Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

Alternatives: 16

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-asin.f64 N/A

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

   4. sin-asin N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. distribute-lft-in N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqr-sin-a N/A

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

   6. lower--.f64 N/A

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

   7. cos-2 N/A

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

   8. cos-sum N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-+.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

6. Applied rewrites 99.7%

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

Alternative 3: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

4. Applied rewrites 99.7%

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

5. Final simplification 99.7%

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

Alternative 4: 94.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. 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 96.5

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

5. Applied rewrites 96.5%

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

7. Applied rewrites 96.5%

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

8. Final simplification 96.5%

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

Alternative 5: 94.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. 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 96.5

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

5. Applied rewrites 96.5%

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

6. Final simplification 96.5%

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

Alternative 6: 92.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 94.0

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

5. Applied rewrites 94.0%

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

6. Final simplification 94.0%

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

Alternative 7: 90.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\\ \mathbf{if}\;delta \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{t\_1}{\cos delta - \phi_1 \cdot \sin delta} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin delta) (sin theta)))))
   (if (<= delta -1.7e+14)
     (+ (atan2 (* (* (cos phi1) (sin theta)) (sin delta)) (cos delta)) lambda1)
     (if (<= delta 8.6e-23)
       (+ (atan2 t_1 (pow (cos phi1) 2.0)) lambda1)
       (+ (atan2 t_1 (- (cos delta) (* phi1 (sin delta)))) lambda1)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(delta) * sin(theta));
	double tmp;
	if (delta <= -1.7e+14) {
		tmp = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	} else if (delta <= 8.6e-23) {
		tmp = atan2(t_1, pow(cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = atan2(t_1, (cos(delta) - (phi1 * sin(delta)))) + lambda1;
	}
	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 = cos(phi1) * (sin(delta) * sin(theta))
    if (delta <= (-1.7d+14)) then
        tmp = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1
    else if (delta <= 8.6d-23) then
        tmp = atan2(t_1, (cos(phi1) ** 2.0d0)) + lambda1
    else
        tmp = atan2(t_1, (cos(delta) - (phi1 * sin(delta)))) + lambda1
    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.cos(phi1) * (Math.sin(delta) * Math.sin(theta));
	double tmp;
	if (delta <= -1.7e+14) {
		tmp = Math.atan2(((Math.cos(phi1) * Math.sin(theta)) * Math.sin(delta)), Math.cos(delta)) + lambda1;
	} else if (delta <= 8.6e-23) {
		tmp = Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = Math.atan2(t_1, (Math.cos(delta) - (phi1 * Math.sin(delta)))) + lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(delta) * math.sin(theta))
	tmp = 0
	if delta <= -1.7e+14:
		tmp = math.atan2(((math.cos(phi1) * math.sin(theta)) * math.sin(delta)), math.cos(delta)) + lambda1
	elif delta <= 8.6e-23:
		tmp = math.atan2(t_1, math.pow(math.cos(phi1), 2.0)) + lambda1
	else:
		tmp = math.atan2(t_1, (math.cos(delta) - (phi1 * math.sin(delta)))) + lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(delta) * sin(theta)))
	tmp = 0.0
	if (delta <= -1.7e+14)
		tmp = Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1);
	elseif (delta <= 8.6e-23)
		tmp = Float64(atan(t_1, (cos(phi1) ^ 2.0)) + lambda1);
	else
		tmp = Float64(atan(t_1, Float64(cos(delta) - Float64(phi1 * sin(delta)))) + lambda1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(delta) * sin(theta));
	tmp = 0.0;
	if (delta <= -1.7e+14)
		tmp = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	elseif (delta <= 8.6e-23)
		tmp = atan2(t_1, (cos(phi1) ^ 2.0)) + lambda1;
	else
		tmp = atan2(t_1, (cos(delta) - (phi1 * sin(delta)))) + lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -1.7e+14], N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], If[LessEqual[delta, 8.6e-23], N[(N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[t$95$1 / N[(N[Cos[delta], $MachinePrecision] - N[(phi1 * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if delta < -1.7e14

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 91.0

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

   5. Applied rewrites 91.0%

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

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 91.0

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

   7. Applied rewrites 91.0%

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

   if -1.7e14 < delta < 8.60000000000000004e-23

   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 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. unpow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 99.9

