Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 18.7s

Alternatives: 17

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), fma(Float64(0.5 + Float64(0.5 * cos(Float64(phi1 * -2.0)))), cos(delta), Float64(Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)) * Float64(-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[(0.5 + N[(0.5 * N[Cos[N[(phi1 * -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[delta], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] * (-N[Sin[phi1], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + lambda1), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sin-asin N/A

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

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

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

   5. distribute-rgt-in N/A

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

   6. associate-+l+ N/A

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

4. Applied egg-rr 99.8%

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

   1. distribute-lft-neg-out N/A

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

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

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

   3. sqr-sin-a N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

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

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

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

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

   9. cos-2 N/A

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

   10. cos-sum N/A

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

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

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

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

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

   13. metadata-eval 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in lambda1 around 0

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 87.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta}\\ t_2 := \cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ t_4 := \cos \phi_1 \cdot \sin delta\\ t_5 := \lambda_1 + \tan^{-1}_* \frac{t\_2}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(t\_4 \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)}\\ \mathbf{if}\;t\_5 \leq -3.14159265358979:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_5 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{\tan^{-1}_* \frac{\sin theta \cdot t\_4}{1}}{\lambda_1} - -1\right)\\ \mathbf{elif}\;t\_5 \leq 3.12:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (atan2 (* (sin delta) (* (cos phi1) (sin theta))) (cos delta)))
        (t_2 (* (cos phi1) (* (sin delta) (sin theta))))
        (t_3 (+ lambda1 (atan2 t_2 (fma delta (* delta -0.5) 1.0))))
        (t_4 (* (cos phi1) (sin delta)))
        (t_5
         (+
          lambda1
          (atan2
           t_2
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin (+ (* t_4 (cos theta)) (* (cos delta) (sin phi1)))))))))))
   (if (<= t_5 -3.14159265358979)
     t_3
     (if (<= t_5 -5e-7)
       t_1
       (if (<= t_5 2e-12)
         (* lambda1 (- (/ (atan2 (* (sin theta) t_4) 1.0) lambda1) -1.0))
         (if (<= t_5 3.12) t_1 t_3))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((sin(delta) * (cos(phi1) * sin(theta))), cos(delta));
	double t_2 = cos(phi1) * (sin(delta) * sin(theta));
	double t_3 = lambda1 + atan2(t_2, fma(delta, (delta * -0.5), 1.0));
	double t_4 = cos(phi1) * sin(delta);
	double t_5 = lambda1 + atan2(t_2, (cos(delta) - (sin(phi1) * sin(asin(((t_4 * cos(theta)) + (cos(delta) * sin(phi1))))))));
	double tmp;
	if (t_5 <= -3.14159265358979) {
		tmp = t_3;
	} else if (t_5 <= -5e-7) {
		tmp = t_1;
	} else if (t_5 <= 2e-12) {
		tmp = lambda1 * ((atan2((sin(theta) * t_4), 1.0) / lambda1) - -1.0);
	} else if (t_5 <= 3.12) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan(Float64(sin(delta) * Float64(cos(phi1) * sin(theta))), cos(delta))
	t_2 = Float64(cos(phi1) * Float64(sin(delta) * sin(theta)))
	t_3 = Float64(lambda1 + atan(t_2, fma(delta, Float64(delta * -0.5), 1.0)))
	t_4 = Float64(cos(phi1) * sin(delta))
	t_5 = Float64(lambda1 + atan(t_2, Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(t_4 * cos(theta)) + Float64(cos(delta) * sin(phi1)))))))))
	tmp = 0.0
	if (t_5 <= -3.14159265358979)
		tmp = t_3;
	elseif (t_5 <= -5e-7)
		tmp = t_1;
	elseif (t_5 <= 2e-12)
		tmp = Float64(lambda1 * Float64(Float64(atan(Float64(sin(theta) * t_4), 1.0) / lambda1) - -1.0));
	elseif (t_5 <= 3.12)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[t$95$2 / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(lambda1 + N[ArcTan[t$95$2 / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(t$95$4 * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -3.14159265358979], t$95$3, If[LessEqual[t$95$5, -5e-7], t$95$1, If[LessEqual[t$95$5, 2e-12], N[(lambda1 * N[(N[(N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$4), $MachinePrecision] / 1.0], $MachinePrecision] / lambda1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 3.12], t$95$1, t$95$3]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -3.14159265358979001 or 3.1200000000000001 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))

   1. Initial program 100.0%

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

      1. sin-asin N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in delta around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

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

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

   7. Simplified 99.4%

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

   8. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.4

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

   10. Simplified 99.4%

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

   if -3.14159265358979001 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -4.99999999999999977e-7 or 1.99999999999999996e-12 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 3.1200000000000001

   1. Initial program 99.3%

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

   3. Taylor expanded in lambda1 around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   5. Simplified 92.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 61.3

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

   8. Simplified 61.3%

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

   if -4.99999999999999977e-7 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 1.99999999999999996e-12

   1. Initial program 99.5%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 88.4%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 80.9%

