Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.8%

Time: 20.2s

Alternatives: 20

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied egg-rr 99.8%

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

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

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

   2. distribute-lft1-in N/A

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

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

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

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

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

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

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

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

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

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

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

   11. cos-2 N/A

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

   12. cos-sum N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. cos-lowering-cos.f64 99.8

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

6. Applied egg-rr 99.8%

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

   1. +-commutative N/A

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

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

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \phi_1 \cdot \phi_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{0.5 \cdot \left(\cos \left(theta - delta\right) - \cos \left(theta + delta\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (- 1.0 (* phi1 phi1)))
        (t_2
         (atan2
          (* (cos phi1) (* (sin theta) (sin delta)))
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (cos delta) (sin phi1))
               (* (cos theta) (* (sin delta) (cos phi1)))))))))))
   (if (<= t_2 -5e-63)
     (+ lambda1 (atan2 (* (sin theta) delta) t_1))
     (if (<= t_2 4e-79)
       lambda1
       (+
        lambda1
        (atan2
         (* 0.5 (- (cos (- theta delta)) (cos (+ theta delta))))
         t_1))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = 1.0 - (phi1 * phi1);
	double t_2 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	double tmp;
	if (t_2 <= -5e-63) {
		tmp = lambda1 + atan2((sin(theta) * delta), t_1);
	} else if (t_2 <= 4e-79) {
		tmp = lambda1;
	} else {
		tmp = lambda1 + atan2((0.5 * (cos((theta - delta)) - cos((theta + delta)))), 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) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 - (phi1 * phi1)
    t_2 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))))
    if (t_2 <= (-5d-63)) then
        tmp = lambda1 + atan2((sin(theta) * delta), t_1)
    else if (t_2 <= 4d-79) then
        tmp = lambda1
    else
        tmp = lambda1 + atan2((0.5d0 * (cos((theta - delta)) - cos((theta + delta)))), 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 = 1.0 - (phi1 * phi1);
	double t_2 = Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.cos(delta) * Math.sin(phi1)) + (Math.cos(theta) * (Math.sin(delta) * Math.cos(phi1)))))))));
	double tmp;
	if (t_2 <= -5e-63) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), t_1);
	} else if (t_2 <= 4e-79) {
		tmp = lambda1;
	} else {
		tmp = lambda1 + Math.atan2((0.5 * (Math.cos((theta - delta)) - Math.cos((theta + delta)))), t_1);
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = 1.0 - (phi1 * phi1)
	t_2 = math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.cos(delta) * math.sin(phi1)) + (math.cos(theta) * (math.sin(delta) * math.cos(phi1)))))))))
	tmp = 0
	if t_2 <= -5e-63:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), t_1)
	elif t_2 <= 4e-79:
		tmp = lambda1
	else:
		tmp = lambda1 + math.atan2((0.5 * (math.cos((theta - delta)) - math.cos((theta + delta)))), t_1)
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(1.0 - Float64(phi1 * phi1))
	t_2 = atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(sin(delta) * cos(phi1)))))))))
	tmp = 0.0
	if (t_2 <= -5e-63)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), t_1));
	elseif (t_2 <= 4e-79)
		tmp = lambda1;
	else
		tmp = Float64(lambda1 + atan(Float64(0.5 * Float64(cos(Float64(theta - delta)) - cos(Float64(theta + delta)))), t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = 1.0 - (phi1 * phi1);
	t_2 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	tmp = 0.0;
	if (t_2 <= -5e-63)
		tmp = lambda1 + atan2((sin(theta) * delta), t_1);
	elseif (t_2 <= 4e-79)
		tmp = lambda1;
	else
		tmp = lambda1 + atan2((0.5 * (cos((theta - delta)) - cos((theta + delta)))), t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e-63], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-79], lambda1, N[(lambda1 + N[ArcTan[N[(0.5 * N[(N[Cos[N[(theta - delta), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(theta + delta), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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)))))))) < -5.0000000000000002e-63

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

   5. Simplified 75.2%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 64.8

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

   8. Simplified 64.8%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 58.0

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

   11. Simplified 58.0%

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

   12. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 59.6

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

   14. Simplified 59.6%

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

   if -5.0000000000000002e-63 < (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)))))))) < 4e-79

