Destination given bearing on a great circle

Percentage Accurate: 99.8% → 99.8%

Time: 16.6s

Alternatives: 18

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in delta around inf

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in theta around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 99.8

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 94.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in theta around 0

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-cos.f64 92.9

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

5. Simplified 92.9%

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

6. Final simplification 92.9%

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

Alternative 3: 93.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in delta around inf

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in theta around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 99.8

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

8. Simplified 99.8%

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

9. Taylor expanded in delta around 0

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

11. Simplified 91.0%

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

12. Taylor expanded in delta around inf

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

    1. sub-neg N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

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

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. lower-cos.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. lower-cos.f64 N/A

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

    11. lower-sin.f64 N/A

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

    12. lower-sin.f64 N/A

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

    13. lower-neg.f64 N/A

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

    14. lower-sin.f64 N/A

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

    15. lower-cos.f64 91.0

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

14. Simplified 91.0%

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

15. Final simplification 91.0%

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

Alternative 4: 93.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Cos[phi1], $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 99.7%

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

3. Taylor expanded in delta around inf

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 99.8

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

5. Simplified 99.8%

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

6. Taylor expanded in theta around inf

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-sin.f64 99.8

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

8. Simplified 99.8%

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

9. Taylor expanded in delta around 0

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

11. Simplified 91.0%

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

12. Taylor expanded in theta around 0

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

    1. sub-neg N/A

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

    2. +-commutative N/A

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

    3. *-commutative N/A

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

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

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

    5. lower-fma.f64 N/A

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

    6. +-commutative N/A

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

    7. lower-fma.f64 N/A

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

    8. lower-cos.f64 N/A

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

    9. lower-sin.f64 N/A

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

    10. lower-sin.f64 N/A

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

    11. lower-neg.f64 N/A

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

    12. lower-sin.f64 N/A

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

    13. lower-cos.f64 90.8

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

14. Simplified 90.8%

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

Alternative 5: 92.7% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* (cos phi1) (* (sin delta) (sin theta)))
   (- (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(delta) * sin(theta))), (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(delta) * sin(theta))), (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(delta) * Math.sin(theta))), (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(delta) * math.sin(theta))), (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(delta) * sin(theta))), Float64(cos(delta) - (sin(phi1) ^ 2.0))))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in delta around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sin.f64 90.1

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

5. Simplified 90.1%

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

6. Final simplification 90.1%

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

Alternative 6: 92.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -0.0045:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (cos phi1) (* (sin delta) (sin theta))) (cos delta)))))
   (if (<= delta -0.0045)
     t_1
     (if (<= delta 8.5e+18)
       (+
        lambda1
        (atan2
         (*
          (* (cos phi1) (sin theta))
          (fma delta (* -0.16666666666666666 (* delta delta)) delta))
         (- (cos delta) (pow (sin phi1) 2.0))))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), cos(delta));
	double tmp;
	if (delta <= -0.0045) {
		tmp = t_1;
	} else if (delta <= 8.5e+18) {
		tmp = lambda1 + atan2(((cos(phi1) * sin(theta)) * fma(delta, (-0.16666666666666666 * (delta * delta)), delta)), (cos(delta) - pow(sin(phi1), 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), cos(delta)))
	tmp = 0.0
	if (delta <= -0.0045)
		tmp = t_1;
	elseif (delta <= 8.5e+18)
		tmp = Float64(lambda1 + atan(Float64(Float64(cos(phi1) * sin(theta)) * fma(delta, Float64(-0.16666666666666666 * Float64(delta * delta)), delta)), Float64(cos(delta) - (sin(phi1) ^ 2.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -0.0045], t$95$1, If[LessEqual[delta, 8.5e+18], N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision]), $MachinePrecision] + delta), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -0.00449999999999999966 or 8.5e18 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.6

