Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.7%

Time: 10.6s

Alternatives: 11

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, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. sin-asin N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate--r+ N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. sin-asin N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate--r+ N/A

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

4. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. sin-cos-mult N/A

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

   5. div-inv N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

   8. +-inverses N/A

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

   9. sin-0 N/A

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

   10. lower-+.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. lower-sin.f64 99.8

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

   13. lift-+.f64 N/A

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

   14. count-2 N/A

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

   15. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 94.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-asin.f64 N/A

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

   5. sin-asin N/A

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

   6. lift-+.f64 N/A

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

   7. distribute-rgt-in N/A

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

   8. associate--r+ N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in delta around 0

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

   1. unpow2 N/A

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

   2. 1-sub-sin N/A

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

   3. unpow2 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-cos.f64 82.7

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

7. Applied rewrites 82.7%

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

8. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-cos.f64 82.5

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

10. Applied rewrites 82.5%

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

11. Taylor expanded in phi1 around inf

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

    1. *-commutative N/A

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

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

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

    3. distribute-rgt1-in N/A

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

    4. +-commutative N/A

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

    5. sub-neg N/A

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

    6. lower-*.f64 N/A

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

    7. unpow2 N/A

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

    8. 1-sub-sin N/A

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

    9. unpow2 N/A

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

    10. lower-pow.f64 N/A

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

    11. lower-cos.f64 N/A

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

    12. lower-cos.f64 94.8

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

13. Applied rewrites 94.8%

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

14. Final simplification 94.8%

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

Alternative 4: 92.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 92.6

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

5. Applied rewrites 92.6%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 92.6

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

7. Applied rewrites 92.6%

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

8. Final simplification 92.6%

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

Alternative 5: 92.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\\ t_2 := \tan^{-1}_* \frac{t\_1}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;delta \leq 110000:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos \phi_1 \cdot \cos \phi_1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (* (sin theta) (sin delta)) (cos phi1)))
        (t_2 (+ (atan2 t_1 (cos delta)) lambda1)))
   (if (<= delta -9.5e-7)
     t_2
     (if (<= delta 110000.0)
       (+ (atan2 t_1 (* (cos phi1) (cos phi1))) lambda1)
       t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	double t_2 = atan2(t_1, cos(delta)) + lambda1;
	double tmp;
	if (delta <= -9.5e-7) {
		tmp = t_2;
	} else if (delta <= 110000.0) {
		tmp = atan2(t_1, (cos(phi1) * cos(phi1))) + lambda1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (sin(theta) * sin(delta)) * cos(phi1)
    t_2 = atan2(t_1, cos(delta)) + lambda1
    if (delta <= (-9.5d-7)) then
        tmp = t_2
    else if (delta <= 110000.0d0) then
        tmp = atan2(t_1, (cos(phi1) * cos(phi1))) + lambda1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = (Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1);
	double t_2 = Math.atan2(t_1, Math.cos(delta)) + lambda1;
	double tmp;
	if (delta <= -9.5e-7) {
		tmp = t_2;
	} else if (delta <= 110000.0) {
		tmp = Math.atan2(t_1, (Math.cos(phi1) * Math.cos(phi1))) + lambda1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = (math.sin(theta) * math.sin(delta)) * math.cos(phi1)
	t_2 = math.atan2(t_1, math.cos(delta)) + lambda1
	tmp = 0
	if delta <= -9.5e-7:
		tmp = t_2
	elif delta <= 110000.0:
		tmp = math.atan2(t_1, (math.cos(phi1) * math.cos(phi1))) + lambda1
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(Float64(sin(theta) * sin(delta)) * cos(phi1))
	t_2 = Float64(atan(t_1, cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -9.5e-7)
		tmp = t_2;
	elseif (delta <= 110000.0)
		tmp = Float64(atan(t_1, Float64(cos(phi1) * cos(phi1))) + lambda1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = (sin(theta) * sin(delta)) * cos(phi1);
	t_2 = atan2(t_1, cos(delta)) + lambda1;
	tmp = 0.0;
	if (delta <= -9.5e-7)
		tmp = t_2;
	elseif (delta <= 110000.0)
		tmp = atan2(t_1, (cos(phi1) * cos(phi1))) + lambda1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -9.5e-7], t$95$2, If[LessEqual[delta, 110000.0], N[(N[ArcTan[t$95$1 / N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if delta < -9.5000000000000001e-7 or 1.1e5 < delta

