp = Float64(lambda1 + atan(Float64(delta * Float64(sin(theta) * fma(-0.16666666666666666, Float64(delta * delta), 1.0))), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -8500000000.0], t$95$1, If[LessEqual[delta, 0.078], N[(lambda1 + N[ArcTan[N[(delta * N[(N[Sin[theta], $MachinePrecision] * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -8.5e9 or 0.0779999999999999999 < delta

   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 81.4

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

   5. Applied rewrites 81.4%

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

   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 78.8

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

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 71.2%

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

   if -8.5e9 < delta < 0.0779999999999999999

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 92.9

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

   5. Applied rewrites 92.9%

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

   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 92.7

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

   8. Applied rewrites 92.7%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 92.3%

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

   12. Taylor expanded in delta around 0

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

   14. Applied rewrites 92.7%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -8500000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.078:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\sin theta \cdot \mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right)\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{if}\;theta \leq -0.00094:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))))
   (if (<= theta -0.00094)
     t_1
     (if (<= theta 8e-10)
       (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(theta) * delta), cos(delta));
	double tmp;
	if (theta <= -0.00094) {
		tmp = t_1;
	} else if (theta <= 8e-10) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = lambda1 + atan2((sin(theta) * delta), cos(delta))
    if (theta <= (-0.00094d0)) then
        tmp = t_1
    else if (theta <= 8d-10) then
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	double tmp;
	if (theta <= -0.00094) {
		tmp = t_1;
	} else if (theta <= 8e-10) {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	tmp = 0
	if theta <= -0.00094:
		tmp = t_1
	elif theta <= 8e-10:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)))
	tmp = 0.0
	if (theta <= -0.00094)
		tmp = t_1;
	elseif (theta <= 8e-10)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((sin(theta) * delta), cos(delta));
	tmp = 0.0;
	if (theta <= -0.00094)
		tmp = t_1;
	elseif (theta <= 8e-10)
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -0.00094], t$95$1, If[LessEqual[theta, 8e-10], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -9.39999999999999972e-4 or 8.00000000000000029e-10 < 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 79.6

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

   5. Applied rewrites 79.6%

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

   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 77.5

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

   8. Applied rewrites 77.5%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 70.2%

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

   if -9.39999999999999972e-4 < theta < 8.00000000000000029e-10

   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 95.2

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

   5. Applied rewrites 95.2%

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

   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 94.5

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

   8. Applied rewrites 94.5%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 94.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -0.00094:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{elif}\;theta \leq 8 \cdot 10^{-10}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.8% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. 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 87.2

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

5. Applied rewrites 87.2%

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

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 85.7

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

8. Applied rewrites 85.7%

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

9. Taylor expanded in delta around 0

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

11. Applied rewrites 75.0%

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

12. Final simplification 75.0%

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

Reproduce

herbie shell --seed 2024233 
(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))))))))))