Destination given bearing on a great circle

Percentage Accurate: 99.7% → 99.7%

Time: 20.8s

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: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.8%

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

   1. sin-asin 99.8%

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

   2. fma-undefine 99.8%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. add-sqr-sqrt 47.9%

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

   6. sqrt-unprod 91.6%

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

   7. sqr-neg 91.6%

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

   8. sqrt-unprod 43.7%

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

   9. add-sqr-sqrt 85.8%

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

   10. distribute-rgt-in 85.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in delta around inf 99.8%

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

   1. *-commutative 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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) + ((cos(delta) * ((cos((phi1 * 2.0d0)) / 2.0d0) - 0.5d0)) - ((cos(phi1) * (sin(delta) * cos(theta))) * sin(phi1)))))
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.cos(delta) * ((Math.cos((phi1 * 2.0)) / 2.0) - 0.5)) - ((Math.cos(phi1) * (Math.sin(delta) * Math.cos(theta))) * Math.sin(phi1)))));
}

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.cos(delta) * ((math.cos((phi1 * 2.0)) / 2.0) - 0.5)) - ((math.cos(phi1) * (math.sin(delta) * math.cos(theta))) * math.sin(phi1)))))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (cos(delta) + ((cos(delta) * ((cos((phi1 * 2.0)) / 2.0) - 0.5)) - ((cos(phi1) * (sin(delta) * cos(theta))) * sin(phi1)))));
end

Wolfram

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.8%

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

   1. sin-asin 99.8%

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

   2. fma-undefine 99.8%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. add-sqr-sqrt 47.9%

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

   6. sqrt-unprod 91.6%

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

   7. sqr-neg 91.6%

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

   8. sqrt-unprod 43.7%

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

   9. add-sqr-sqrt 85.8%

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

   10. distribute-rgt-in 85.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in delta around inf 99.8%

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

   1. *-commutative 99.8%

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

9. Simplified 99.8%

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

    1. unpow2 99.8%

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

    2. sin-mult 99.8%

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

11. Applied egg-rr 99.8%

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

    1. div-sub 99.8%

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

    2. +-inverses 99.8%

       \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right) \cdot \cos delta\right)} \]

    3. cos-0 99.8%

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

    4. metadata-eval 99.8%

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

    5. count-2 99.8%

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

    6. *-commutative 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

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) * ((cos(phi1) * (sin(delta) * cos(theta))) + (cos(delta) * sin(phi1))))))
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.cos(phi1) * (Math.sin(delta) * Math.cos(theta))) + (Math.cos(delta) * Math.sin(phi1))))));
}

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.cos(phi1) * (math.sin(delta) * math.cos(theta))) + (math.cos(delta) * math.sin(phi1))))))

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.8%

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

5. Taylor expanded in delta around inf 99.8%

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

6. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.8%

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

5. Taylor expanded in delta around inf 99.8%

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

6. Taylor expanded in theta around inf 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 94.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

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
    t_1 = sin(delta) * cos(phi1)
    code = lambda1 + atan2((sin(theta) * t_1), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + t_1))))
end function

Java

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

Python

def code(lambda1, phi1, phi2, delta, theta):
	t_1 = math.sin(delta) * math.cos(phi1)
	return lambda1 + math.atan2((math.sin(theta) * t_1), (math.cos(delta) - (math.sin(phi1) * ((math.cos(delta) * math.sin(phi1)) + t_1))))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	t_1 = sin(delta) * cos(phi1);
	tmp = lambda1 + atan2((sin(theta) * t_1), (cos(delta) - (sin(phi1) * ((cos(delta) * sin(phi1)) + t_1))));
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := Block[{t$95$1 = N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * t$95$1), $MachinePrecision] / N[(N[Cos[delta], $MachinePrecision] - N[(N[Sin[phi1], $MachinePrecision] * N[(N[(N[Cos[delta], $MachinePrecision] * N[Sin[phi1], $MachinePrecision]), $MachinePrecision] + t$95$1), $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. Step-by-step derivation

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.8%

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

5. Taylor expanded in theta around 0 93.0%

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

6. Final simplification 93.0%

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

Alternative 6: 91.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $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. Step-by-step derivation

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.8%

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

5. Taylor expanded in delta around 0 90.8%

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

Alternative 7: 91.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -2.5 \cdot 10^{-5} \lor \neg \left(delta \leq 1.45 \cdot 10^{-10}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -2.5e-5) (not (<= delta 1.45e-10)))
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))
   (+
    lambda1
    (atan2 (* (sin theta) (* (sin delta) (cos phi1))) (pow (cos phi1) 2.0)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -2.5e-5) || !(delta <= 1.45e-10)) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), pow(cos(phi1), 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((delta <= (-2.5d-5)) .or. (.not. (delta <= 1.45d-10))) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    else
        tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (cos(phi1) ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -2.5e-5) || !(delta <= 1.45e-10)) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), Math.pow(Math.cos(phi1), 2.0));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -2.5e-5) or not (delta <= 1.45e-10):
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), math.pow(math.cos(phi1), 2.0))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -2.5e-5) || !(delta <= 1.45e-10))
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), (cos(phi1) ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -2.5e-5) || ~((delta <= 1.45e-10)))
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	else
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (cos(phi1) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -2.5e-5], N[Not[LessEqual[delta, 1.45e-10]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[phi1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -2.50000000000000012e-5 or 1.4499999999999999e-10 < delta