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

   9. Taylor expanded in lambda1 around -inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

   11. Simplified 80.9%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -3.14159265358979:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{\tan^{-1}_* \frac{\sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)}{1}}{\lambda_1} - -1\right)\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 3.12:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \left(\cos \phi_1 \cdot \sin theta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ t_2 := \tan^{-1}_* \frac{t\_1}{\cos delta}\\ t_3 := \cos \phi_1 \cdot t\_1\\ t_4 := \lambda_1 + \tan^{-1}_* \frac{t\_3}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ t_5 := \cos \phi_1 \cdot \sin delta\\ t_6 := \lambda_1 + \tan^{-1}_* \frac{t\_3}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(t\_5 \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)}\\ \mathbf{if}\;t\_6 \leq -3.14159265358979:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_6 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_6 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{\tan^{-1}_* \frac{\sin theta \cdot t\_5}{1}}{\lambda_1} - -1\right)\\ \mathbf{elif}\;t\_6 \leq 3:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta)))
        (t_2 (atan2 t_1 (cos delta)))
        (t_3 (* (cos phi1) t_1))
        (t_4 (+ lambda1 (atan2 t_3 (fma delta (* delta -0.5) 1.0))))
        (t_5 (* (cos phi1) (sin delta)))
        (t_6
         (+
          lambda1
          (atan2
           t_3
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin (+ (* t_5 (cos theta)) (* (cos delta) (sin phi1)))))))))))
   (if (<= t_6 -3.14159265358979)
     t_4
     (if (<= t_6 -5e-7)
       t_2
       (if (<= t_6 2e-12)
         (* lambda1 (- (/ (atan2 (* (sin theta) t_5) 1.0) lambda1) -1.0))
         (if (<= t_6 3.0) t_2 t_4))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double t_2 = atan2(t_1, cos(delta));
	double t_3 = cos(phi1) * t_1;
	double t_4 = lambda1 + atan2(t_3, fma(delta, (delta * -0.5), 1.0));
	double t_5 = cos(phi1) * sin(delta);
	double t_6 = lambda1 + atan2(t_3, (cos(delta) - (sin(phi1) * sin(asin(((t_5 * cos(theta)) + (cos(delta) * sin(phi1))))))));
	double tmp;
	if (t_6 <= -3.14159265358979) {
		tmp = t_4;
	} else if (t_6 <= -5e-7) {
		tmp = t_2;
	} else if (t_6 <= 2e-12) {
		tmp = lambda1 * ((atan2((sin(theta) * t_5), 1.0) / lambda1) - -1.0);
	} else if (t_6 <= 3.0) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	t_2 = atan(t_1, cos(delta))
	t_3 = Float64(cos(phi1) * t_1)
	t_4 = Float64(lambda1 + atan(t_3, fma(delta, Float64(delta * -0.5), 1.0)))
	t_5 = Float64(cos(phi1) * sin(delta))
	t_6 = Float64(lambda1 + atan(t_3, Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(t_5 * cos(theta)) + Float64(cos(delta) * sin(phi1)))))))))
	tmp = 0.0
	if (t_6 <= -3.14159265358979)
		tmp = t_4;
	elseif (t_6 <= -5e-7)
		tmp = t_2;
	elseif (t_6 <= 2e-12)
		tmp = Float64(lambda1 * Float64(Float64(atan(Float64(sin(theta) * t_5), 1.0) / lambda1) - -1.0));
	elseif (t_6 <= 3.0)
		tmp = t_2;
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(lambda1 + N[ArcTan[t$95$3 / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(lambda1 + N[ArcTan[t$95$3 / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(t$95$5 * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -3.14159265358979], t$95$4, If[LessEqual[t$95$6, -5e-7], t$95$2, If[LessEqual[t$95$6, 2e-12], N[(lambda1 * N[(N[(N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$5), $MachinePrecision] / 1.0], $MachinePrecision] / lambda1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 3.0], t$95$2, t$95$4]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -3.14159265358979001 or 3 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))

   1. Initial program 100.0%

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

      1. sin-asin N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in delta around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

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

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

   7. Simplified 98.9%

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

   8. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.9

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

   10. Simplified 98.9%

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

   if -3.14159265358979001 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -4.99999999999999977e-7 or 1.99999999999999996e-12 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 3

   1. Initial program 99.3%

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

   3. Taylor expanded in lambda1 around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   5. Simplified 92.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 62.0

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

   8. Simplified 62.0%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 58.3

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

   11. Simplified 58.3%

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

   if -4.99999999999999977e-7 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 1.99999999999999996e-12

   1. Initial program 99.5%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 88.4%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 80.9%