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

   5. Simplified 82.6%

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

   if -0.00449999999999999966 < delta < 8.5e18

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 98.7

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

   5. Simplified 98.7%

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

   6. Taylor expanded in delta around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 98.7

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

   8. Simplified 98.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -0.0045:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\cos delta - {\sin \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)\\ t_2 := \lambda_1 + \tan^{-1}_* \frac{t\_1}{\cos delta}\\ \mathbf{if}\;delta \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{t\_1}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (* (cos phi1) (* (sin delta) (sin theta))))
        (t_2 (+ lambda1 (atan2 t_1 (cos delta)))))
   (if (<= delta -2.7e-6)
     t_2
     (if (<= delta 8.5e+18)
       (+ lambda1 (atan2 t_1 (pow (cos phi1) 2.0)))
       t_2))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = cos(phi1) * (sin(delta) * sin(theta));
	double t_2 = lambda1 + atan2(t_1, cos(delta));
	double tmp;
	if (delta <= -2.7e-6) {
		tmp = t_2;
	} else if (delta <= 8.5e+18) {
		tmp = lambda1 + atan2(t_1, pow(cos(phi1), 2.0));
	} 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 = cos(phi1) * (sin(delta) * sin(theta))
    t_2 = lambda1 + atan2(t_1, cos(delta))
    if (delta <= (-2.7d-6)) then
        tmp = t_2
    else if (delta <= 8.5d+18) then
        tmp = lambda1 + atan2(t_1, (cos(phi1) ** 2.0d0))
    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.cos(phi1) * (Math.sin(delta) * Math.sin(theta));
	double t_2 = lambda1 + Math.atan2(t_1, Math.cos(delta));
	double tmp;
	if (delta <= -2.7e-6) {
		tmp = t_2;
	} else if (delta <= 8.5e+18) {
		tmp = lambda1 + Math.atan2(t_1, Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.cos(phi1) * (math.sin(delta) * math.sin(theta))
	t_2 = lambda1 + math.atan2(t_1, math.cos(delta))
	tmp = 0
	if delta <= -2.7e-6:
		tmp = t_2
	elif delta <= 8.5e+18:
		tmp = lambda1 + math.atan2(t_1, math.pow(math.cos(phi1), 2.0))
	else:
		tmp = t_2
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(cos(phi1) * Float64(sin(delta) * sin(theta)))
	t_2 = Float64(lambda1 + atan(t_1, cos(delta)))
	tmp = 0.0
	if (delta <= -2.7e-6)
		tmp = t_2;
	elseif (delta <= 8.5e+18)
		tmp = Float64(lambda1 + atan(t_1, (cos(phi1) ^ 2.0)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = cos(phi1) * (sin(delta) * sin(theta));
	t_2 = lambda1 + atan2(t_1, cos(delta));
	tmp = 0.0;
	if (delta <= -2.7e-6)
		tmp = t_2;
	elseif (delta <= 8.5e+18)
		tmp = lambda1 + atan2(t_1, (cos(phi1) ^ 2.0));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(lambda1 + N[ArcTan[t$95$1 / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -2.7e-6], t$95$2, If[LessEqual[delta, 8.5e+18], N[(lambda1 + N[ArcTan[t$95$1 / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.69999999999999998e-6 or 8.5e18 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.3

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

   5. Simplified 82.3%

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

   if -2.69999999999999998e-6 < delta < 8.5e18

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around inf

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. unpow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 99.2

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

   8. Simplified 99.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \mathsf{fma}\left(delta, delta \cdot \left(delta \cdot -0.16666666666666666\right), delta\right)\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (cos phi1) (* (sin delta) (sin theta))) (cos delta)))))
   (if (<= delta -2.7e-6)
     t_1
     (if (<= delta 8.5e+18)
       (+
        lambda1
        (atan2
         (*
          (cos phi1)
          (*
           (sin theta)
           (fma delta (* delta (* delta -0.16666666666666666)) delta)))
         (pow (cos phi1) 2.0)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), cos(delta));
	double tmp;
	if (delta <= -2.7e-6) {
		tmp = t_1;
	} else if (delta <= 8.5e+18) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * fma(delta, (delta * (delta * -0.16666666666666666)), delta))), pow(cos(phi1), 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), cos(delta)))
	tmp = 0.0
	if (delta <= -2.7e-6)
		tmp = t_1;
	elseif (delta <= 8.5e+18)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * fma(delta, Float64(delta * Float64(delta * -0.16666666666666666)), delta))), (cos(phi1) ^ 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -2.7e-6], t$95$1, If[LessEqual[delta, 8.5e+18], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(delta * N[(delta * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + delta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.69999999999999998e-6 or 8.5e18 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.3