   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 87.7%

      \[\leadsto \color{blue}{\lambda_1} \]

   if 4e-79 < (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 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

   5. Simplified 81.4%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 70.0

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

   8. Simplified 70.0%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 62.3

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

   11. Simplified 62.3%

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

       1. *-commutative N/A

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

       2. sin-mult N/A

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

       3. div-inv N/A

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

       4. metadata-eval N/A

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

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

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

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

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

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

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

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

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

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

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

       10. +-lowering-+.f64 65.4

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

   13. Applied egg-rr 65.4%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{0.5 \cdot \left(\cos \left(theta - delta\right) - \cos \left(theta + delta\right)\right)}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \phi_1 \cdot \phi_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 0.45:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (- 1.0 (* phi1 phi1)))
        (t_2
         (atan2
          (* (cos phi1) (* (sin theta) (sin delta)))
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (cos delta) (sin phi1))
               (* (cos theta) (* (sin delta) (cos phi1)))))))))))
   (if (<= t_2 -5e-63)
     (+ lambda1 (atan2 (* (sin theta) delta) t_1))
     (if (<= t_2 0.45)
       lambda1
       (+ lambda1 (atan2 (* (cos phi1) (* theta (sin delta))) t_1))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = 1.0 - (phi1 * phi1);
	double t_2 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	double tmp;
	if (t_2 <= -5e-63) {
		tmp = lambda1 + atan2((sin(theta) * delta), t_1);
	} else if (t_2 <= 0.45) {
		tmp = lambda1;
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (theta * sin(delta))), 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) :: t_2
    real(8) :: tmp
    t_1 = 1.0d0 - (phi1 * phi1)
    t_2 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))))
    if (t_2 <= (-5d-63)) then
        tmp = lambda1 + atan2((sin(theta) * delta), t_1)
    else if (t_2 <= 0.45d0) then
        tmp = lambda1
    else
        tmp = lambda1 + atan2((cos(phi1) * (theta * sin(delta))), 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 = 1.0 - (phi1 * phi1);
	double t_2 = Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.cos(delta) * Math.sin(phi1)) + (Math.cos(theta) * (Math.sin(delta) * Math.cos(phi1)))))))));
	double tmp;
	if (t_2 <= -5e-63) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), t_1);
	} else if (t_2 <= 0.45) {
		tmp = lambda1;
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (theta * Math.sin(delta))), t_1);
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = 1.0 - (phi1 * phi1)
	t_2 = math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.cos(delta) * math.sin(phi1)) + (math.cos(theta) * (math.sin(delta) * math.cos(phi1)))))))))
	tmp = 0
	if t_2 <= -5e-63:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), t_1)
	elif t_2 <= 0.45:
		tmp = lambda1
	else:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (theta * math.sin(delta))), t_1)
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(1.0 - Float64(phi1 * phi1))
	t_2 = atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(sin(delta) * cos(phi1)))))))))
	tmp = 0.0
	if (t_2 <= -5e-63)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), t_1));
	elseif (t_2 <= 0.45)
		tmp = lambda1;
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(theta * sin(delta))), t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = 1.0 - (phi1 * phi1);
	t_2 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	tmp = 0.0;
	if (t_2 <= -5e-63)
		tmp = lambda1 + atan2((sin(theta) * delta), t_1);
	elseif (t_2 <= 0.45)
		tmp = lambda1;
	else
		tmp = lambda1 + atan2((cos(phi1) * (theta * sin(delta))), t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e-63], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.45], lambda1, N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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)))))))) < -5.0000000000000002e-63

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

   5. Simplified 75.2%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 64.8

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

   8. Simplified 64.8%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 58.0

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

   11. Simplified 58.0%

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

   12. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 59.6

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

   14. Simplified 59.6%

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

   if -5.0000000000000002e-63 < (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)))))))) < 0.450000000000000011

   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 84.4%

      \[\leadsto \color{blue}{\lambda_1} \]

   if 0.450000000000000011 < (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 99.9%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