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

   9. Taylor expanded in lambda1 around -inf

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

       1. associate-*r* N/A

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

       2. *-commutative N/A

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

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

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

   11. Simplified 80.9%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -3.14159265358979:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{\tan^{-1}_* \frac{\sin theta \cdot \left(\cos \phi_1 \cdot \sin delta\right)}{1}}{\lambda_1} - -1\right)\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 3:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ t_2 := \cos \phi_1 \cdot t\_1\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{t\_2}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ t_4 := \lambda_1 + \tan^{-1}_* \frac{t\_2}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)}\\ t_5 := \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;t\_4 \leq -3.14159265358979:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{1}\\ \mathbf{elif}\;t\_4 \leq 3:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta)))
        (t_2 (* (cos phi1) t_1))
        (t_3 (+ lambda1 (atan2 t_2 (fma delta (* delta -0.5) 1.0))))
        (t_4
         (+
          lambda1
          (atan2
           t_2
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin
               (+
                (* (* (cos phi1) (sin delta)) (cos theta))
                (* (cos delta) (sin phi1))))))))))
        (t_5 (atan2 t_1 (cos delta))))
   (if (<= t_4 -3.14159265358979)
     t_3
     (if (<= t_4 -5e-7)
       t_5
       (if (<= t_4 2e-12)
         (+ lambda1 (atan2 t_2 1.0))
         (if (<= t_4 3.0) t_5 t_3))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double t_2 = cos(phi1) * t_1;
	double t_3 = lambda1 + atan2(t_2, fma(delta, (delta * -0.5), 1.0));
	double t_4 = lambda1 + atan2(t_2, (cos(delta) - (sin(phi1) * sin(asin((((cos(phi1) * sin(delta)) * cos(theta)) + (cos(delta) * sin(phi1))))))));
	double t_5 = atan2(t_1, cos(delta));
	double tmp;
	if (t_4 <= -3.14159265358979) {
		tmp = t_3;
	} else if (t_4 <= -5e-7) {
		tmp = t_5;
	} else if (t_4 <= 2e-12) {
		tmp = lambda1 + atan2(t_2, 1.0);
	} else if (t_4 <= 3.0) {
		tmp = t_5;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	t_2 = Float64(cos(phi1) * t_1)
	t_3 = Float64(lambda1 + atan(t_2, fma(delta, Float64(delta * -0.5), 1.0)))
	t_4 = Float64(lambda1 + atan(t_2, Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)) + Float64(cos(delta) * sin(phi1)))))))))
	t_5 = atan(t_1, cos(delta))
	tmp = 0.0
	if (t_4 <= -3.14159265358979)
		tmp = t_3;
	elseif (t_4 <= -5e-7)
		tmp = t_5;
	elseif (t_4 <= 2e-12)
		tmp = Float64(lambda1 + atan(t_2, 1.0));
	elseif (t_4 <= 3.0)
		tmp = t_5;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[t$95$2 / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(lambda1 + N[ArcTan[t$95$2 / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -3.14159265358979], t$95$3, If[LessEqual[t$95$4, -5e-7], t$95$5, If[LessEqual[t$95$4, 2e-12], N[(lambda1 + N[ArcTan[t$95$2 / 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 3.0], t$95$5, t$95$3]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -3.14159265358979001 or 3 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))))

   1. Initial program 100.0%

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

      1. sin-asin N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in delta around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

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

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

   7. Simplified 98.9%

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

   8. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 98.9

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

   10. Simplified 98.9%

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

   if -3.14159265358979001 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -4.99999999999999977e-7 or 1.99999999999999996e-12 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 3

   1. Initial program 99.3%

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

   3. Taylor expanded in lambda1 around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   5. Simplified 92.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 62.0

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

   8. Simplified 62.0%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 58.3

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

   11. Simplified 58.3%

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

   if -4.99999999999999977e-7 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 1.99999999999999996e-12

   1. Initial program 99.5%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 88.4%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 80.9%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -3.14159265358979:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{1}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 3:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ t_2 := \tan^{-1}_* \frac{t\_1}{\cos delta}\\ t_3 := \cos \phi_1 \cdot t\_1\\ t_4 := \lambda_1 + \tan^{-1}_* \frac{t\_3}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)}\\ \mathbf{if}\;t\_4 \leq -500:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{1}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_3}{1}\\ \mathbf{elif}\;t\_4 \leq 5:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta)))
        (t_2 (atan2 t_1 (cos delta)))
        (t_3 (* (cos phi1) t_1))
        (t_4
         (+
          lambda1
          (atan2
           t_3
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin
               (+
                (* (* (cos phi1) (sin delta)) (cos theta))
                (* (cos delta) (sin phi1)))))))))))
   (if (<= t_4 -500.0)
     (+
      lambda1
      (atan2
       (*
        (sin theta)
        (fma delta (* -0.16666666666666666 (* delta delta)) delta))
       1.0))
     (if (<= t_4 -5e-7)
       t_2
       (if (<= t_4 2e-12)
         (+ lambda1 (atan2 t_3 1.0))
         (if (<= t_4 5.0)
           t_2
           (+ lambda1 (atan2 (* delta (sin theta)) 1.0))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double t_2 = atan2(t_1, cos(delta));
	double t_3 = cos(phi1) * t_1;
	double t_4 = lambda1 + atan2(t_3, (cos(delta) - (sin(phi1) * sin(asin((((cos(phi1) * sin(delta)) * cos(theta)) + (cos(delta) * sin(phi1))))))));
	double tmp;
	if (t_4 <= -500.0) {
		tmp = lambda1 + atan2((sin(theta) * fma(delta, (-0.16666666666666666 * (delta * delta)), delta)), 1.0);
	} else if (t_4 <= -5e-7) {
		tmp = t_2;
	} else if (t_4 <= 2e-12) {
		tmp = lambda1 + atan2(t_3, 1.0);
	} else if (t_4 <= 5.0) {
		tmp = t_2;
	} else {
		tmp = lambda1 + atan2((delta * sin(theta)), 1.0);
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	t_2 = atan(t_1, cos(delta))
	t_3 = Float64(cos(phi1) * t_1)
	t_4 = Float64(lambda1 + atan(t_3, Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)) + Float64(cos(delta) * sin(phi1)))))))))
	tmp = 0.0
	if (t_4 <= -500.0)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * fma(delta, Float64(-0.16666666666666666 * Float64(delta * delta)), delta)), 1.0));
	elseif (t_4 <= -5e-7)
		tmp = t_2;
	elseif (t_4 <= 2e-12)
		tmp = Float64(lambda1 + atan(t_3, 1.0));
	elseif (t_4 <= 5.0)
		tmp = t_2;
	else
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), 1.0));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(lambda1 + N[ArcTan[t$95$3 / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -500.0], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision]), $MachinePrecision] + delta), $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-7], t$95$2, If[LessEqual[t$95$4, 2e-12], N[(lambda1 + N[ArcTan[t$95$3 / 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5.0], t$95$2, N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 99.4%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 99.4