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

   5. Simplified 82.3%

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

   if -2.69999999999999998e-6 < delta < 8.5e18

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around inf

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. unpow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 99.2

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

   8. Simplified 99.2%

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

   9. Taylor expanded in delta around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r* N/A

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

       4. distribute-lft-out N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. associate-*r* N/A

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

       8. associate-*r* N/A

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

       9. *-commutative N/A

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

       10. associate-*r* N/A

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

       11. unpow2 N/A

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

       12. cube-mult N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\left(\color{blue}{{delta}^{3}} \cdot \frac{-1}{6}\right) \cdot \sin theta + delta \cdot \sin theta\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}} \]

       13. *-commutative N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\color{blue}{\left(\frac{-1}{6} \cdot {delta}^{3}\right)} \cdot \sin theta + delta \cdot \sin theta\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}} \]

       14. distribute-rgt-in N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\sin theta \cdot \left(\frac{-1}{6} \cdot {delta}^{3} + delta\right)\right)} \cdot \cos \phi_1}{{\cos \phi_1}^{2}} \]

       15. +-commutative N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\sin theta \cdot \color{blue}{\left(delta + \frac{-1}{6} \cdot {delta}^{3}\right)}\right) \cdot \cos \phi_1}{{\cos \phi_1}^{2}} \]

       16. lower-*.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\left(\sin theta \cdot \left(delta + \frac{-1}{6} \cdot {delta}^{3}\right)\right)} \cdot \cos \phi_1}{{\cos \phi_1}^{2}} \]

   11. Simplified 99.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \mathsf{fma}\left(delta, delta \cdot \left(delta \cdot -0.16666666666666666\right), delta\right)\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{if}\;delta \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 1.15 \cdot 10^{-47}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1
         (+
          lambda1
          (atan2 (* (cos phi1) (* (sin delta) (sin theta))) (cos delta)))))
   (if (<= delta -2.7e-6)
     t_1
     (if (<= delta 1.15e-47)
       (+
        lambda1
        (atan2 (* (cos phi1) (* delta (sin theta))) (pow (cos phi1) 2.0)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), cos(delta));
	double tmp;
	if (delta <= -2.7e-6) {
		tmp = t_1;
	} else if (delta <= 1.15e-47) {
		tmp = lambda1 + atan2((cos(phi1) * (delta * sin(theta))), pow(cos(phi1), 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: t_1
    real(8) :: tmp
    t_1 = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), cos(delta))
    if (delta <= (-2.7d-6)) then
        tmp = t_1
    else if (delta <= 1.15d-47) then
        tmp = lambda1 + atan2((cos(phi1) * (delta * sin(theta))), (cos(phi1) ** 2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(delta) * Math.sin(theta))), Math.cos(delta));
	double tmp;
	if (delta <= -2.7e-6) {
		tmp = t_1;
	} else if (delta <= 1.15e-47) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (delta * Math.sin(theta))), Math.pow(Math.cos(phi1), 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = lambda1 + math.atan2((math.cos(phi1) * (math.sin(delta) * math.sin(theta))), math.cos(delta))
	tmp = 0
	if delta <= -2.7e-6:
		tmp = t_1
	elif delta <= 1.15e-47:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (delta * math.sin(theta))), math.pow(math.cos(phi1), 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(delta) * sin(theta))), cos(delta)))
	tmp = 0.0
	if (delta <= -2.7e-6)
		tmp = t_1;
	elseif (delta <= 1.15e-47)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(delta * sin(theta))), (cos(phi1) ^ 2.0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	t_1 = lambda1 + atan2((cos(phi1) * (sin(delta) * sin(theta))), cos(delta));
	tmp = 0.0;
	if (delta <= -2.7e-6)
		tmp = t_1;
	elseif (delta <= 1.15e-47)
		tmp = lambda1 + atan2((cos(phi1) * (delta * sin(theta))), (cos(phi1) ^ 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -2.7e-6], t$95$1, If[LessEqual[delta, 1.15e-47], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.69999999999999998e-6 or 1.14999999999999991e-47 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.9