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

   8. Applied rewrites 99.9%

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

   if 8.60000000000000004e-23 < 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 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 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 87.8%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \phi_1 \cdot \sin delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2 (* (* (cos phi1) (sin theta)) (sin delta)) (cos delta))
          lambda1)))
   (if (<= delta -1.7e+14)
     t_1
     (if (<= delta 8.6e-23)
       (+
        (atan2 (* (cos phi1) (* (sin delta) (sin theta))) (pow (cos phi1) 2.0))
        lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -1.7e+14) {
		tmp = t_1;
	} else if (delta <= 8.6e-23) {
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), pow(cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1
    if (delta <= (-1.7d+14)) then
        tmp = t_1
    else if (delta <= 8.6d-23) then
        tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) ** 2.0d0)) + lambda1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.atan2(((Math.cos(phi1) * Math.sin(theta)) * Math.sin(delta)), Math.cos(delta)) + lambda1;
	double tmp;
	if (delta <= -1.7e+14) {
		tmp = t_1;
	} else if (delta <= 8.6e-23) {
		tmp = Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), Math.pow(Math.cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2(((math.cos(phi1) * math.sin(theta)) * math.sin(delta)), math.cos(delta)) + lambda1
	tmp = 0
	if delta <= -1.7e+14:
		tmp = t_1
	elif delta <= 8.6e-23:
		tmp = math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), math.pow(math.cos(phi1), 2.0)) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -1.7e+14)
		tmp = t_1;
	elseif (delta <= 8.6e-23)
		tmp = Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), (cos(phi1) ^ 2.0)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	tmp = 0.0;
	if (delta <= -1.7e+14)
		tmp = t_1;
	elseif (delta <= 8.6e-23)
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), (cos(phi1) ^ 2.0)) + lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -1.7e+14], t$95$1, If[LessEqual[delta, 8.6e-23], 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], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.7e14 or 8.60000000000000004e-23 < 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 88.3

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

   5. Applied rewrites 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if -1.7e14 < delta < 8.60000000000000004e-23

   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 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. unpow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 99.9

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

   8. Applied rewrites 99.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\left(\mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right) \cdot \sin theta\right) \cdot delta\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2 (* (* (cos phi1) (sin theta)) (sin delta)) (cos delta))
          lambda1)))
   (if (<= delta -1.7e+14)
     t_1
     (if (<= delta 8.6e-23)
       (+
        (atan2
         (*
          (*
           (* (fma -0.16666666666666666 (* delta delta) 1.0) (sin theta))
           delta)
          (cos phi1))
         (pow (cos phi1) 2.0))
        lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -1.7e+14) {
		tmp = t_1;
	} else if (delta <= 8.6e-23) {
		tmp = atan2((((fma(-0.16666666666666666, (delta * delta), 1.0) * sin(theta)) * delta) * cos(phi1)), pow(cos(phi1), 2.0)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -1.7e+14)
		tmp = t_1;
	elseif (delta <= 8.6e-23)
		tmp = Float64(atan(Float64(Float64(Float64(fma(-0.16666666666666666, Float64(delta * delta), 1.0) * sin(theta)) * delta) * cos(phi1)), (cos(phi1) ^ 2.0)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -1.7e+14], t$95$1, If[LessEqual[delta, 8.6e-23], N[(N[ArcTan[N[(N[(N[(N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision] + 1.0), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * delta), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.7e14 or 8.60000000000000004e-23 < 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 88.3

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

   5. Applied rewrites 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if -1.7e14 < delta < 8.60000000000000004e-23

   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 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in delta around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt1-in N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 90.0

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\left(\mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right) \cdot \color{blue}{\sin theta}\right) \cdot delta\right) \cdot \cos \phi_1}{\cos delta} \]

   8. Applied rewrites 90.0%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\left(\mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right) \cdot \sin theta\right) \cdot delta\right)} \cdot \cos \phi_1}{\cos delta} \]

   9. Taylor expanded in delta around 0

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

       1. unpow2 N/A

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

       2. 1-sub-sin N/A

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

       3. unpow2 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-cos.f64 99.9

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\left(\mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right) \cdot \sin theta\right) \cdot delta\right) \cdot \cos \phi_1}{{\color{blue}{\cos \phi_1}}^{2}} \]