   5. Simplified 83.0%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 68.9

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

   8. Simplified 68.9%

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

   9. Taylor expanded in theta around 0

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

       1. *-commutative N/A

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

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

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

       3. sin-lowering-sin.f64 66.6

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

   11. Simplified 66.6%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq 0.45:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(theta \cdot \sin delta\right)}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \phi_1 \cdot \phi_1\\ t_2 := \sin theta \cdot \sin delta\\ t_3 := \tan^{-1}_* \frac{\cos \phi_1 \cdot t\_2}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_2}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (- 1.0 (* phi1 phi1)))
        (t_2 (* (sin theta) (sin delta)))
        (t_3
         (atan2
          (* (cos phi1) t_2)
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (cos delta) (sin phi1))
               (* (cos theta) (* (sin delta) (cos phi1)))))))))))
   (if (<= t_3 -5e-63)
     (+ lambda1 (atan2 (* (sin theta) delta) t_1))
     (if (<= t_3 4e-79) lambda1 (+ lambda1 (atan2 t_2 t_1))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = 1.0 - (phi1 * phi1);
	double t_2 = sin(theta) * sin(delta);
	double t_3 = atan2((cos(phi1) * t_2), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	double tmp;
	if (t_3 <= -5e-63) {
		tmp = lambda1 + atan2((sin(theta) * delta), t_1);
	} else if (t_3 <= 4e-79) {
		tmp = lambda1;
	} else {
		tmp = lambda1 + atan2(t_2, 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = 1.0d0 - (phi1 * phi1)
    t_2 = sin(theta) * sin(delta)
    t_3 = atan2((cos(phi1) * t_2), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))))
    if (t_3 <= (-5d-63)) then
        tmp = lambda1 + atan2((sin(theta) * delta), t_1)
    else if (t_3 <= 4d-79) then
        tmp = lambda1
    else
        tmp = lambda1 + atan2(t_2, 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 = 1.0 - (phi1 * phi1);
	double t_2 = Math.sin(theta) * Math.sin(delta);
	double t_3 = Math.atan2((Math.cos(phi1) * t_2), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.cos(delta) * Math.sin(phi1)) + (Math.cos(theta) * (Math.sin(delta) * Math.cos(phi1)))))))));
	double tmp;
	if (t_3 <= -5e-63) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), t_1);
	} else if (t_3 <= 4e-79) {
		tmp = lambda1;
	} else {
		tmp = lambda1 + Math.atan2(t_2, t_1);
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = 1.0 - (phi1 * phi1)
	t_2 = math.sin(theta) * math.sin(delta)
	t_3 = math.atan2((math.cos(phi1) * t_2), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.cos(delta) * math.sin(phi1)) + (math.cos(theta) * (math.sin(delta) * math.cos(phi1)))))))))
	tmp = 0
	if t_3 <= -5e-63:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), t_1)
	elif t_3 <= 4e-79:
		tmp = lambda1
	else:
		tmp = lambda1 + math.atan2(t_2, t_1)
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(1.0 - Float64(phi1 * phi1))
	t_2 = Float64(sin(theta) * sin(delta))
	t_3 = atan(Float64(cos(phi1) * t_2), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(sin(delta) * cos(phi1)))))))))
	tmp = 0.0
	if (t_3 <= -5e-63)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), t_1));
	elseif (t_3 <= 4e-79)
		tmp = lambda1;
	else
		tmp = Float64(lambda1 + atan(t_2, t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = 1.0 - (phi1 * phi1);
	t_2 = sin(theta) * sin(delta);
	t_3 = atan2((cos(phi1) * t_2), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	tmp = 0.0;
	if (t_3 <= -5e-63)
		tmp = lambda1 + atan2((sin(theta) * delta), t_1);
	elseif (t_3 <= 4e-79)
		tmp = lambda1;
	else
		tmp = lambda1 + atan2(t_2, t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * t$95$2), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -5e-63], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e-79], lambda1, N[(lambda1 + N[ArcTan[t$95$2 / t$95$1], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 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)))))))) < -5.0000000000000002e-63

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

   5. Simplified 75.2%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 64.8

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

   8. Simplified 64.8%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 58.0

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

   11. Simplified 58.0%

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

   12. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 59.6

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

   14. Simplified 59.6%

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

   if -5.0000000000000002e-63 < (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)))))))) < 4e-79