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

   11. Simplified 99.4%

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

   12. Taylor expanded in delta around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 99.4

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

   14. Simplified 99.4%

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

   if -500 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -4.99999999999999977e-7 or 1.99999999999999996e-12 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 5

   1. Initial program 99.5%

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

   3. Taylor expanded in lambda1 around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   5. Simplified 94.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 67.5

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

   8. Simplified 67.5%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 55.6

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

   11. Simplified 55.6%

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

   if -4.99999999999999977e-7 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 1.99999999999999996e-12

   1. Initial program 99.5%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 88.4%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 80.9%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 98.7%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 98.7%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 98.7

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

   11. Simplified 98.7%

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

   12. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 98.8

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

   14. Simplified 98.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -500:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{1}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{1}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 5:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \sin theta\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot t\_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)}\\ t_3 := \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;t\_2 \leq -500:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{1}\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{\tan^{-1}_* \frac{t\_1}{1}}{\lambda_1} - -1\right)\\ \mathbf{elif}\;t\_2 \leq 5:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (sin theta)))
        (t_2
         (+
          lambda1
          (atan2
           (* (cos phi1) t_1)
           (-
            (cos delta)
            (*
             (sin phi1)
             (sin
              (asin
               (+
                (* (* (cos phi1) (sin delta)) (cos theta))
                (* (cos delta) (sin phi1))))))))))
        (t_3 (atan2 t_1 (cos delta))))
   (if (<= t_2 -500.0)
     (+
      lambda1
      (atan2
       (*
        (sin theta)
        (fma delta (* -0.16666666666666666 (* delta delta)) delta))
       1.0))
     (if (<= t_2 -5e-7)
       t_3
       (if (<= t_2 2e-12)
         (* lambda1 (- (/ (atan2 t_1 1.0) lambda1) -1.0))
         (if (<= t_2 5.0)
           t_3
           (+ lambda1 (atan2 (* delta (sin theta)) 1.0))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * sin(theta);
	double t_2 = lambda1 + atan2((cos(phi1) * t_1), (cos(delta) - (sin(phi1) * sin(asin((((cos(phi1) * sin(delta)) * cos(theta)) + (cos(delta) * sin(phi1))))))));
	double t_3 = atan2(t_1, cos(delta));
	double tmp;
	if (t_2 <= -500.0) {
		tmp = lambda1 + atan2((sin(theta) * fma(delta, (-0.16666666666666666 * (delta * delta)), delta)), 1.0);
	} else if (t_2 <= -5e-7) {
		tmp = t_3;
	} else if (t_2 <= 2e-12) {
		tmp = lambda1 * ((atan2(t_1, 1.0) / lambda1) - -1.0);
	} else if (t_2 <= 5.0) {
		tmp = t_3;
	} else {
		tmp = lambda1 + atan2((delta * sin(theta)), 1.0);
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * sin(theta))
	t_2 = Float64(lambda1 + atan(Float64(cos(phi1) * t_1), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)) + Float64(cos(delta) * sin(phi1)))))))))
	t_3 = atan(t_1, cos(delta))
	tmp = 0.0
	if (t_2 <= -500.0)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * fma(delta, Float64(-0.16666666666666666 * Float64(delta * delta)), delta)), 1.0));
	elseif (t_2 <= -5e-7)
		tmp = t_3;
	elseif (t_2 <= 2e-12)
		tmp = Float64(lambda1 * Float64(Float64(atan(t_1, 1.0) / lambda1) - -1.0));
	elseif (t_2 <= 5.0)
		tmp = t_3;
	else
		tmp = Float64(lambda1 + atan(Float64(delta * sin(theta)), 1.0));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -500.0], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision]), $MachinePrecision] + delta), $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -5e-7], t$95$3, If[LessEqual[t$95$2, 2e-12], N[(lambda1 * N[(N[(N[ArcTan[t$95$1 / 1.0], $MachinePrecision] / lambda1), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5.0], t$95$3, N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -500

   1. Initial program 100.0%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 100.0%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 99.4%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 99.4

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

   11. Simplified 99.4%

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

   12. Taylor expanded in delta around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

       4. accelerator-lowering-fma.f64 N/A

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 99.4

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

   14. Simplified 99.4%

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

   if -500 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < -4.99999999999999977e-7 or 1.99999999999999996e-12 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 5

