Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 13.0s

Alternatives: 13

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.7

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.7

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. sin-asin N/A

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

   5. lift-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. lower-fma.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. lower-*.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. distribute-rgt-out N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. +-commutative N/A

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

8. Applied rewrites 99.7%

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

Alternative 2: 96.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin \phi_1 \cdot \cos delta\\ t_2 := \left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{t\_2}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)}\\ \mathbf{if}\;t\_3 \leq 6 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{\cos delta - \mathsf{fma}\left(\sin delta, \cos \phi_1, t\_1\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\tan^{-1}_* \frac{t\_2}{\cos delta - \mathsf{fma}\left(\cos theta \cdot \sin delta, \cos \phi_1, t\_1\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin phi1) (cos delta)))
        (t_2 (* (* (sin theta) (sin delta)) (cos phi1)))
        (t_3
         (+
          lambda1
          (atan2
           t_2
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin (+ t_1 (* (* (cos phi1) (sin delta)) (cos theta)))))))))))
   (if (<= t_3 6e-12)
     (+
      lambda1
      (atan2
       t_2
       (- (cos delta) (* (fma (sin delta) (cos phi1) t_1) (sin phi1)))))
     (if (<= t_3 5.0)
       (atan2
        t_2
        (-
         (cos delta)
         (* (fma (* (cos theta) (sin delta)) (cos phi1) t_1) (sin phi1))))
       (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(phi1) * cos(delta);
	double t_2 = (sin(theta) * sin(delta)) * cos(phi1);
	double t_3 = lambda1 + atan2(t_2, (cos(delta) - (sin(phi1) * sin(asin((t_1 + ((cos(phi1) * sin(delta)) * cos(theta))))))));
	double tmp;
	if (t_3 <= 6e-12) {
		tmp = lambda1 + atan2(t_2, (cos(delta) - (fma(sin(delta), cos(phi1), t_1) * sin(phi1))));
	} else if (t_3 <= 5.0) {
		tmp = atan2(t_2, (cos(delta) - (fma((cos(theta) * sin(delta)), cos(phi1), t_1) * sin(phi1))));
	} else {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(phi1) * cos(delta))
	t_2 = Float64(Float64(sin(theta) * sin(delta)) * cos(phi1))
	t_3 = Float64(lambda1 + atan(t_2, Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(t_1 + Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)))))))))
	tmp = 0.0
	if (t_3 <= 6e-12)
		tmp = Float64(lambda1 + atan(t_2, Float64(cos(delta) - Float64(fma(sin(delta), cos(phi1), t_1) * sin(phi1)))));
	elseif (t_3 <= 5.0)
		tmp = atan(t_2, Float64(cos(delta) - Float64(fma(Float64(cos(theta) * sin(delta)), cos(phi1), t_1) * sin(phi1))));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Cos[delta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[t$95$2 / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, 6e-12], N[(lambda1 + N[ArcTan[t$95$2 / N[(N[Cos[delta], $MachinePrecision] - N[(N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[ArcTan[t$95$2 / N[(N[Cos[delta], $MachinePrecision] - N[(N[(N[(N[Cos[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 6.0000000000000003e-12

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in theta around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sin.f64 96.2

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

   7. Applied rewrites 96.2%

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

   if 6.0000000000000003e-12 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 5

   1. Initial program 99.1%

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

   3. Taylor expanded in lambda1 around 0

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

      1. lower-atan2.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-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 100.0

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

   8. Applied rewrites 100.0%

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

   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 100.0%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq 6 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \mathsf{fma}\left(\sin delta, \cos \phi_1, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \leq 5:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \mathsf{fma}\left(\cos theta \cdot \sin delta, \cos \phi_1, \sin \phi_1 \cdot \cos delta\right) \cdot \sin \phi_1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 99.7

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.7

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.7

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 93.5

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

7. Applied rewrites 93.5%

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

8. Final simplification 93.5%

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

Alternative 4: 94.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in theta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-sin.f64 93.5