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

   5. Simplified 82.9%

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

   if -2.69999999999999998e-6 < delta < 1.14999999999999991e-47

   1. Initial program 99.7%

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

   3. Taylor expanded in delta around inf

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 99.7

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

   5. Simplified 99.7%

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

   6. Taylor expanded in delta around 0

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

      1. unpow2 N/A

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

      2. 1-sub-sin N/A

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

      3. unpow2 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-cos.f64 99.8

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

   8. Simplified 99.8%

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

   9. Taylor expanded in delta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 99.8

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

   11. Simplified 99.8%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \mathbf{elif}\;delta \leq 1.15 \cdot 10^{-47}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(delta \cdot \sin theta\right)}{{\cos \phi_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin delta \cdot \sin theta\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.1% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 86.5

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

5. Simplified 86.5%

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

6. Final simplification 86.5%

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

Alternative 11: 86.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 86.5

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

5. Simplified 86.5%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 83.8

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

8. Simplified 83.8%

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

Alternative 12: 80.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{if}\;delta \leq -2.05 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, delta \cdot delta, 0.041666666666666664\right), -0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))
   (if (<= delta -2.05e+41)
     t_1
     (if (<= delta 4.8e-23)
       (+
        lambda1
        (atan2
         (*
          (* (cos phi1) (sin theta))
          (fma delta (* -0.16666666666666666 (* delta delta)) delta))
         (fma
          (* delta delta)
          (fma
           (* delta delta)
           (fma -0.001388888888888889 (* delta delta) 0.041666666666666664)
           -0.5)
          1.0)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * theta), cos(delta));
	double tmp;
	if (delta <= -2.05e+41) {
		tmp = t_1;
	} else if (delta <= 4.8e-23) {
		tmp = lambda1 + atan2(((cos(phi1) * sin(theta)) * fma(delta, (-0.16666666666666666 * (delta * delta)), delta)), fma((delta * delta), fma((delta * delta), fma(-0.001388888888888889, (delta * delta), 0.041666666666666664), -0.5), 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)))
	tmp = 0.0
	if (delta <= -2.05e+41)
		tmp = t_1;
	elseif (delta <= 4.8e-23)
		tmp = Float64(lambda1 + atan(Float64(Float64(cos(phi1) * sin(theta)) * fma(delta, Float64(-0.16666666666666666 * Float64(delta * delta)), delta)), fma(Float64(delta * delta), fma(Float64(delta * delta), fma(-0.001388888888888889, Float64(delta * delta), 0.041666666666666664), -0.5), 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -2.05e+41], t$95$1, If[LessEqual[delta, 4.8e-23], N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision]), $MachinePrecision] + delta), $MachinePrecision]), $MachinePrecision] / N[(N[(delta * delta), $MachinePrecision] * N[(N[(delta * delta), $MachinePrecision] * N[(-0.001388888888888889 * N[(delta * delta), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.0500000000000002e41 or 4.79999999999999993e-23 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.0

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

   5. Simplified 82.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 77.3

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

   8. Simplified 77.3%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 68.6

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

   11. Simplified 68.6%

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

   if -2.0500000000000002e41 < delta < 4.79999999999999993e-23

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in delta around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 89.7

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

   8. Simplified 89.7%

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

   9. Taylor expanded in delta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\color{blue}{1 + {delta}^{2} \cdot \left({delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\color{blue}{{delta}^{2} \cdot \left({delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\color{blue}{\mathsf{fma}\left({delta}^{2}, {delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}, 1\right)}} \]

