Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.7%

Time: 20.2s

Alternatives: 12

Speedup: N/A×

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: 99.7% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (sin delta) (* (cos phi1) (sin theta)))
   (-
    (fma (- (* (cos (* phi1 2.0)) (- 0.0 -0.5)) 0.5) (cos delta) (cos delta))
    (* (sin phi1) (* (sin delta) (* (cos phi1) (cos theta))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(delta) * (cos(phi1) * sin(theta))), (fma(((cos((phi1 * 2.0)) * (0.0 - -0.5)) - 0.5), cos(delta), cos(delta)) - (sin(phi1) * (sin(delta) * (cos(phi1) * cos(theta))))));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(sin(delta) * Float64(cos(phi1) * sin(theta))), Float64(fma(Float64(Float64(cos(Float64(phi1 * 2.0)) * Float64(0.0 - -0.5)) - 0.5), cos(delta), cos(delta)) - Float64(sin(phi1) * Float64(sin(delta) * Float64(cos(phi1) * cos(theta)))))))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.0 - -0.5), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[Cos[delta], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

   1. sin-asin N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

6. Applied egg-rr 99.8%

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

   1. associate-/r/ N/A

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

   2. /-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate--r+ N/A

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

   7. *-commutative N/A

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

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

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

8. Applied egg-rr 99.9%

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

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

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

   2. +-commutative N/A

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

   3. sqr-sin-a N/A

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

   4. unpow2 N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. fma-define N/A

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

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

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.sin(delta) * (math.cos(phi1) * math.sin(theta))), ((math.cos(delta) * ((-0.5 + (0.5 * math.cos((phi1 * 2.0)))) + 1.0)) - (math.sin(delta) * (math.sin(phi1) * (math.cos(phi1) * math.cos(theta))))))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((sin(delta) * (cos(phi1) * sin(theta))), ((cos(delta) * ((-0.5 + (0.5 * cos((phi1 * 2.0)))) + 1.0)) - (sin(delta) * (sin(phi1) * (cos(phi1) * cos(theta))))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

   1. sin-asin N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

6. Applied egg-rr 99.8%

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

   1. associate-/r/ N/A

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

   2. /-rgt-identity N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. associate--r+ N/A

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

   7. *-commutative N/A

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

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

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

8. Applied egg-rr 99.9%

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

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

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

   2. +-commutative N/A

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

   3. sqr-sin-a N/A

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

   4. unpow2 N/A

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

   5. *-commutative N/A

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

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

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

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

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

   8. fma-define N/A

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

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

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

10. Applied egg-rr 99.9%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

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

Alternative 3: 94.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

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

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in theta around 0

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

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

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

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

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

   3. sin-lowering-sin.f64 96.0%

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

9. Simplified 96.0%

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

10. Final simplification 96.0%

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

Alternative 4: 94.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

5. Taylor expanded in theta around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. sin-lowering-sin.f64 96.0%

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

7. Simplified 96.0%

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

8. Final simplification 96.0%

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

Alternative 5: 92.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(delta) * (cos(phi1) * sin(theta))), (1.0 / (1.0 / (cos(delta) + ((cos((phi1 * 2.0)) + -1.0) / 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((sin(delta) * (cos(phi1) * sin(theta))), (1.0d0 / (1.0d0 / (cos(delta) + ((cos((phi1 * 2.0d0)) + (-1.0d0)) / 2.0d0)))))
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.cos(phi1) * Math.sin(theta))), (1.0 / (1.0 / (Math.cos(delta) + ((Math.cos((phi1 * 2.0)) + -1.0) / 2.0)))));
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.sin(delta) * (math.cos(phi1) * math.sin(theta))), (1.0 / (1.0 / (math.cos(delta) + ((math.cos((phi1 * 2.0)) + -1.0) / 2.0)))))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((sin(delta) * (cos(phi1) * sin(theta))), (1.0 / (1.0 / (cos(delta) + ((cos((phi1 * 2.0)) + -1.0) / 2.0)))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