   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 87.7%

      \[\leadsto \color{blue}{\lambda_1} \]

   if 4e-79 < (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 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

   5. Simplified 81.4%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 70.0

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

   8. Simplified 70.0%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 62.3

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

   11. Simplified 62.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 - \phi_1 \cdot \phi_1\\ t_2 := \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (- 1.0 (* phi1 phi1)))
        (t_2
         (atan2
          (* (cos phi1) (* (sin theta) (sin delta)))
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (cos delta) (sin phi1))
               (* (cos theta) (* (sin delta) (cos phi1)))))))))))
   (if (<= t_2 -5e-63)
     (+ lambda1 (atan2 (* (sin theta) delta) t_1))
     (if (<= t_2 4e-79)
       lambda1
       (+
        lambda1
        (atan2
         (*
          (sin theta)
          (fma delta (* -0.16666666666666666 (* delta delta)) delta))
         t_1))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = 1.0 - (phi1 * phi1);
	double t_2 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	double tmp;
	if (t_2 <= -5e-63) {
		tmp = lambda1 + atan2((sin(theta) * delta), t_1);
	} else if (t_2 <= 4e-79) {
		tmp = lambda1;
	} else {
		tmp = lambda1 + atan2((sin(theta) * fma(delta, (-0.16666666666666666 * (delta * delta)), delta)), t_1);
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(1.0 - Float64(phi1 * phi1))
	t_2 = atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(sin(delta) * cos(phi1)))))))))
	tmp = 0.0
	if (t_2 <= -5e-63)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), t_1));
	elseif (t_2 <= 4e-79)
		tmp = lambda1;
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * fma(delta, Float64(-0.16666666666666666 * Float64(delta * delta)), delta)), t_1));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, -5e-63], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-79], lambda1, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision]), $MachinePrecision] + delta), $MachinePrecision]), $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 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)))))))) < -5.0000000000000002e-63

   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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

   5. Simplified 75.2%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 64.8

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

   8. Simplified 64.8%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 58.0

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

   11. Simplified 58.0%

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

   12. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 59.6

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

   14. Simplified 59.6%

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

   if -5.0000000000000002e-63 < (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)))))))) < 4e-79

   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 87.7%

      \[\leadsto \color{blue}{\lambda_1} \]

   if 4e-79 < (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 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