       3. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, {delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}, 1\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, {delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}, 1\right)} \]

       5. sub-neg N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{{delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)} \]

       6. metadata-eval N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, {delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right)} \]

       7. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{\mathsf{fma}\left({delta}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}, \frac{-1}{2}\right)}, 1\right)} \]

       8. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}, \frac{-1}{2}\right), 1\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}, \frac{-1}{2}\right), 1\right)} \]

       10. +-commutative N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \color{blue}{\frac{-1}{720} \cdot {delta}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right)} \]

       11. lower-fma.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {delta}^{2}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right)} \]

       12. unpow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, \color{blue}{delta \cdot delta}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       13. lower-*.f64 89.8

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, \color{blue}{delta \cdot delta}, 0.041666666666666664\right), -0.5\right), 1\right)} \]

   11. Simplified 89.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\color{blue}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, delta \cdot delta, 0.041666666666666664\right), -0.5\right), 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.05 \cdot 10^{+41}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, delta \cdot delta, 0.041666666666666664\right), -0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{if}\;delta \leq -9.6 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))
   (if (<= delta -9.6e+40)
     t_1
     (if (<= delta 4.8e-23)
       (+
        lambda1
        (atan2
         (*
          (* (cos phi1) (sin theta))
          (fma delta (* -0.16666666666666666 (* delta delta)) delta))
         (fma delta (* delta -0.5) 1.0)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * theta), cos(delta));
	double tmp;
	if (delta <= -9.6e+40) {
		tmp = t_1;
	} else if (delta <= 4.8e-23) {
		tmp = lambda1 + atan2(((cos(phi1) * sin(theta)) * fma(delta, (-0.16666666666666666 * (delta * delta)), delta)), fma(delta, (delta * -0.5), 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)))
	tmp = 0.0
	if (delta <= -9.6e+40)
		tmp = t_1;
	elseif (delta <= 4.8e-23)
		tmp = Float64(lambda1 + atan(Float64(Float64(cos(phi1) * sin(theta)) * fma(delta, Float64(-0.16666666666666666 * Float64(delta * delta)), delta)), fma(delta, Float64(delta * -0.5), 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -9.6e+40], t$95$1, If[LessEqual[delta, 4.8e-23], N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision]), $MachinePrecision] + delta), $MachinePrecision]), $MachinePrecision] / N[(delta * N[(delta * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -9.5999999999999999e40 or 4.79999999999999993e-23 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.0

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

   5. Simplified 82.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 77.3

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

   8. Simplified 77.3%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 68.6

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

   11. Simplified 68.6%

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

   if -9.5999999999999999e40 < delta < 4.79999999999999993e-23

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in delta around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 89.7

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

   8. Simplified 89.7%

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

   9. Taylor expanded in delta around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. unpow2 N/A

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

       4. associate-*l* N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 89.7

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta, \color{blue}{delta \cdot -0.5}, 1\right)} \]

   11. Simplified 89.7%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\color{blue}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -9.6 \cdot 10^{+40}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta, delta \cdot -0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 80.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{if}\;delta \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))
   (if (<= delta -3.8e-5)
     t_1
     (if (<= delta 4.8e-23)
       (+
        lambda1
        (atan2
         (*
          (* (cos phi1) (sin theta))
          (fma delta (* -0.16666666666666666 (* delta delta)) delta))
         1.0))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * theta), cos(delta));
	double tmp;
	if (delta <= -3.8e-5) {
		tmp = t_1;
	} else if (delta <= 4.8e-23) {
		tmp = lambda1 + atan2(((cos(phi1) * sin(theta)) * fma(delta, (-0.16666666666666666 * (delta * delta)), delta)), 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)))
	tmp = 0.0
	if (delta <= -3.8e-5)
		tmp = t_1;
	elseif (delta <= 4.8e-23)
		tmp = Float64(lambda1 + atan(Float64(Float64(cos(phi1) * sin(theta)) * fma(delta, Float64(-0.16666666666666666 * Float64(delta * delta)), delta)), 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -3.8e-5], t$95$1, If[LessEqual[delta, 4.8e-23], N[(lambda1 + N[ArcTan[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[theta], $MachinePrecision]), $MachinePrecision] * N[(delta * N[(-0.16666666666666666 * N[(delta * delta), $MachinePrecision]), $MachinePrecision] + delta), $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -3.8000000000000002e-5 or 4.79999999999999993e-23 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.4