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

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in delta around 0

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

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

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

   2. sin-lowering-sin.f64 93.0%

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

9. Simplified 93.0%

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

    1. unpow2 N/A

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

    2. sin-mult N/A

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

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

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

    4. cos-diff N/A

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

    5. cos-sin-sum N/A

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

    6. cos-sum N/A

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

    7. cos-2 N/A

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

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

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

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

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

    10. *-commutative N/A

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

    11. *-lowering-*.f64 93.1%

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

11. Applied egg-rr 93.1%

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

12. Final simplification 93.1%

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

Alternative 6: 92.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

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

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in delta around 0

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

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

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

   2. sin-lowering-sin.f64 93.0%

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

9. Simplified 93.0%

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

    1. +-commutative N/A

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

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

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

11. Applied egg-rr 93.1%

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

12. Final simplification 93.1%

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

Alternative 7: 88.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

5. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 89.7%

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

7. Simplified 89.7%

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

Alternative 8: 86.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

5. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 89.7%

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

7. Simplified 89.7%

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

8. Taylor expanded in phi1 around 0

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

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

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

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

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

   3. sin-lowering-sin.f64 87.6%

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

10. Simplified 87.6%

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

Alternative 9: 79.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \left(1 + -0.16666666666666666 \cdot \left(theta \cdot theta\right)\right)\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -5 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 0.0215:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2
           (*
            theta
            (* (sin delta) (+ 1.0 (* -0.16666666666666666 (* theta theta)))))
           (cos delta)))))
   (if (<= delta -5e+98)
     t_1
     (if (<= delta 0.0215)
       (+ lambda1 (atan2 (* delta (sin theta)) (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) * (1.0 + (-0.16666666666666666 * (theta * theta))))), cos(delta));
	double tmp;
	if (delta <= -5e+98) {
		tmp = t_1;
	} else if (delta <= 0.0215) {
		tmp = lambda1 + atan2((delta * sin(theta)), 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((theta * (sin(delta) * (1.0d0 + ((-0.16666666666666666d0) * (theta * theta))))), cos(delta))
    if (delta <= (-5d+98)) then
        tmp = t_1
    else if (delta <= 0.0215d0) then
        tmp = lambda1 + atan2((delta * sin(theta)), 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((theta * (Math.sin(delta) * (1.0 + (-0.16666666666666666 * (theta * theta))))), Math.cos(delta));
	double tmp;
	if (delta <= -5e+98) {
		tmp = t_1;
	} else if (delta <= 0.0215) {
		tmp = lambda1 + Math.atan2((delta * Math.sin(theta)), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((theta * (math.sin(delta) * (1.0 + (-0.16666666666666666 * (theta * theta))))), math.cos(delta))
	tmp = 0
	if delta <= -5e+98:
		tmp = t_1
	elif delta <= 0.0215:
		tmp = lambda1 + math.atan2((delta * math.sin(theta)), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(theta * Float64(sin(delta) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(theta * theta))))), cos(delta)))
	tmp = 0.0
	if (delta <= -5e+98)
		tmp = t_1;
	elseif (delta <= 0.0215)
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((theta * (sin(delta) * (1.0 + (-0.16666666666666666 * (theta * theta))))), cos(delta));
	tmp = 0.0;
	if (delta <= -5e+98)
		tmp = t_1;
	elseif (delta <= 0.0215)
		tmp = lambda1 + atan2((delta * sin(theta)), 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[(theta * N[(N[Sin[delta], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(theta * theta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -5e+98], t$95$1, If[LessEqual[delta, 0.0215], N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.9999999999999998e98 or 0.021499999999999998 < 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. Step-by-step derivation

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

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

      2. atan2-lowering-atan2.f64 N/A

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in phi1 around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 78.9%