   5. Simplified 81.4%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 70.0

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

   8. Simplified 70.0%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 62.3

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

   11. Simplified 62.3%

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

   12. Taylor expanded in delta around 0

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. *-rgt-identity N/A

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

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

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

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

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

       6. unpow2 N/A

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

       7. *-lowering-*.f64 61.0

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

   14. Simplified 61.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)}\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-63}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (atan2
          (* (cos phi1) (* (sin theta) (sin delta)))
          (-
           (cos delta)
           (*
            (sin phi1)
            (sin
             (asin
              (+
               (* (cos delta) (sin phi1))
               (* (cos theta) (* (sin delta) (cos phi1))))))))))
        (t_2 (+ lambda1 (atan2 (* (sin theta) delta) (- 1.0 (* phi1 phi1))))))
   (if (<= t_1 -5e-63) t_2 (if (<= t_1 4e-79) lambda1 t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	double t_2 = lambda1 + atan2((sin(theta) * delta), (1.0 - (phi1 * phi1)));
	double tmp;
	if (t_1 <= -5e-63) {
		tmp = t_2;
	} else if (t_1 <= 4e-79) {
		tmp = lambda1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))))
    t_2 = lambda1 + atan2((sin(theta) * delta), (1.0d0 - (phi1 * phi1)))
    if (t_1 <= (-5d-63)) then
        tmp = t_2
    else if (t_1 <= 4d-79) then
        tmp = lambda1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), (Math.cos(delta) - (Math.sin(phi1) * Math.sin(Math.asin(((Math.cos(delta) * Math.sin(phi1)) + (Math.cos(theta) * (Math.sin(delta) * Math.cos(phi1)))))))));
	double t_2 = lambda1 + Math.atan2((Math.sin(theta) * delta), (1.0 - (phi1 * phi1)));
	double tmp;
	if (t_1 <= -5e-63) {
		tmp = t_2;
	} else if (t_1 <= 4e-79) {
		tmp = lambda1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), (math.cos(delta) - (math.sin(phi1) * math.sin(math.asin(((math.cos(delta) * math.sin(phi1)) + (math.cos(theta) * (math.sin(delta) * math.cos(phi1)))))))))
	t_2 = lambda1 + math.atan2((math.sin(theta) * delta), (1.0 - (phi1 * phi1)))
	tmp = 0
	if t_1 <= -5e-63:
		tmp = t_2
	elif t_1 <= 4e-79:
		tmp = lambda1
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), Float64(cos(delta) - Float64(sin(phi1) * sin(asin(Float64(Float64(cos(delta) * sin(phi1)) + Float64(cos(theta) * Float64(sin(delta) * cos(phi1)))))))))
	t_2 = Float64(lambda1 + atan(Float64(sin(theta) * delta), Float64(1.0 - Float64(phi1 * phi1))))
	tmp = 0.0
	if (t_1 <= -5e-63)
		tmp = t_2;
	elseif (t_1 <= 4e-79)
		tmp = lambda1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = atan2((cos(phi1) * (sin(theta) * sin(delta))), (cos(delta) - (sin(phi1) * sin(asin(((cos(delta) * sin(phi1)) + (cos(theta) * (sin(delta) * cos(phi1)))))))));
	t_2 = lambda1 + atan2((sin(theta) * delta), (1.0 - (phi1 * phi1)));
	tmp = 0.0;
	if (t_1 <= -5e-63)
		tmp = t_2;
	elseif (t_1 <= 4e-79)
		tmp = lambda1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[Sin[N[ArcSin[N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[(1.0 - N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-63], t$95$2, If[LessEqual[t$95$1, 4e-79], lambda1, t$95$2]]]]

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)))))))) < -5.0000000000000002e-63 or 4e-79 < (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

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

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

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

   5. Simplified 78.2%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 67.3

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

   8. Simplified 67.3%

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

   9. Taylor expanded in phi1 around 0

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

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

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

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

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

       3. sin-lowering-sin.f64 60.1

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

   11. Simplified 60.1%

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

   12. Taylor expanded in delta around 0

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

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

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

       2. sin-lowering-sin.f64 60.0

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

   14. Simplified 60.0%

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

   if -5.0000000000000002e-63 < (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)))))))) < 4e-79

   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 87.7%

      \[\leadsto \color{blue}{\lambda_1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq -5 \cdot 10^{-63}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1 - \phi_1 \cdot \phi_1}\\ \mathbf{elif}\;\tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\cos delta \cdot \sin \phi_1 + \cos theta \cdot \left(\sin delta \cdot \cos \phi_1\right)\right)} \leq 4 \cdot 10^{-79}:\\ \;\;\;\;\lambda_1\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{1 - \phi_1 \cdot \phi_1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied egg-rr 99.8%

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

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

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

   2. distribute-lft1-in N/A

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

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

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

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

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

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

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. *-commutative N/A

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

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

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

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

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

   11. cos-2 N/A

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

   12. cos-sum N/A

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

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

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

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

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

   15. metadata-eval N/A

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

   16. cos-lowering-cos.f64 99.8

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

6. Applied egg-rr 99.8%

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

   1. distribute-neg-in N/A

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

   2. metadata-eval N/A

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

   3. associate-+l+ N/A

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

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

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

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

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

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

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

   9. +-lowering-+.f64 99.8

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 8: 94.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin delta \cdot \cos \phi_1\\ \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot t\_1}{\mathsf{fma}\left(-\sin \phi_1, \mathsf{fma}\left(\cos delta, \sin \phi_1, t\_1\right), \cos delta\right)} \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (sin delta) (cos phi1))))
   (+
    lambda1
    (atan2
     (* (sin theta) t_1)
     (fma (- (sin phi1)) (fma (cos delta) (sin phi1) t_1) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = sin(delta) * cos(phi1);
	return lambda1 + atan2((sin(theta) * t_1), fma(-sin(phi1), fma(cos(delta), sin(phi1), t_1), cos(delta)));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(sin(delta) * cos(phi1))
	return Float64(lambda1 + atan(Float64(sin(theta) * t_1), fma(Float64(-sin(phi1)), fma(cos(delta), sin(phi1), t_1), cos(delta))))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$1), $MachinePrecision] / N[((-N[Sin[phi1], $MachinePrecision]) * N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + t$95$1), $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. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.8%

