Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.7%

Time: 18.4s

Alternatives: 10

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (* (sin theta) (sin delta)) (cos phi1))
   (+
    (- (cos delta) (* (sin delta) (* (cos phi1) (* (cos theta) (sin phi1)))))
    (* (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(theta) * sin(delta)) * cos(phi1)), ((cos(delta) - (sin(delta) * (cos(phi1) * (cos(theta) * sin(phi1))))) + (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(theta) * sin(delta)) * cos(phi1)), ((cos(delta) - (sin(delta) * (cos(phi1) * (cos(theta) * sin(phi1))))) + (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(theta) * Math.sin(delta)) * Math.cos(phi1)), ((Math.cos(delta) - (Math.sin(delta) * (Math.cos(phi1) * (Math.cos(theta) * Math.sin(phi1))))) + (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(theta) * math.sin(delta)) * math.cos(phi1)), ((math.cos(delta) - (math.sin(delta) * (math.cos(phi1) * (math.cos(theta) * math.sin(phi1))))) + (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(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(Float64(cos(delta) - Float64(sin(delta) * Float64(cos(phi1) * Float64(cos(theta) * sin(phi1))))) + Float64(cos(delta) * Float64(Float64(0.5 * cos(Float64(phi1 * 2.0))) - 0.5)))))
end

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. sin-asin N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 99.9%

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

   1. +-commutative N/A

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

   2. associate--r+ N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

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.cos(delta) * ((0.5 * math.cos((phi1 * 2.0))) - 0.5))) - (math.sin(delta) * (math.cos(phi1) * (math.cos(theta) * math.sin(phi1))))))

Julia

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

MATLAB

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

Derivation

1. Initial program 99.9%

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

   1. sin-asin N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

4. Applied egg-rr 99.9%

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

   1. associate--r+ N/A

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

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

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (sin delta) (* (sin theta) (cos phi1)))
   (-
    (cos delta)
    (*
     (sin phi1)
     (+
      (* (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) * (sin(theta) * cos(phi1))), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * (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) * (sin(theta) * cos(phi1))), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + (sin(delta) * (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.sin(theta) * Math.cos(phi1))), (Math.cos(delta) - (Math.sin(phi1) * ((Math.cos(delta) * Math.sin(phi1)) + (Math.sin(delta) * (Math.cos(phi1) * Math.cos(theta)))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 94.7% accurate, 1.3× speedup

Math

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

FPCore

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

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(delta) * cos(phi1)) + (cos(delta) * sin(phi1))))));
}

Fortran

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

Java

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

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(delta) * math.cos(phi1)) + (math.cos(delta) * math.sin(phi1))))))

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) * Float64(Float64(sin(delta) * cos(phi1)) + Float64(cos(delta) * sin(phi1)))))))
end

MATLAB

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in theta around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   11. sin-lowering-sin.f64 94.4%

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

5. Simplified 94.4%

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

6. Final simplification 94.4%

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

Alternative 5: 92.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 99.8%

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

5. Taylor expanded in delta around 0

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

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

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

   2. sin-lowering-sin.f64 91.6%

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

7. Simplified 91.6%

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

8. Final simplification 91.6%

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

Alternative 6: 89.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.22 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -1.22e+16)
   (+
    lambda1
    (atan2
     (* (* (sin theta) (sin delta)) (cos phi1))
     (+ 0.5 (* 0.5 (cos (* phi1 2.0))))))
   (+ lambda1 (atan2 (* (sin delta) (* (sin theta) (cos phi1))) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -1.22e+16) {
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	} else {
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-1.22d+16)) then
        tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0)))))
    else
        tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -1.22e+16) {
		tmp = lambda1 + Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), (0.5 + (0.5 * Math.cos((phi1 * 2.0)))));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -1.22e+16:
		tmp = lambda1 + math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), (0.5 + (0.5 * math.cos((phi1 * 2.0)))))
	else:
		tmp = lambda1 + math.atan2((math.sin(delta) * (math.sin(theta) * math.cos(phi1))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -1.22e+16)
		tmp = Float64(lambda1 + atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), Float64(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -1.22e+16)
		tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	else
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -1.22e+16], N[(lambda1 + N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.22e16

   1. Initial program 99.8%

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

      1. sin-asin N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt-in N/A

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

      8. fma-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 99.8%

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in delta around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 79.6%

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

   9. Simplified 79.6%

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

   if -1.22e16 < phi1

   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 93.3%

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

   7. Simplified 93.3%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.22 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2(((sin(theta) * sin(delta)) * cos(phi1)), cos(delta));
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[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 87.9%

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

5. Simplified 87.9%

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

Alternative 8: 88.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
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 99.8%

   \[\leadsto \color{blue}{\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \sin delta \cdot \left(\cos \phi_1 \cdot \cos theta\right)\right)}} \]
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 87.9%

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

7. Simplified 87.9%

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

8. Final simplification 87.9%

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

Alternative 9: 86.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 87.9%

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

5. Simplified 87.9%

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

6. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{+.f64}\left(\lambda_1, \mathsf{atan2.f64}\left(\color{blue}{\left(\sin delta \cdot \sin theta\right)}, \mathsf{cos.f64}\left(delta\right)\right)\right) \]
7. 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 85.8%

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

8. Simplified 85.8%

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

9. Final simplification 85.8%

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

Alternative 10: 69.9% 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 99.9%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
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 99.8%

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

5. Taylor expanded in lambda1 around inf

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

7. Simplified 67.0%

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

Reproduce

herbie shell --seed 2024150 
(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))))))))))