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

   10. Simplified 78.9%

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

   11. Taylor expanded in theta around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-*r* N/A

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

       7. *-commutative N/A

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

       8. associate-*r* N/A

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

       9. distribute-rgt1-in N/A

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

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

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

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

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

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

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

       13. unpow2 N/A

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

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

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

       15. sin-lowering-sin.f64 65.8%

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

   13. Simplified 65.8%

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

   if -4.9999999999999998e98 < delta < 0.021499999999999998

   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. Step-by-step derivation

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

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

      2. atan2-lowering-atan2.f64 N/A

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in phi1 around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 93.8%

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

   10. Simplified 93.8%

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

   11. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 92.7%

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

   13. Simplified 92.7%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -5 \cdot 10^{+98}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \left(1 + -0.16666666666666666 \cdot \left(theta \cdot theta\right)\right)\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 0.0215:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \left(1 + -0.16666666666666666 \cdot \left(theta \cdot theta\right)\right)\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta}\\ \mathbf{if}\;theta \leq -1.35 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 4.6 \cdot 10^{-50}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* delta (sin theta)) (cos delta)))))
   (if (<= theta -1.35e+16)
     t_1
     (if (<= theta 4.6e-50)
       (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((delta * sin(theta)), cos(delta));
	double tmp;
	if (theta <= -1.35e+16) {
		tmp = t_1;
	} else if (theta <= 4.6e-50) {
		tmp = lambda1 + atan2((sin(delta) * theta), 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((delta * sin(theta)), cos(delta))
    if (theta <= (-1.35d+16)) then
        tmp = t_1
    else if (theta <= 4.6d-50) then
        tmp = lambda1 + atan2((sin(delta) * theta), 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((delta * Math.sin(theta)), Math.cos(delta));
	double tmp;
	if (theta <= -1.35e+16) {
		tmp = t_1;
	} else if (theta <= 4.6e-50) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((delta * math.sin(theta)), math.cos(delta))
	tmp = 0
	if theta <= -1.35e+16:
		tmp = t_1
	elif theta <= 4.6e-50:
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(delta * sin(theta)), cos(delta)))
	tmp = 0.0
	if (theta <= -1.35e+16)
		tmp = t_1;
	elseif (theta <= 4.6e-50)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((delta * sin(theta)), cos(delta));
	tmp = 0.0;
	if (theta <= -1.35e+16)
		tmp = t_1;
	elseif (theta <= 4.6e-50)
		tmp = lambda1 + atan2((sin(delta) * theta), 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[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[theta, -1.35e+16], t$95$1, If[LessEqual[theta, 4.6e-50], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -1.35e16 or 4.60000000000000039e-50 < 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. Step-by-step derivation

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

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

      2. atan2-lowering-atan2.f64 N/A

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 90.4%

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

   7. Simplified 90.4%

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

   8. Taylor expanded in phi1 around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 88.4%

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

   10. Simplified 88.4%

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

   11. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 77.0%

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

   13. Simplified 77.0%

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

   if -1.35e16 < theta < 4.60000000000000039e-50

   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. Step-by-step derivation

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

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

      2. atan2-lowering-atan2.f64 N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 89.0%

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

   7. Simplified 89.0%

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

   8. Taylor expanded in phi1 around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 86.7%

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

   10. Simplified 86.7%

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

   11. Taylor expanded in theta around 0

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

       1. *-commutative N/A

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

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

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

       3. sin-lowering-sin.f64 86.3%

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

   13. Simplified 86.3%

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

Alternative 11: 74.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

5. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 89.7%

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

7. Simplified 89.7%

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

8. Taylor expanded in phi1 around 0

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

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

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

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

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

   3. sin-lowering-sin.f64 87.6%

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

10. Simplified 87.6%

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

11. Taylor expanded in delta around 0

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

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

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

    2. sin-lowering-sin.f64 76.2%

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

13. Simplified 76.2%

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

Alternative 12: 69.6% accurate, 1320.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

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

   2. atan2-lowering-atan2.f64 N/A

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

3. Simplified 99.8%

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

5. Taylor expanded in lambda1 around inf

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

7. Simplified 70.9%

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

Reproduce

herbie shell --seed 2024147 
(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))))))))))