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 84.1

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

   5. Applied rewrites 84.1%

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

   if -9.5000000000000001e-7 < delta < 1.1e5

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 99.5

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

   5. Applied rewrites 99.5%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 110000:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos \phi_1 \cdot \cos \phi_1} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin theta \cdot delta\right) \cdot \cos \phi_1}{\cos delta - {\sin \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2 (* (* (sin theta) (sin delta)) (cos phi1)) (cos delta))
          lambda1)))
   (if (<= delta -9.5e-7)
     t_1
     (if (<= delta 5.8e-5)
       (+
        (atan2
         (* (* (sin theta) delta) (cos phi1))
         (- (cos delta) (pow (sin phi1) 2.0)))
        lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -9.5e-7) {
		tmp = t_1;
	} else if (delta <= 5.8e-5) {
		tmp = atan2(((sin(theta) * delta) * cos(phi1)), (cos(delta) - pow(sin(phi1), 2.0))) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), cos(delta)) + lambda1
    if (delta <= (-9.5d-7)) then
        tmp = t_1
    else if (delta <= 5.8d-5) then
        tmp = atan2(((sin(theta) * delta) * cos(phi1)), (cos(delta) - (sin(phi1) ** 2.0d0))) + lambda1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.atan2(((Math.sin(theta) * Math.sin(delta)) * Math.cos(phi1)), Math.cos(delta)) + lambda1;
	double tmp;
	if (delta <= -9.5e-7) {
		tmp = t_1;
	} else if (delta <= 5.8e-5) {
		tmp = Math.atan2(((Math.sin(theta) * delta) * Math.cos(phi1)), (Math.cos(delta) - Math.pow(Math.sin(phi1), 2.0))) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2(((math.sin(theta) * math.sin(delta)) * math.cos(phi1)), math.cos(delta)) + lambda1
	tmp = 0
	if delta <= -9.5e-7:
		tmp = t_1
	elif delta <= 5.8e-5:
		tmp = math.atan2(((math.sin(theta) * delta) * math.cos(phi1)), (math.cos(delta) - math.pow(math.sin(phi1), 2.0))) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(sin(theta) * sin(delta)) * cos(phi1)), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -9.5e-7)
		tmp = t_1;
	elseif (delta <= 5.8e-5)
		tmp = Float64(atan(Float64(Float64(sin(theta) * delta) * cos(phi1)), Float64(cos(delta) - (sin(phi1) ^ 2.0))) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2(((sin(theta) * sin(delta)) * cos(phi1)), cos(delta)) + lambda1;
	tmp = 0.0;
	if (delta <= -9.5e-7)
		tmp = t_1;
	elseif (delta <= 5.8e-5)
		tmp = atan2(((sin(theta) * delta) * cos(phi1)), (cos(delta) - (sin(phi1) ^ 2.0))) + lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -9.5e-7], t$95$1, If[LessEqual[delta, 5.8e-5], N[(N[ArcTan[N[(N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -9.5000000000000001e-7 or 5.8e-5 < delta

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -9.5000000000000001e-7 < delta < 5.8e-5

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in delta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 99.4

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

   8. Applied rewrites 99.4%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 5.8 \cdot 10^{-5}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin theta \cdot delta\right) \cdot \cos \phi_1}{\cos delta - {\sin \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.8