   1. Initial program 99.5%

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

   3. Taylor expanded in lambda1 around 0

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

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

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

   5. Simplified 94.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 67.5

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

   8. Simplified 67.5%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 55.6

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

   11. Simplified 55.6%

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

   if -4.99999999999999977e-7 < (+.f64 lambda1 (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))) < 1.99999999999999996e-12

   1. Initial program 99.5%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 88.4%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 80.9%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 79.6

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

   11. Simplified 79.6%

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

   12. Taylor expanded in lambda1 around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

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

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

       4. mul-1-neg N/A

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

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

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

   14. Simplified 79.6%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 98.7%

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

   6. Taylor expanded in phi1 around 0

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

   8. Simplified 98.7%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 98.7

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

   11. Simplified 98.7%

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

   12. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 98.8

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

   14. Simplified 98.8%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -500:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{1}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -5 \cdot 10^{-7}:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 2 \cdot 10^{-12}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}}{\lambda_1} - -1\right)\\ \mathbf{elif}\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 5:\\ \;\;\;\;\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\\ t_2 := \tan^{-1}_* \frac{t\_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)}\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;t\_2 \leq -0.002:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin delta) (sin theta))))
        (t_2
         (atan2
          t_1
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (* (cos phi1) (sin delta)) (cos theta))
               (* (cos delta) (sin phi1)))))))))
        (t_3 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= t_2 -0.002)
     t_3
     (if (<= t_2 2e-19) (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0))) t_3))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(delta) * sin(theta));
	double t_2 = atan2(t_1, (cos(delta) - (sin(phi1) * sin(asin((((cos(phi1) * sin(delta)) * cos(theta)) + (cos(delta) * sin(phi1))))))));
	double t_3 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (t_2 <= -0.002) {
		tmp = t_3;
	} else if (t_2 <= 2e-19) {
		tmp = lambda1 + atan2(t_1, pow(cos(phi1), 2.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = cos(phi1) * (sin(delta) * sin(theta))
    t_2 = atan2(t_1, (cos(delta) - (sin(phi1) * sin(asin((((cos(phi1) * sin(delta)) * cos(theta)) + (cos(delta) * sin(phi1))))))))
    t_3 = lambda1 + atan2(t_1, cos(delta))
    if (t_2 <= (-0.002d0)) then
        tmp = t_3
    else if (t_2 <= 2d-19) then
        tmp = lambda1 + atan2(t_1, (cos(phi1) ** 2.0d0))
    else
        tmp = t_3
    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 t_2 = Math.atan2(t_1, (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin((((Math.cos(phi1) * Math.sin(delta)) * Math.cos(theta)) + (Math.cos(delta) * Math.sin(phi1))))))));
	double t_3 = lambda1 + Math.atan2(t_1, Math.cos(delta));
	double tmp;
	if (t_2 <= -0.002) {
		tmp = t_3;
	} else if (t_2 <= 2e-19) {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(delta) * math.sin(theta))
	t_2 = math.atan2(t_1, (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin((((math.cos(phi1) * math.sin(delta)) * math.cos(theta)) + (math.cos(delta) * math.sin(phi1))))))))
	t_3 = lambda1 + math.atan2(t_1, math.cos(delta))
	tmp = 0
	if t_2 <= -0.002:
		tmp = t_3
	elif t_2 <= 2e-19:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	else:
		tmp = t_3
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(delta) * sin(theta)))
	t_2 = atan(t_1, Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)) + Float64(cos(delta) * sin(phi1))))))))
	t_3 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (t_2 <= -0.002)
		tmp = t_3;
	elseif (t_2 <= 2e-19)
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(delta) * sin(theta));
	t_2 = atan2(t_1, (cos(delta) - (sin(phi1) * sin(asin((((cos(phi1) * sin(delta)) * cos(theta)) + (cos(delta) * sin(phi1))))))));
	t_3 = lambda1 + atan2(t_1, cos(delta));
	tmp = 0.0;
	if (t_2 <= -0.002)
		tmp = t_3;
	elseif (t_2 <= 2e-19)
		tmp = lambda1 + atan2(t_1, (cos(phi1) ^ 2.0));
	else
		tmp = t_3;
	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]}, Block[{t$95$2 = N[ArcTan[t$95$1 / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.002], t$95$3, If[LessEqual[t$95$2, 2e-19], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -2e-3 or 2e-19 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 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. cos-lowering-cos.f64 85.3

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

   5. Simplified 85.3%

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

   if -2e-3 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 2e-19

   1. Initial program 99.8%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. sin-asin N/A

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

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

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

      5. distribute-rgt-in N/A

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

      6. associate-+l+ N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. unpow2 N/A