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

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. sin-lowering-sin.f64 99.8

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in theta around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. cos-lowering-cos.f64 94.3

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

10. Simplified 94.3%

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

11. Final simplification 94.3%

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

Alternative 9: 94.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-N[Sin[phi1], $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. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.8%

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

6. Taylor expanded in theta around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

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

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

   5. mul-1-neg N/A

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

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

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

8. Simplified 94.2%

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

9. Final simplification 94.2%

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

Alternative 10: 94.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $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. Taylor expanded in theta around 0

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

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

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

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

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

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

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

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

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   12. sin-lowering-sin.f64 94.2

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

5. Simplified 94.2%

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

6. Final simplification 94.2%

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

Alternative 11: 91.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision] + N[Sin[phi1], $MachinePrecision]), $MachinePrecision] * (-N[Sin[phi1], $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. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.8%

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

6. Taylor expanded in delta around 0

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

   1. sin-lowering-sin.f64 93.3

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

8. Simplified 93.3%

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

9. Taylor expanded in phi1 around 0

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

11. Simplified 93.3%

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

12. Final simplification 93.3%

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

Alternative 12: 92.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-N[Sin[phi1], $MachinePrecision]) * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision] + N[Sin[phi1], $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. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.8%

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

6. Taylor expanded in delta around 0

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

   1. sin-lowering-sin.f64 93.3

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

8. Simplified 93.3%

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

9. Taylor expanded in theta around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

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

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

    4. mul-1-neg N/A

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

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

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

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

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

    7. +-commutative N/A

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

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

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

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

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

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

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

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

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

    12. cos-lowering-cos.f64 93.1

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

11. Simplified 93.1%

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

12. Final simplification 93.1%

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

Alternative 13: 91.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[N[(delta + phi1), $MachinePrecision]], $MachinePrecision] * (-N[Sin[phi1], $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. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.8%

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

   1. associate-*r* N/A

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

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

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

   3. *-commutative N/A

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

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

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

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

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

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

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

   7. sin-lowering-sin.f64 99.8

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

7. Applied egg-rr 99.8%

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

8. Taylor expanded in theta around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

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

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

   4. mul-1-neg N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   13. cos-lowering-cos.f64 94.3

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

10. Simplified 94.3%

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

    1. *-commutative N/A

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

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

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

    3. *-commutative N/A

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

    4. sin-sum N/A

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

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

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

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

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

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

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

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

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

    9. cos-lowering-cos.f64 92.1

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

12. Applied egg-rr 92.1%

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

13. Final simplification 92.1%

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

Alternative 14: 91.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \sin delta\right) \cdot \cos \phi_1}{\cos delta - \sin \phi_1 \cdot \sin \sin^{-1} \left(\sin \phi_1 \cdot \cos delta + \left(\cos \phi_1 \cdot \sin delta\right) \cdot \cos theta\right)} \]
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 92.0

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

5. Simplified 92.0%

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

6. Final simplification 92.0%

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

Alternative 15: 88.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -1.3e+16)
   (+
    lambda1
    (atan2 (* (sin delta) (* (sin theta) (cos phi1))) (pow (cos phi1) 2.0)))
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -1.3e+16) {
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), pow(cos(phi1), 2.0));
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-1.3d+16)) then
        tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), (cos(phi1) ** 2.0d0))
    else
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -1.3e+16) {
		tmp = lambda1 + Math.atan2((Math.sin(delta) * (Math.sin(theta) * Math.cos(phi1))), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -1.3e+16:
		tmp = lambda1 + math.atan2((math.sin(delta) * (math.sin(theta) * math.cos(phi1))), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -1.3e+16)
		tmp = Float64(lambda1 + atan(Float64(sin(delta) * Float64(sin(theta) * cos(phi1))), (cos(phi1) ^ 2.0)));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -1.3e+16)
		tmp = lambda1 + atan2((sin(delta) * (sin(theta) * cos(phi1))), (cos(phi1) ^ 2.0));
	else
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -1.3e+16], N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.3e16