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

   5. Simplified 82.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 78.0

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

   8. Simplified 78.0%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 68.9

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

   11. Simplified 68.9%

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

   if -3.8000000000000002e-5 < delta < 4.79999999999999993e-23

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.4

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

   5. Simplified 90.4%

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

   6. Taylor expanded in delta around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 90.4

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

   8. Simplified 90.4%

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

   9. Taylor expanded in delta around 0

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

   11. Simplified 90.4%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -3.8 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{1}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{if}\;delta \leq -1.4 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;delta \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, delta \cdot \left(delta \cdot -0.16666666666666666\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, delta \cdot delta, 0.041666666666666664\right), -0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (let* ((t_1 (+ lambda1 (atan2 (* (sin delta) theta) (cos delta)))))
   (if (<= delta -1.4e+41)
     t_1
     (if (<= delta 4.8e-23)
       (+
        lambda1
        (atan2
         (*
          (sin theta)
          (fma delta (* delta (* delta -0.16666666666666666)) delta))
         (fma
          (* delta delta)
          (fma
           (* delta delta)
           (fma -0.001388888888888889 (* delta delta) 0.041666666666666664)
           -0.5)
          1.0)))
       t_1))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double t_1 = lambda1 + atan2((sin(delta) * theta), cos(delta));
	double tmp;
	if (delta <= -1.4e+41) {
		tmp = t_1;
	} else if (delta <= 4.8e-23) {
		tmp = lambda1 + atan2((sin(theta) * fma(delta, (delta * (delta * -0.16666666666666666)), delta)), fma((delta * delta), fma((delta * delta), fma(-0.001388888888888889, (delta * delta), 0.041666666666666664), -0.5), 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	t_1 = Float64(lambda1 + atan(Float64(sin(delta) * theta), cos(delta)))
	tmp = 0.0
	if (delta <= -1.4e+41)
		tmp = t_1;
	elseif (delta <= 4.8e-23)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * fma(delta, Float64(delta * Float64(delta * -0.16666666666666666)), delta)), fma(Float64(delta * delta), fma(Float64(delta * delta), fma(-0.001388888888888889, Float64(delta * delta), 0.041666666666666664), -0.5), 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(lambda1 + N[ArcTan[N[(N[Sin[delta], $MachinePrecision] * theta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[delta, -1.4e+41], t$95$1, If[LessEqual[delta, 4.8e-23], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(delta * N[(delta * N[(delta * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + delta), $MachinePrecision]), $MachinePrecision] / N[(N[(delta * delta), $MachinePrecision] * N[(N[(delta * delta), $MachinePrecision] * N[(-0.001388888888888889 * N[(delta * delta), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if delta < -1.4e41 or 4.79999999999999993e-23 < delta

   1. Initial program 99.8%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 82.0

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

   5. Simplified 82.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-sin.f64 77.3

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

   8. Simplified 77.3%

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

   9. Taylor expanded in theta around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 68.6

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

   11. Simplified 68.6%

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

   if -1.4e41 < delta < 4.79999999999999993e-23

   1. Initial program 99.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. lower-cos.f64 90.3

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

   5. Simplified 90.3%

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

   6. Taylor expanded in delta around 0

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

      1. distribute-lft-in N/A

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

      2. associate-*r* N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 89.7

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

   8. Simplified 89.7%

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

   9. Taylor expanded in delta around 0

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\color{blue}{1 + {delta}^{2} \cdot \left({delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}\right)}} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\color{blue}{{delta}^{2} \cdot \left({delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}\right) + 1}} \]