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

      4. 1-sub-sin N/A

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

      5. unpow2 N/A

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

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

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

      7. cos-lowering-cos.f64 96.5

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

   7. Simplified 96.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -0.002:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\\ t_2 := \tan^{-1}_* \frac{t\_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)}\\ t_3 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;t\_2 \leq -0.002:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot 2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin delta) (sin theta))))
        (t_2
         (atan2
          t_1
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (* (cos phi1) (sin delta)) (cos theta))
               (* (cos delta) (sin phi1)))))))))
        (t_3 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= t_2 -0.002)
     t_3
     (if (<= t_2 2e-19)
       (+ lambda1 (atan2 t_1 (fma 0.5 (cos (* phi1 2.0)) 0.5)))
       t_3))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(delta) * sin(theta));
	double t_2 = atan2(t_1, (cos(delta) - (sin(phi1) * sin(asin((((cos(phi1) * sin(delta)) * cos(theta)) + (cos(delta) * sin(phi1))))))));
	double t_3 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (t_2 <= -0.002) {
		tmp = t_3;
	} else if (t_2 <= 2e-19) {
		tmp = lambda1 + atan2(t_1, fma(0.5, cos((phi1 * 2.0)), 0.5));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(delta) * sin(theta)))
	t_2 = atan(t_1, Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(Float64(cos(phi1) * sin(delta)) * cos(theta)) + Float64(cos(delta) * sin(phi1))))))))
	t_3 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (t_2 <= -0.002)
		tmp = t_3;
	elseif (t_2 <= 2e-19)
		tmp = Float64(lambda1 + atan(t_1, fma(0.5, cos(Float64(phi1 * 2.0)), 0.5)));
	else
		tmp = t_3;
	end
	return 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]}, Block[{t$95$2 = N[ArcTan[t$95$1 / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Cos[theta], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.002], t$95$3, If[LessEqual[t$95$2, 2e-19], N[(lambda1 + N[ArcTan[t$95$1 / N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < -2e-3 or 2e-19 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta))))))))

   1. Initial program 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. cos-lowering-cos.f64 85.3

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

   5. Simplified 85.3%

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

   if -2e-3 < (atan2.f64 (*.f64 (*.f64 (sin.f64 theta) (sin.f64 delta)) (cos.f64 phi1)) (-.f64 (cos.f64 delta) (*.f64 (sin.f64 phi1) (sin.f64 (asin.f64 (+.f64 (*.f64 (sin.f64 phi1) (cos.f64 delta)) (*.f64 (*.f64 (cos.f64 phi1) (sin.f64 delta)) (cos.f64 theta)))))))) < 2e-19

   1. Initial program 99.8%

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

      1. sin-asin N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sqr-sin-a N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

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

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

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

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

      4. *-lowering-*.f64 96.4

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

   7. Simplified 96.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq -0.002:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta + \cos delta \cdot \sin \phi_1\right)} \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot 2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sin-asin N/A

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

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

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

   5. distribute-rgt-in N/A

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

   6. associate-+l+ N/A

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in theta around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

7. Simplified 94.1%

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

8. Final simplification 94.1%

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

Alternative 10: 94.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. sin-asin N/A

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

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

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

   5. distribute-rgt-in N/A

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

   6. associate-+l+ N/A

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

4. Applied egg-rr 99.8%

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

   1. distribute-lft-neg-out N/A

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

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

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

   3. sqr-sin-a N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

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

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

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

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

   9. cos-2 N/A

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

   10. cos-sum N/A

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

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

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

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

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

   13. metadata-eval 99.8

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in theta around 0

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

   1. associate-+r+ N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. *-commutative N/A

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

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

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

   6. mul-1-neg N/A

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

   7. distribute-rgt1-in N/A

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

   8. +-commutative N/A

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

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

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

9. Simplified 94.0%

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

10. Final simplification 94.0%

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

Alternative 11: 92.2% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. 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. pow-lowering-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. sin-lowering-sin.f64 90.0

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

5. Simplified 90.0%

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

   1. +-commutative N/A

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

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

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

7. Applied egg-rr 90.0%

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

8. Final simplification 90.0%

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

Alternative 12: 91.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (cos phi1) (* (sin delta) (sin theta))) (cos delta)))))
   (if (<= delta -4.5e-6)
     t_1
     (if (<= delta 8.6e-18)
       (+
        lambda1
        (atan2 (* (cos phi1) (* delta (sin theta))) (pow (cos phi1) 2.0)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), cos(delta));
	double tmp;
	if (delta <= -4.5e-6) {
		tmp = t_1;
	} else if (delta <= 8.6e-18) {
		tmp = lambda1 + atan2((cos(phi1) * (delta * sin(theta))), pow(cos(phi1), 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), cos(delta))
    if (delta <= (-4.5d-6)) then
        tmp = t_1
    else if (delta <= 8.6d-18) then
        tmp = lambda1 + atan2((cos(phi1) * (delta * sin(theta))), (cos(phi1) ** 2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), Math.cos(delta));
	double tmp;
	if (delta <= -4.5e-6) {
		tmp = t_1;
	} else if (delta <= 8.6e-18) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (delta * Math.sin(theta))), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), math.cos(delta))
	tmp = 0
	if delta <= -4.5e-6:
		tmp = t_1
	elif delta <= 8.6e-18:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (delta * math.sin(theta))), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), cos(delta)))
	tmp = 0.0
	if (delta <= -4.5e-6)
		tmp = t_1;
	elseif (delta <= 8.6e-18)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(delta * sin(theta))), (cos(phi1) ^ 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), cos(delta));
	tmp = 0.0;
	if (delta <= -4.5e-6)
		tmp = t_1;
	elseif (delta <= 8.6e-18)
		tmp = lambda1 + atan2((cos(phi1) * (delta * sin(theta))), (cos(phi1) ^ 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -4.5e-6], t$95$1, If[LessEqual[delta, 8.6e-18], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.50000000000000011e-6 or 8.6000000000000005e-18 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. cos-lowering-cos.f64 82.9