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

5. Applied rewrites 87.8%

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

6. Final simplification 87.8%

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

Alternative 8: 86.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.8

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

5. Applied rewrites 87.8%

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

6. Taylor expanded in phi1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 84.6

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

8. Applied rewrites 84.6%

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

9. Final simplification 84.6%

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

Alternative 9: 80.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;theta \leq -2.8 \cdot 10^{-28}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 0.018:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot \sin delta\right) \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2
           (*
            (* (fma (* -0.16666666666666666 delta) delta 1.0) (sin theta))
            delta)
           (cos delta))
          lambda1)))
   (if (<= theta -2.8e-28)
     t_1
     (if (<= theta 0.018)
       (+
        (atan2
         (*
          (* (fma (* theta theta) -0.16666666666666666 1.0) (sin delta))
          theta)
         (cos delta))
        lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((fma((-0.16666666666666666 * delta), delta, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1;
	double tmp;
	if (theta <= -2.8e-28) {
		tmp = t_1;
	} else if (theta <= 0.018) {
		tmp = atan2(((fma((theta * theta), -0.16666666666666666, 1.0) * sin(delta)) * theta), cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(fma(Float64(-0.16666666666666666 * delta), delta, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1)
	tmp = 0.0
	if (theta <= -2.8e-28)
		tmp = t_1;
	elseif (theta <= 0.018)
		tmp = Float64(atan(Float64(Float64(fma(Float64(theta * theta), -0.16666666666666666, 1.0) * sin(delta)) * theta), cos(delta)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[(N[(-0.16666666666666666 * delta), $MachinePrecision] * delta + 1.0), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[theta, -2.8e-28], t$95$1, If[LessEqual[theta, 0.018], N[(N[ArcTan[N[(N[(N[(N[(theta * theta), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -2.7999999999999998e-28 or 0.0179999999999999986 < theta

   1. Initial program 99.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 80.0

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

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 70.5%

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

   if -2.7999999999999998e-28 < theta < 0.0179999999999999986

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 92.7

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 89.5%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -2.8 \cdot 10^{-28}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 0.018:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot \sin delta\right) \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -510000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.34 \cdot 10^{-30}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ (atan2 (* theta (sin delta)) (cos delta)) lambda1)))
   (if (<= delta -510000.0)
     t_1
     (if (<= delta 1.34e-30)
       (+ (atan2 (* (sin theta) delta) (cos delta)) lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((theta * sin(delta)), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -510000.0) {
		tmp = t_1;
	} else if (delta <= 1.34e-30) {
		tmp = atan2((sin(theta) * delta), cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan2((theta * sin(delta)), cos(delta)) + lambda1
    if (delta <= (-510000.0d0)) then
        tmp = t_1
    else if (delta <= 1.34d-30) then
        tmp = atan2((sin(theta) * delta), cos(delta)) + lambda1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.atan2((theta * Math.sin(delta)), Math.cos(delta)) + lambda1;
	double tmp;
	if (delta <= -510000.0) {
		tmp = t_1;
	} else if (delta <= 1.34e-30) {
		tmp = Math.atan2((Math.sin(theta) * delta), Math.cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2((theta * math.sin(delta)), math.cos(delta)) + lambda1
	tmp = 0
	if delta <= -510000.0:
		tmp = t_1
	elif delta <= 1.34e-30:
		tmp = math.atan2((math.sin(theta) * delta), math.cos(delta)) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(theta * sin(delta)), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -510000.0)
		tmp = t_1;
	elseif (delta <= 1.34e-30)
		tmp = Float64(atan(Float64(sin(theta) * delta), cos(delta)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2((theta * sin(delta)), cos(delta)) + lambda1;
	tmp = 0.0;
	if (delta <= -510000.0)
		tmp = t_1;
	elseif (delta <= 1.34e-30)
		tmp = atan2((sin(theta) * delta), cos(delta)) + lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -510000.0], t$95$1, If[LessEqual[delta, 1.34e-30], N[(N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -5.1e5 or 1.34000000000000001e-30 < delta

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 84.0

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 78.8

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

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 68.7%

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

   if -5.1e5 < delta < 1.34000000000000001e-30

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 91.4

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

   5. Applied rewrites 91.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 90.0

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 90.0%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -510000:\\ \;\;\;\;\tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 1.34 \cdot 10^{-30}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 87.8

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

5. Applied rewrites 87.8%

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

6. Taylor expanded in phi1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-sin.f64 84.6

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

8. Applied rewrites 84.6%

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

9. Taylor expanded in theta around 0

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

11. Applied rewrites 73.0%

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

12. Final simplification 73.0%

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

Reproduce

herbie shell --seed 2024332 
(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))))))))))