       2. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\color{blue}{\mathsf{fma}\left({delta}^{2}, {delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}, 1\right)}} \]

       3. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, {delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}, 1\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, {delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) - \frac{1}{2}, 1\right)} \]

       5. sub-neg N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{{delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)} \]

       6. metadata-eval N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, {delta}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right)} \]

       7. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{\mathsf{fma}\left({delta}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}, \frac{-1}{2}\right)}, 1\right)} \]

       8. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}, \frac{-1}{2}\right), 1\right)} \]

       9. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} + \frac{-1}{720} \cdot {delta}^{2}, \frac{-1}{2}\right), 1\right)} \]

       10. +-commutative N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \color{blue}{\frac{-1}{720} \cdot {delta}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right)} \]

       11. lower-fma.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \color{blue}{\mathsf{fma}\left(\frac{-1}{720}, {delta}^{2}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right)} \]

       12. unpow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, \frac{-1}{6} \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, \color{blue}{delta \cdot delta}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       13. lower-*.f64 89.8

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, \color{blue}{delta \cdot delta}, 0.041666666666666664\right), -0.5\right), 1\right)} \]

   11. Simplified 89.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\left(\cos \phi_1 \cdot \sin theta\right) \cdot \mathsf{fma}\left(delta, -0.16666666666666666 \cdot \left(delta \cdot delta\right), delta\right)}{\color{blue}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, delta \cdot delta, 0.041666666666666664\right), -0.5\right), 1\right)}} \]

   12. Taylor expanded in phi1 around 0

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta \cdot \left(delta + \frac{-1}{6} \cdot {delta}^{3}\right)}}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta \cdot \left(delta + \frac{-1}{6} \cdot {delta}^{3}\right)}}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       2. lower-sin.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta} \cdot \left(delta + \frac{-1}{6} \cdot {delta}^{3}\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       3. +-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \color{blue}{\left(\frac{-1}{6} \cdot {delta}^{3} + delta\right)}}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       4. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\color{blue}{{delta}^{3} \cdot \frac{-1}{6}} + delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       5. cube-mult N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\color{blue}{\left(delta \cdot \left(delta \cdot delta\right)\right)} \cdot \frac{-1}{6} + delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       6. unpow2 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\left(delta \cdot \color{blue}{{delta}^{2}}\right) \cdot \frac{-1}{6} + delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       7. associate-*l* N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\color{blue}{delta \cdot \left({delta}^{2} \cdot \frac{-1}{6}\right)} + delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       8. *-commutative N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(delta \cdot \color{blue}{\left(\frac{-1}{6} \cdot {delta}^{2}\right)} + delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       9. lower-fma.f64 N/A

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \color{blue}{\mathsf{fma}\left(delta, \frac{-1}{6} \cdot {delta}^{2}, delta\right)}}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       10. *-commutative N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, \color{blue}{{delta}^{2} \cdot \frac{-1}{6}}, delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       11. unpow2 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, \color{blue}{\left(delta \cdot delta\right)} \cdot \frac{-1}{6}, delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       12. associate-*l* N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, \color{blue}{delta \cdot \left(delta \cdot \frac{-1}{6}\right)}, delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       13. *-commutative N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, delta \cdot \color{blue}{\left(\frac{-1}{6} \cdot delta\right)}, delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       14. lower-*.f64 N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, \color{blue}{delta \cdot \left(\frac{-1}{6} \cdot delta\right)}, delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       15. *-commutative N/A

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, delta \cdot \color{blue}{\left(delta \cdot \frac{-1}{6}\right)}, delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{-1}{720}, delta \cdot delta, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right)} \]

       16. lower-*.f64 88.7

           \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, delta \cdot \color{blue}{\left(delta \cdot -0.16666666666666666\right)}, delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, delta \cdot delta, 0.041666666666666664\right), -0.5\right), 1\right)} \]