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

   5. Simplified 82.9%

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

   if -4.50000000000000011e-6 < delta < 8.6000000000000005e-18

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. 1-sub-sin N/A

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

      8. distribute-rgt-out N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. distribute-lft-neg-out N/A

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

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

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

      14. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.8%

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

   6. Taylor expanded in delta around 0

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

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

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

      2. sin-lowering-sin.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in delta around 0

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

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

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

       2. cos-lowering-cos.f64 99.5

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

   11. Simplified 99.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -4.5 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 8.6 \cdot 10^{-18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 88.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. 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. cos-lowering-cos.f64 87.3

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

5. Simplified 87.3%

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

6. Final simplification 87.3%

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

Alternative 14: 76.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in delta around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. 1-sub-sin N/A

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

   8. distribute-rgt-out N/A

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

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

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

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

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

   11. mul-1-neg N/A

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

   12. distribute-lft-neg-out N/A

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

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

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

   14. accelerator-lowering-fma.f64 N/A

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

5. Simplified 79.7%

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

6. Taylor expanded in phi1 around 0

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

8. Simplified 76.1%

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

9. Taylor expanded in phi1 around 0

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

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

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

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

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

    3. sin-lowering-sin.f64 75.1

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

11. Simplified 75.1%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1} \]

12. Taylor expanded in lambda1 around -inf

    \[\leadsto \color{blue}{-1 \cdot \left(\lambda_1 \cdot \left(-1 \cdot \frac{\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}}{\lambda_1} - 1\right)\right)} \]
13. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\lambda_1 \cdot \left(-1 \cdot \frac{\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}}{\lambda_1} - 1\right)\right)} \]

    2. *-commutative N/A

       \[\leadsto \mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}}{\lambda_1} - 1\right) \cdot \lambda_1}\right) \]

    3. distribute-rgt-neg-in N/A

       \[\leadsto \color{blue}{\left(-1 \cdot \frac{\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}}{\lambda_1} - 1\right) \cdot \left(\mathsf{neg}\left(\lambda_1\right)\right)} \]

    4. mul-1-neg N/A

       \[\leadsto \left(-1 \cdot \frac{\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}}{\lambda_1} - 1\right) \cdot \color{blue}{\left(-1 \cdot \lambda_1\right)} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \color{blue}{\left(-1 \cdot \frac{\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}}{\lambda_1} - 1\right) \cdot \left(-1 \cdot \lambda_1\right)} \]

14. Simplified 75.1%

    \[\leadsto \color{blue}{\left(-1 - \frac{\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}}{\lambda_1}\right) \cdot \left(-\lambda_1\right)} \]

15. Final simplification 75.1%

    \[\leadsto \lambda_1 \cdot \left(\frac{\tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1}}{\lambda_1} - -1\right) \]
16. Add Preprocessing

Alternative 15: 76.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \sin theta}{1} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+ lambda1 (atan2 (* (sin delta) (sin theta)) 1.0)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((sin(delta) * sin(theta)), 1.0);
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2((sin(delta) * sin(theta)), 1.0d0)
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2((Math.sin(delta) * Math.sin(theta)), 1.0);
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((math.sin(delta) * math.sin(theta)), 1.0)

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(sin(delta) * sin(theta)), 1.0))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((sin(delta) * sin(theta)), 1.0);
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. Add Preprocessing

3. Taylor expanded in delta around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(1 + -1 \cdot \left(delta \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)\right) - {\sin \phi_1}^{2}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(-1 \cdot \left(delta \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right) + 1\right)} - {\sin \phi_1}^{2}} \]

   2. associate--l+ N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{-1 \cdot \left(delta \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right) + \left(1 - {\sin \phi_1}^{2}\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(-1 \cdot delta\right) \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)} + \left(1 - {\sin \phi_1}^{2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(-1 \cdot delta\right) \cdot \color{blue}{\left(\left(\cos theta \cdot \sin \phi_1\right) \cdot \cos \phi_1\right)} + \left(1 - {\sin \phi_1}^{2}\right)} \]

   5. associate-*r* N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) \cdot \cos \phi_1} + \left(1 - {\sin \phi_1}^{2}\right)} \]

   6. unpow2 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) \cdot \cos \phi_1 + \left(1 - \color{blue}{\sin \phi_1 \cdot \sin \phi_1}\right)} \]

   7. 1-sub-sin N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) \cdot \cos \phi_1 + \color{blue}{\cos \phi_1 \cdot \cos \phi_1}} \]

   8. distribute-rgt-out N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos \phi_1 \cdot \left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right) + \cos \phi_1\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos \phi_1 \cdot \left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right) + \cos \phi_1\right)}} \]

   10. cos-lowering-cos.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos \phi_1} \cdot \left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right) + \cos \phi_1\right)} \]

   11. mul-1-neg N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos \phi_1 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(delta\right)\right)} \cdot \left(\cos theta \cdot \sin \phi_1\right) + \cos \phi_1\right)} \]

   12. distribute-lft-neg-out N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos \phi_1 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)} + \cos \phi_1\right)} \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos \phi_1 \cdot \left(\color{blue}{delta \cdot \left(\mathsf{neg}\left(\cos theta \cdot \sin \phi_1\right)\right)} + \cos \phi_1\right)} \]