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

5. Applied rewrites 93.5%

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

Alternative 5: 91.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -0.00047:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{\cos \phi_1}{{\left(\sin theta \cdot \sin delta\right)}^{-1}}}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.8 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{{\left(\frac{{\sin theta}^{-1}}{\cos \phi_1 \cdot \sin delta}\right)}^{-1}}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -0.00047)
   (+
    lambda1
    (atan2 (/ (cos phi1) (pow (* (sin theta) (sin delta)) -1.0)) (cos delta)))
   (if (<= delta 1.8e-18)
     (+
      (atan2 (* (cos phi1) (* (sin delta) (sin theta))) (pow (cos phi1) 2.0))
      lambda1)
     (+
      lambda1
      (atan2
       (pow (/ (pow (sin theta) -1.0) (* (cos phi1) (sin delta))) -1.0)
       (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -0.00047) {
		tmp = lambda1 + atan2((cos(phi1) / pow((sin(theta) * sin(delta)), -1.0)), cos(delta));
	} else if (delta <= 1.8e-18) {
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), pow(cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = lambda1 + atan2(pow((pow(sin(theta), -1.0) / (cos(phi1) * sin(delta))), -1.0), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (delta <= (-0.00047d0)) then
        tmp = lambda1 + atan2((cos(phi1) / ((sin(theta) * sin(delta)) ** (-1.0d0))), cos(delta))
    else if (delta <= 1.8d-18) then
        tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) ** 2.0d0)) + lambda1
    else
        tmp = lambda1 + atan2((((sin(theta) ** (-1.0d0)) / (cos(phi1) * sin(delta))) ** (-1.0d0)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -0.00047) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) / Math.pow((Math.sin(theta) * Math.sin(delta)), -1.0)), Math.cos(delta));
	} else if (delta <= 1.8e-18) {
		tmp = Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), Math.pow(Math.cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = lambda1 + Math.atan2(Math.pow((Math.pow(Math.sin(theta), -1.0) / (Math.cos(phi1) * Math.sin(delta))), -1.0), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -0.00047:
		tmp = lambda1 + math.atan2((math.cos(phi1) / math.pow((math.sin(theta) * math.sin(delta)), -1.0)), math.cos(delta))
	elif delta <= 1.8e-18:
		tmp = math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), math.pow(math.cos(phi1), 2.0)) + lambda1
	else:
		tmp = lambda1 + math.atan2(math.pow((math.pow(math.sin(theta), -1.0) / (math.cos(phi1) * math.sin(delta))), -1.0), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -0.00047)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) / (Float64(sin(theta) * sin(delta)) ^ -1.0)), cos(delta)));
	elseif (delta <= 1.8e-18)
		tmp = Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), (cos(phi1) ^ 2.0)) + lambda1);
	else
		tmp = Float64(lambda1 + atan((Float64((sin(theta) ^ -1.0) / Float64(cos(phi1) * sin(delta))) ^ -1.0), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -0.00047)
		tmp = lambda1 + atan2((cos(phi1) / ((sin(theta) * sin(delta)) ^ -1.0)), cos(delta));
	elseif (delta <= 1.8e-18)
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) ^ 2.0)) + lambda1;
	else
		tmp = lambda1 + atan2((((sin(theta) ^ -1.0) / (cos(phi1) * sin(delta))) ^ -1.0), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -0.00047], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] / N[Power[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 1.8e-18], N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[Power[N[(N[Power[N[Sin[theta], $MachinePrecision], -1.0], $MachinePrecision] / N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -4.69999999999999986e-4

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. remove-double-div N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 84.1

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 84.1

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

   7. Applied rewrites 84.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 84.1

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

   9. Applied rewrites 84.1%

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

   if -4.69999999999999986e-4 < delta < 1.80000000000000005e-18

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in delta around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. unpow2 N/A

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

      14. 1-sub-sin N/A

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in delta around 0

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

       1. mul-1-neg N/A

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

       2. sub-neg N/A

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

       3. unpow2 N/A

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

       4. 1-sub-sin N/A

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

       5. unpow2 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-cos.f64 99.6

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

   12. Applied rewrites 99.6%

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

   if 1.80000000000000005e-18 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 84.2

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

   5. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. remove-double-div N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. inv-pow N/A

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

      10. associate-/l/ N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-pow.f64 84.2

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 84.2

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

   7. Applied rewrites 84.2%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.00047:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{\cos \phi_1}{{\left(\sin theta \cdot \sin delta\right)}^{-1}}}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.8 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{{\left(\frac{{\sin theta}^{-1}}{\cos \phi_1 \cdot \sin delta}\right)}^{-1}}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 91.5