   11. Applied rewrites 99.9%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\left(\mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right) \cdot \sin theta\right) \cdot delta\right) \cdot \cos \phi_1}{\color{blue}{{\cos \phi_1}^{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\left(\mathsf{fma}\left(-0.16666666666666666, delta \cdot delta, 1\right) \cdot \sin theta\right) \cdot delta\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(\cos \left(2 \cdot \phi_1\right), 0.5, 0.5\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2 (* (* (cos phi1) (sin theta)) (sin delta)) (cos delta))
          lambda1)))
   (if (<= delta -1.7e+14)
     t_1
     (if (<= delta 8.6e-23)
       (+
        (atan2
         (* (cos phi1) (* (sin delta) (sin theta)))
         (fma (cos (* 2.0 phi1)) 0.5 0.5))
        lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -1.7e+14) {
		tmp = t_1;
	} else if (delta <= 8.6e-23) {
		tmp = atan2((cos(phi1) * (sin(delta) * sin(theta))), fma(cos((2.0 * phi1)), 0.5, 0.5)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(cos(phi1) * sin(theta)) * sin(delta)), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -1.7e+14)
		tmp = t_1;
	elseif (delta <= 8.6e-23)
		tmp = Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), fma(cos(Float64(2.0 * phi1)), 0.5, 0.5)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -1.7e+14], t$95$1, If[LessEqual[delta, 8.6e-23], N[(N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(2.0 * phi1), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.7e14 or 8.60000000000000004e-23 < 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 88.3

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

   5. Applied rewrites 88.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 88.3

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

   7. Applied rewrites 88.3%

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

   if -1.7e14 < delta < 8.60000000000000004e-23

   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

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

      2. lift-sin.f64 N/A

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

      3. lift-asin.f64 N/A

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

      4. sin-asin N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-cos.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. lower-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.7 \cdot 10^{+14}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-23}:\\ \;\;\;\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(\cos \left(2 \cdot \phi_1\right), 0.5, 0.5\right)} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \sin delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 89.1

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

5. Applied rewrites 89.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 89.1

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

7. Applied rewrites 89.1%

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

8. Final simplification 89.1%

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

Alternative 12: 86.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 89.1

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

5. Applied rewrites 89.1%

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

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 87.6

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

8. Applied rewrites 87.6%

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

9. Final simplification 87.6%

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

Alternative 13: 79.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq -2400000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 7.5 \cdot 10^{-50}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot \sin delta\right) \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= theta -2400000000000.0)
   (+
    (atan2
     (* (* (fma (* -0.16666666666666666 delta) delta 1.0) (sin theta)) delta)
     (cos delta))
    lambda1)
   (if (<= theta 7.5e-50)
     (+
      (atan2
       (* (* (fma (* theta theta) -0.16666666666666666 1.0) (sin delta)) theta)
       (cos delta))
      lambda1)
     (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= -2400000000000.0) {
		tmp = atan2(((fma((-0.16666666666666666 * delta), delta, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1;
	} else if (theta <= 7.5e-50) {
		tmp = atan2(((fma((theta * theta), -0.16666666666666666, 1.0) * sin(delta)) * theta), cos(delta)) + lambda1;
	} else {
		tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (theta <= -2400000000000.0)
		tmp = Float64(atan(Float64(Float64(fma(Float64(-0.16666666666666666 * delta), delta, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1);
	elseif (theta <= 7.5e-50)
		tmp = Float64(atan(Float64(Float64(fma(Float64(theta * theta), -0.16666666666666666, 1.0) * sin(delta)) * theta), cos(delta)) + lambda1);
	else
		tmp = Float64(atan(Float64(delta * sin(theta)), cos(delta)) + lambda1);
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[theta, -2400000000000.0], N[(N[ArcTan[N[(N[(N[(N[(-0.16666666666666666 * delta), $MachinePrecision] * delta + 1.0), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], If[LessEqual[theta, 7.5e-50], N[(N[ArcTan[N[(N[(N[(N[(theta * theta), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], N[(N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if theta < -2.4e12

   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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 87.0

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

   8. Applied rewrites 87.0%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 76.5%

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

   if -2.4e12 < theta < 7.5e-50

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 91.1

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

   5. Applied rewrites 91.1%

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

   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 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 89.5%

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

   if 7.5e-50 < theta

   1. Initial program 99.7%

      \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
   2. 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 86.1