   14. Simplified 88.7%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\color{blue}{\sin theta \cdot \mathsf{fma}\left(delta, delta \cdot \left(delta \cdot -0.16666666666666666\right), delta\right)}}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, delta \cdot delta, 0.041666666666666664\right), -0.5\right), 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -1.4 \cdot 10^{+41}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \mathsf{fma}\left(delta, delta \cdot \left(delta \cdot -0.16666666666666666\right), delta\right)}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(-0.001388888888888889, delta \cdot delta, 0.041666666666666664\right), -0.5\right), 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin delta \cdot theta}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, delta \cdot delta, -0.5\right), 1\right)} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (+
  lambda1
  (atan2
   (* delta (sin theta))
   (fma (* delta delta) (fma 0.041666666666666664 (* delta delta) -0.5) 1.0))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1 + atan2((delta * sin(theta)), fma((delta * delta), fma(0.041666666666666664, (delta * delta), -0.5), 1.0));
}

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return Float64(lambda1 + atan(Float64(delta * sin(theta)), fma(Float64(delta * delta), fma(0.041666666666666664, Float64(delta * delta), -0.5), 1.0)))
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(delta * N[Sin[theta], $MachinePrecision]), $MachinePrecision] / N[(N[(delta * delta), $MachinePrecision] * N[(0.041666666666666664 * N[(delta * delta), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 86.5

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

5. Simplified 86.5%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 83.8

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

8. Simplified 83.8%

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

9. Taylor expanded in delta around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sin.f64 72.2

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

11. Simplified 72.2%

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

12. Taylor expanded in delta around 0

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{1 + {delta}^{2} \cdot \left(\frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}\right)}} \]
13. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{{delta}^{2} \cdot \left(\frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}\right) + 1}} \]

    2. lower-fma.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\mathsf{fma}\left({delta}^{2}, \frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}, 1\right)}} \]

    3. unpow2 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}, 1\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(\color{blue}{delta \cdot delta}, \frac{1}{24} \cdot {delta}^{2} - \frac{1}{2}, 1\right)} \]

    5. sub-neg N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{\frac{1}{24} \cdot {delta}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right)} \]

    6. metadata-eval N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \frac{1}{24} \cdot {delta}^{2} + \color{blue}{\frac{-1}{2}}, 1\right)} \]

    7. lower-fma.f64 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \color{blue}{\mathsf{fma}\left(\frac{1}{24}, {delta}^{2}, \frac{-1}{2}\right)}, 1\right)} \]

    8. unpow2 N/A

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(\frac{1}{24}, \color{blue}{delta \cdot delta}, \frac{-1}{2}\right), 1\right)} \]

    9. lower-*.f64 73.9

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, \color{blue}{delta \cdot delta}, -0.5\right), 1\right)} \]

14. Simplified 73.9%

    \[\leadsto \lambda_1 + \tan^{-1}_* \frac{delta \cdot \sin theta}{\color{blue}{\mathsf{fma}\left(delta \cdot delta, \mathsf{fma}\left(0.041666666666666664, delta \cdot delta, -0.5\right), 1\right)}} \]
15. Add Preprocessing

Alternative 17: 73.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 86.5

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

5. Simplified 86.5%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 83.8

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

8. Simplified 83.8%

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

9. Taylor expanded in delta around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sin.f64 72.2

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

11. Simplified 72.2%

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

12. Taylor expanded in delta around 0

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

14. Simplified 71.8%

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

Alternative 18: 67.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in phi1 around 0

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

   1. lower-cos.f64 86.5

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

5. Simplified 86.5%

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

6. Taylor expanded in phi1 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-sin.f64 83.8

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

8. Simplified 83.8%

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

9. Taylor expanded in delta around 0

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

    1. lower-*.f64 N/A

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

    2. lower-sin.f64 72.2

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

11. Simplified 72.2%

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

12. Taylor expanded in theta around 0

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

    1. lower-*.f64 66.2

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

14. Simplified 66.2%

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

Reproduce

herbie shell --seed 2024215 
(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))))))))))