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

5. Applied rewrites 91.5%

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

Alternative 7: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -0.00047:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{\cos \phi_1}{{\left(\sin theta \cdot \sin delta\right)}^{-1}}}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.8 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -0.00047)
   (+
    lambda1
    (atan2 (/ (cos phi1) (pow (* (sin theta) (sin delta)) -1.0)) (cos delta)))
   (if (<= delta 1.8e-18)
     (+
      (atan2 (* (cos phi1) (* (sin delta) (sin theta))) (pow (cos phi1) 2.0))
      lambda1)
     (+
      lambda1
      (atan2 (* (* (cos phi1) (sin delta)) (sin theta)) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -0.00047) {
		tmp = lambda1 + atan2((cos(phi1) / pow((sin(theta) * sin(delta)), -1.0)), cos(delta));
	} else if (delta <= 1.8e-18) {
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), pow(cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = lambda1 + atan2(((cos(phi1) * sin(delta)) * sin(theta)), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (delta <= (-0.00047d0)) then
        tmp = lambda1 + atan2((cos(phi1) / ((sin(theta) * sin(delta)) ** (-1.0d0))), cos(delta))
    else if (delta <= 1.8d-18) then
        tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) ** 2.0d0)) + lambda1
    else
        tmp = lambda1 + atan2(((cos(phi1) * sin(delta)) * sin(theta)), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -0.00047) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) / Math.pow((Math.sin(theta) * Math.sin(delta)), -1.0)), Math.cos(delta));
	} else if (delta <= 1.8e-18) {
		tmp = Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), Math.pow(Math.cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = lambda1 + Math.atan2(((Math.cos(phi1) * Math.sin(delta)) * Math.sin(theta)), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -0.00047:
		tmp = lambda1 + math.atan2((math.cos(phi1) / math.pow((math.sin(theta) * math.sin(delta)), -1.0)), math.cos(delta))
	elif delta <= 1.8e-18:
		tmp = math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), math.pow(math.cos(phi1), 2.0)) + lambda1
	else:
		tmp = lambda1 + math.atan2(((math.cos(phi1) * math.sin(delta)) * math.sin(theta)), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -0.00047)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) / (Float64(sin(theta) * sin(delta)) ^ -1.0)), cos(delta)));
	elseif (delta <= 1.8e-18)
		tmp = Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), (cos(phi1) ^ 2.0)) + lambda1);
	else
		tmp = Float64(lambda1 + atan(Float64(Float64(cos(phi1) * sin(delta)) * sin(theta)), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -0.00047)
		tmp = lambda1 + atan2((cos(phi1) / ((sin(theta) * sin(delta)) ^ -1.0)), cos(delta));
	elseif (delta <= 1.8e-18)
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) ^ 2.0)) + lambda1;
	else
		tmp = lambda1 + atan2(((cos(phi1) * sin(delta)) * sin(theta)), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -0.00047], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] / N[Power[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 1.8e-18], N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -4.69999999999999986e-4

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. remove-double-div N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 84.1

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 84.1

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

   7. Applied rewrites 84.1%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 84.1

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

   9. Applied rewrites 84.1%

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

   if -4.69999999999999986e-4 < delta < 1.80000000000000005e-18

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in delta around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-+l+ N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. unpow2 N/A

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

      14. 1-sub-sin N/A

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

   9. Applied rewrites 99.8%

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

   10. Taylor expanded in delta around 0

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

       1. mul-1-neg N/A

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

       2. sub-neg N/A

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

       3. unpow2 N/A

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

       4. 1-sub-sin N/A

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

       5. unpow2 N/A

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

       6. lower-pow.f64 N/A

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

       7. lower-cos.f64 99.6

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

   12. Applied rewrites 99.6%

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

   if 1.80000000000000005e-18 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 84.2

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

   5. Applied rewrites 84.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 84.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 84.2

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

   7. Applied rewrites 84.2%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.00047:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{\cos \phi_1}{{\left(\sin theta \cdot \sin delta\right)}^{-1}}}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.8 \cdot 10^{-18}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin delta\right) \cdot \sin theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.6% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 86.8

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

5. Applied rewrites 86.8%

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

   1. lift-*.f64 N/A

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

   2. remove-double-div N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/r/ N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 86.8

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

   7. lift-/.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. inv-pow N/A

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

   12. lower-/.f64 N/A

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

   13. lower-pow.f64 86.8

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

7. Applied rewrites 86.8%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow-1 N/A

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

   4. associate-/l/ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-/.f64 86.8

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

9. Applied rewrites 86.8%

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

10. Final simplification 86.8%

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

Alternative 9: 88.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 86.8

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

5. Applied rewrites 86.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 86.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 86.8