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

   5. Applied rewrites 86.1%

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

   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 85.1

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

   8. Applied rewrites 85.1%

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

   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 77.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -2400000000000:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 7.5 \cdot 10^{-50}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(theta \cdot theta, -0.16666666666666666, 1\right) \cdot \sin delta\right) \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{if}\;theta \leq -2.66 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;theta \leq 5.7 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          (atan2
           (*
            (* (fma (* -0.16666666666666666 delta) delta 1.0) (sin theta))
            delta)
           (cos delta))
          lambda1)))
   (if (<= theta -2.66e+26)
     t_1
     (if (<= theta 5.7e+20)
       (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2(((fma((-0.16666666666666666 * delta), delta, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1;
	double tmp;
	if (theta <= -2.66e+26) {
		tmp = t_1;
	} else if (theta <= 5.7e+20) {
		tmp = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(Float64(fma(Float64(-0.16666666666666666 * delta), delta, 1.0) * sin(theta)) * delta), cos(delta)) + lambda1)
	tmp = 0.0
	if (theta <= -2.66e+26)
		tmp = t_1;
	elseif (theta <= 5.7e+20)
		tmp = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[(N[(N[(-0.16666666666666666 * delta), $MachinePrecision] * delta + 1.0), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[theta, -2.66e+26], t$95$1, If[LessEqual[theta, 5.7e+20], N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if theta < -2.65999999999999994e26 or 5.7e20 < theta

   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 86.1

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

   5. Applied rewrites 86.1%

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

   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 84.6

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

   8. Applied rewrites 84.6%

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

   9. Taylor expanded in delta around 0

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

   11. Applied rewrites 76.2%

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

   if -2.65999999999999994e26 < theta < 5.7e20

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 90.5

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

   8. Applied rewrites 90.5%

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

   9. Taylor expanded in theta around 0

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

   11. Applied rewrites 88.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq -2.66 \cdot 10^{+26}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;theta \leq 5.7 \cdot 10^{+20}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\left(\mathsf{fma}\left(-0.16666666666666666 \cdot delta, delta, 1\right) \cdot \sin theta\right) \cdot delta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{if}\;delta \leq -4.6 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 4.3 \cdot 10^{+24}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ (atan2 (* (sin delta) theta) (cos delta)) lambda1)))
   (if (<= delta -4.6e+21)
     t_1
     (if (<= delta 4.3e+24)
       (+ (atan2 (* delta (sin theta)) (cos delta)) lambda1)
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	double tmp;
	if (delta <= -4.6e+21) {
		tmp = t_1;
	} else if (delta <= 4.3e+24) {
		tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan2((sin(delta) * theta), cos(delta)) + lambda1
    if (delta <= (-4.6d+21)) then
        tmp = t_1
    else if (delta <= 4.3d+24) then
        tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.atan2((Math.sin(delta) * theta), Math.cos(delta)) + lambda1;
	double tmp;
	if (delta <= -4.6e+21) {
		tmp = t_1;
	} else if (delta <= 4.3e+24) {
		tmp = Math.atan2((delta * Math.sin(theta)), Math.cos(delta)) + lambda1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2((math.sin(delta) * theta), math.cos(delta)) + lambda1
	tmp = 0
	if delta <= -4.6e+21:
		tmp = t_1
	elif delta <= 4.3e+24:
		tmp = math.atan2((delta * math.sin(theta)), math.cos(delta)) + lambda1
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(atan(Float64(sin(delta) * theta), cos(delta)) + lambda1)
	tmp = 0.0
	if (delta <= -4.6e+21)
		tmp = t_1;
	elseif (delta <= 4.3e+24)
		tmp = Float64(atan(Float64(delta * sin(theta)), cos(delta)) + lambda1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2((sin(delta) * theta), cos(delta)) + lambda1;
	tmp = 0.0;
	if (delta <= -4.6e+21)
		tmp = t_1;
	elseif (delta <= 4.3e+24)
		tmp = atan2((delta * sin(theta)), cos(delta)) + lambda1;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]}, If[LessEqual[delta, -4.6e+21], t$95$1, If[LessEqual[delta, 4.3e+24], N[(N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.6e21 or 4.29999999999999987e24 < delta

   1. Initial program 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 88.2

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

   5. Applied rewrites 88.2%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-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 86.3

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

   8. Applied rewrites 86.3%

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

   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 76.8%

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

   if -4.6e21 < delta < 4.29999999999999987e24

   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 89.8

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

   5. Applied rewrites 89.8%

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

   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 88.8

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

   8. Applied rewrites 88.8%

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

   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 87.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -4.6 \cdot 10^{+21}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \mathbf{elif}\;delta \leq 4.3 \cdot 10^{+24}:\\ \;\;\;\;\tan^{-1}_* \frac{delta \cdot \sin theta}{\cos delta} + \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta} + \lambda_1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 89.1

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

5. Applied rewrites 89.1%

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

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 87.6

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

8. Applied rewrites 87.6%

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

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 72.7%

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

12. Final simplification 72.7%

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

Reproduce

herbie shell --seed 2024325 
(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))))))))))