   14. accelerator-lowering-fma.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(delta, \mathsf{neg}\left(\cos theta \cdot \sin \phi_1\right), \cos \phi_1\right)}} \]

5. Simplified 79.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos \phi_1 \cdot \mathsf{fma}\left(delta, \cos theta \cdot \left(-\sin \phi_1\right), \cos \phi_1\right)}} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{1}} \]
7. Step-by-step derivation

8. Simplified 76.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{1}} \]

9. Taylor expanded in phi1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1} \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1} \]

    2. sin-lowering-sin.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta} \cdot \sin theta}{1} \]

    3. sin-lowering-sin.f64 75.1

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\sin theta}}{1} \]

11. Simplified 75.1%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1} \]
12. Add Preprocessing

Alternative 16: 73.6% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{1} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+ lambda1 (atan2 (* delta (sin theta)) 1.0)))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((delta * sin(theta)), 1.0);
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2((delta * sin(theta)), 1.0d0)
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + Math.atan2((delta * Math.sin(theta)), 1.0);
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1 + math.atan2((delta * math.sin(theta)), 1.0)

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(delta * sin(theta)), 1.0))
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((delta * sin(theta)), 1.0);
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. Add Preprocessing

3. Taylor expanded in delta around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(1 + -1 \cdot \left(delta \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)\right) - {\sin \phi_1}^{2}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(-1 \cdot \left(delta \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right) + 1\right)} - {\sin \phi_1}^{2}} \]

   2. associate--l+ N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{-1 \cdot \left(delta \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right) + \left(1 - {\sin \phi_1}^{2}\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(-1 \cdot delta\right) \cdot \left(\cos \phi_1 \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)} + \left(1 - {\sin \phi_1}^{2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(-1 \cdot delta\right) \cdot \color{blue}{\left(\left(\cos theta \cdot \sin \phi_1\right) \cdot \cos \phi_1\right)} + \left(1 - {\sin \phi_1}^{2}\right)} \]

   5. associate-*r* N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) \cdot \cos \phi_1} + \left(1 - {\sin \phi_1}^{2}\right)} \]

   6. unpow2 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) \cdot \cos \phi_1 + \left(1 - \color{blue}{\sin \phi_1 \cdot \sin \phi_1}\right)} \]

   7. 1-sub-sin N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right)\right) \cdot \cos \phi_1 + \color{blue}{\cos \phi_1 \cdot \cos \phi_1}} \]

   8. distribute-rgt-out N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos \phi_1 \cdot \left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right) + \cos \phi_1\right)}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos \phi_1 \cdot \left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right) + \cos \phi_1\right)}} \]

   10. cos-lowering-cos.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos \phi_1} \cdot \left(\left(-1 \cdot delta\right) \cdot \left(\cos theta \cdot \sin \phi_1\right) + \cos \phi_1\right)} \]

   11. mul-1-neg N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos \phi_1 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(delta\right)\right)} \cdot \left(\cos theta \cdot \sin \phi_1\right) + \cos \phi_1\right)} \]

   12. distribute-lft-neg-out N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos \phi_1 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(delta \cdot \left(\cos theta \cdot \sin \phi_1\right)\right)\right)} + \cos \phi_1\right)} \]

   13. distribute-rgt-neg-in N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos \phi_1 \cdot \left(\color{blue}{delta \cdot \left(\mathsf{neg}\left(\cos theta \cdot \sin \phi_1\right)\right)} + \cos \phi_1\right)} \]

   14. accelerator-lowering-fma.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos \phi_1 \cdot \color{blue}{\mathsf{fma}\left(delta, \mathsf{neg}\left(\cos theta \cdot \sin \phi_1\right), \cos \phi_1\right)}} \]

5. Simplified 79.7%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{\cos \phi_1 \cdot \mathsf{fma}\left(delta, \cos theta \cdot \left(-\sin \phi_1\right), \cos \phi_1\right)}} \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{1}} \]
7. Step-by-step derivation

8. Simplified 76.1%

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\color{blue}{1}} \]

9. Taylor expanded in phi1 around 0

   \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1} \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1} \]

    2. sin-lowering-sin.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta} \cdot \sin theta}{1} \]

    3. sin-lowering-sin.f64 75.1

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \color{blue}{\sin theta}}{1} \]

11. Simplified 75.1%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin delta \cdot \sin theta}}{1} \]

12. Taylor expanded in delta around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{delta \cdot \sin theta}}{1} \]
13. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{delta \cdot \sin theta}}{1} \]

    2. sin-lowering-sin.f64 72.9

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \color{blue}{\sin theta}}{1} \]

14. Simplified 72.9%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{delta \cdot \sin theta}}{1} \]
15. Add Preprocessing

Alternative 17: 70.3% accurate, 1341.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta) :precision binary64 lambda1)

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return lambda1
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := lambda1

Derivation

1. Initial program 99.8%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
2. Add Preprocessing

3. Taylor expanded in lambda1 around inf

   \[\leadsto \color{blue}{\lambda_1} \]
4. Step-by-step derivation

5. Simplified 69.3%

   \[\leadsto \color{blue}{\lambda_1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(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))))))))))