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

7. Applied rewrites 86.8%

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

Alternative 10: 80.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot theta\\ \mathbf{if}\;delta \leq -6.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{1}{{t\_1}^{-1}}}{\cos delta}\\ \mathbf{elif}\;delta \leq 20000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) theta)))
   (if (<= delta -6.5)
     (+ lambda1 (atan2 (/ 1.0 (pow t_1 -1.0)) (cos delta)))
     (if (<= delta 20000000.0)
       (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))
       (+ lambda1 (atan2 t_1 (cos delta)))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * theta;
	double tmp;
	if (delta <= -6.5) {
		tmp = lambda1 + atan2((1.0 / pow(t_1, -1.0)), cos(delta));
	} else if (delta <= 20000000.0) {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	} else {
		tmp = lambda1 + atan2(t_1, cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(delta) * theta
    if (delta <= (-6.5d0)) then
        tmp = lambda1 + atan2((1.0d0 / (t_1 ** (-1.0d0))), cos(delta))
    else if (delta <= 20000000.0d0) then
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    else
        tmp = lambda1 + atan2(t_1, cos(delta))
    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(delta) * theta;
	double tmp;
	if (delta <= -6.5) {
		tmp = lambda1 + Math.atan2((1.0 / Math.pow(t_1, -1.0)), Math.cos(delta));
	} else if (delta <= 20000000.0) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2(t_1, Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(delta) * theta
	tmp = 0
	if delta <= -6.5:
		tmp = lambda1 + math.atan2((1.0 / math.pow(t_1, -1.0)), math.cos(delta))
	elif delta <= 20000000.0:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2(t_1, math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * theta)
	tmp = 0.0
	if (delta <= -6.5)
		tmp = Float64(lambda1 + atan(Float64(1.0 / (t_1 ^ -1.0)), cos(delta)));
	elseif (delta <= 20000000.0)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(delta) * theta;
	tmp = 0.0;
	if (delta <= -6.5)
		tmp = lambda1 + atan2((1.0 / (t_1 ^ -1.0)), cos(delta));
	elseif (delta <= 20000000.0)
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	else
		tmp = lambda1 + atan2(t_1, cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision]}, If[LessEqual[delta, -6.5], N[(lambda1 + N[ArcTan[N[(1.0 / N[Power[t$95$1, -1.0], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 20000000.0], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -6.5

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      2. remove-double-div N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 84.1

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lower-pow.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 82.5%

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

   11. Taylor expanded in theta around 0

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

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 N/A

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

       4. lower-sin.f64 70.7

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

   13. Applied rewrites 70.7%

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

   if -6.5 < delta < 2e7

   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 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-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 88.9

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

   8. Applied rewrites 88.9%

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

   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 89.1%

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

   if 2e7 < 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 83.5

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

   5. Applied rewrites 83.5%

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

   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 79.6

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

   8. Applied rewrites 79.6%

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

   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 69.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -6.5:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\frac{1}{{\left(\sin delta \cdot theta\right)}^{-1}}}{\cos delta}\\ \mathbf{elif}\;delta \leq 20000000:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 86.8

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

5. Applied rewrites 86.8%

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

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 84.8

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

8. Applied rewrites 84.8%

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

Alternative 12: 80.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -6.5 \lor \neg \left(delta \leq 20000000\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -6.5) (not (<= delta 20000000.0)))
   (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))
   (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -6.5) || !(delta <= 20000000.0)) {
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((delta <= (-6.5d0)) .or. (.not. (delta <= 20000000.0d0))) then
        tmp = lambda1 + atan2((sin(delta) * theta), cos(delta))
    else
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -6.5) || !(delta <= 20000000.0)) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * theta), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -6.5) or not (delta <= 20000000.0):
		tmp = lambda1 + math.atan2((math.sin(delta) * theta), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -6.5) || !(delta <= 20000000.0))
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -6.5) || ~((delta <= 20000000.0)))
		tmp = lambda1 + atan2((sin(delta) * theta), cos(delta));
	else
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -6.5], N[Not[LessEqual[delta, 20000000.0]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -6.5 or 2e7 < 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 83.8

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

   5. Applied rewrites 83.8%

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

   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 81.1

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

   8. Applied rewrites 81.1%

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

   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 70.1%

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

   if -6.5 < delta < 2e7

   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 90.1

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-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 88.9

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

   8. Applied rewrites 88.9%

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

   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 89.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -6.5 \lor \neg \left(delta \leq 20000000\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.6% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 86.8

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

5. Applied rewrites 86.8%

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

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 84.8

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

8. Applied rewrites 84.8%

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

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 72.1%

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

Reproduce

herbie shell --seed 2024324 
(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))))))))))