   1. Initial program 99.6%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

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

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

   5. Simplified 99.7%

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

   6. Taylor expanded in delta around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. unpow2 N/A

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

      4. 1-sub-sin N/A

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

      5. unpow2 N/A

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

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

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

      7. cos-lowering-cos.f64 84.5

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

   8. Simplified 84.5%

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

   if -1.3e16 < phi1

   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 92.4

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

   5. Simplified 92.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 88.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{+19}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{0.5 + \cos \left(\phi_1 + \phi_1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 -3.5e+19)
   (+
    lambda1
    (atan2
     (* (sin theta) (* (sin delta) (cos phi1)))
     (+ 0.5 (* (cos (+ phi1 phi1)) 0.5))))
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -3.5e+19) {
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (0.5 + (cos((phi1 + phi1)) * 0.5)));
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= (-3.5d+19)) then
        tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (0.5d0 + (cos((phi1 + phi1)) * 0.5d0)))
    else
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= -3.5e+19) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), (0.5 + (Math.cos((phi1 + phi1)) * 0.5)));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= -3.5e+19:
		tmp = lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), (0.5 + (math.cos((phi1 + phi1)) * 0.5)))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= -3.5e+19)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(0.5 + Float64(cos(Float64(phi1 + phi1)) * 0.5))));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= -3.5e+19)
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (0.5 + (cos((phi1 + phi1)) * 0.5)));
	else
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, -3.5e+19], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(N[Cos[N[(phi1 + phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.5e19

   1. Initial program 99.6%

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

      1. flip-- N/A

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

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

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in delta around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. sin-lowering-sin.f64 84.3

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

   7. Simplified 84.3%

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

      1. +-commutative N/A

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

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

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

   9. Applied egg-rr 84.5%

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

   if -3.5e19 < phi1

   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 92.4

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

   5. Simplified 92.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{+19}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{0.5 + \cos \left(\phi_1 + \phi_1\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 88.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)\\ \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{+18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot 2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin theta) (sin delta)))))
   (if (<= phi1 -1.4e+18)
     (+ lambda1 (atan2 t_1 (fma 0.5 (cos (* phi1 2.0)) 0.5)))
     (+ lambda1 (atan2 t_1 (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(theta) * sin(delta));
	double tmp;
	if (phi1 <= -1.4e+18) {
		tmp = lambda1 + atan2(t_1, fma(0.5, cos((phi1 * 2.0)), 0.5));
	} else {
		tmp = lambda1 + atan2(t_1, cos(delta));
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(theta) * sin(delta)))
	tmp = 0.0
	if (phi1 <= -1.4e+18)
		tmp = Float64(lambda1 + atan(t_1, fma(0.5, cos(Float64(phi1 * 2.0)), 0.5)));
	else
		tmp = Float64(lambda1 + atan(t_1, cos(delta)));
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.4e+18], N[(lambda1 + N[ArcTan[t$95$1 / N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.4e18

   1. Initial program 99.6%

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

      1. 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. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   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 84.5

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

   7. Simplified 84.5%

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

   if -1.4e18 < phi1

   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 92.4

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

   5. Simplified 92.4%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{+18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\mathsf{fma}\left(0.5, \cos \left(\phi_1 \cdot 2\right), 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 87.9% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $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 88.1

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

5. Simplified 88.1%

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

6. Final simplification 88.1%

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

Alternative 19: 85.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in lambda1 around 0

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

   1. +-commutative N/A

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

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

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

5. Simplified 99.8%

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

6. Taylor expanded in phi1 around 0

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

   1. cos-lowering-cos.f64 88.1

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

8. Simplified 88.1%

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

9. Taylor expanded in phi1 around 0

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

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

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

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

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

    3. sin-lowering-sin.f64 84.4

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

11. Simplified 84.4%

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

12. Final simplification 84.4%

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

Alternative 20: 69.2% 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 68.8%

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

Reproduce

herbie shell --seed 2024204 
(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))))))))))