   1. Initial program 99.7%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 82.1%

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

   6. Taylor expanded in theta around 0 82.1%

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

   if -2.50000000000000012e-5 < delta < 1.4499999999999999e-10

   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. Step-by-step derivation

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. sin-asin 99.7%

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

      2. fma-undefine 99.7%

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

      3. associate-*r* 99.7%

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

      4. *-commutative 99.7%

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

      5. add-sqr-sqrt 47.0%

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

      6. sqrt-unprod 92.6%

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

      7. sqr-neg 92.6%

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

      8. sqrt-unprod 45.6%

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

      9. add-sqr-sqrt 90.0%

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

      10. distribute-rgt-in 90.0%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in delta around 0 99.7%

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

      1. unpow2 99.7%

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

      2. 1-sub-sin 99.8%

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

      3. unpow2 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -2.5 \cdot 10^{-5} \lor \neg \left(delta \leq 1.45 \cdot 10^{-10}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{{\cos \phi_1}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.7% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    code = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (cos(delta) + ((cos((phi1 * 2.0d0)) / 2.0d0) - 0.5d0)))
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.cos((phi1 * 2.0)) / 2.0) - 0.5)));
}

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.cos((phi1 * 2.0)) / 2.0) - 0.5)))

Julia

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

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (cos(delta) + ((cos((phi1 * 2.0)) / 2.0) - 0.5)));
end

Wolfram

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.8%

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

5. Taylor expanded in delta around 0 90.8%

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

   1. unpow2 99.8%

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

   2. sin-mult 99.8%

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

7. Applied egg-rr 90.8%

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

   1. div-sub 99.8%

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

   2. +-inverses 99.8%

      \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right) \cdot \cos delta\right)} \]

   3. cos-0 99.8%

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

   4. metadata-eval 99.8%

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

   5. count-2 99.8%

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

   6. *-commutative 99.8%

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

9. Simplified 90.8%

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

10. Final simplification 90.8%

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

Alternative 9: 91.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -4.9 \cdot 10^{-5} \lor \neg \left(delta \leq 1.45 \cdot 10^{-10}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -4.9e-5) (not (<= delta 1.45e-10)))
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))
   (+
    lambda1
    (atan2
     (* (sin theta) (* (sin delta) (cos phi1)))
     (+ 0.5 (* 0.5 (cos (* phi1 2.0))))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -4.9e-5) || !(delta <= 1.45e-10)) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((delta <= (-4.9d-5)) .or. (.not. (delta <= 1.45d-10))) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    else
        tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (0.5d0 + (0.5d0 * cos((phi1 * 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -4.9e-5) || !(delta <= 1.45e-10)) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * (Math.sin(delta) * Math.cos(phi1))), (0.5 + (0.5 * Math.cos((phi1 * 2.0)))));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -4.9e-5) or not (delta <= 1.45e-10):
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * (math.sin(delta) * math.cos(phi1))), (0.5 + (0.5 * math.cos((phi1 * 2.0)))))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -4.9e-5) || !(delta <= 1.45e-10))
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * Float64(sin(delta) * cos(phi1))), Float64(0.5 + Float64(0.5 * cos(Float64(phi1 * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -4.9e-5) || ~((delta <= 1.45e-10)))
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	else
		tmp = lambda1 + atan2((sin(theta) * (sin(delta) * cos(phi1))), (0.5 + (0.5 * cos((phi1 * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -4.9e-5], N[Not[LessEqual[delta, 1.45e-10]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -4.9e-5 or 1.4499999999999999e-10 < delta

   1. Initial program 99.7%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 82.1%

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

   6. Taylor expanded in theta around 0 82.1%

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

   if -4.9e-5 < delta < 1.4499999999999999e-10

   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. Step-by-step derivation

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. sin-asin 99.7%

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

      2. fma-undefine 99.7%

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

      3. associate-*r* 99.7%

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

      4. *-commutative 99.7%

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

      5. add-sqr-sqrt 47.0%

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

      6. sqrt-unprod 92.6%

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

      7. sqr-neg 92.6%

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

      8. sqrt-unprod 45.6%

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

      9. add-sqr-sqrt 90.0%

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

      10. distribute-rgt-in 90.0%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in delta around inf 99.7%

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

      1. *-commutative 99.7%

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

   9. Simplified 99.7%

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

       1. unpow2 99.7%

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

       2. sin-mult 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. div-sub 99.7%

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

       2. +-inverses 99.7%

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right) \cdot \cos delta\right)} \]

       3. cos-0 99.7%

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

       4. metadata-eval 99.7%

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

       5. count-2 99.7%

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

       6. *-commutative 99.7%

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

   13. Simplified 99.7%

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

   14. Taylor expanded in delta around 0 99.8%

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

       1. *-commutative 99.8%

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

   16. Simplified 99.8%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -4.9 \cdot 10^{-5} \lor \neg \left(delta \leq 1.45 \cdot 10^{-10}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{0.5 + 0.5 \cdot \cos \left(\phi_1 \cdot 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -6.6 \cdot 10^{-7} \lor \neg \left(delta \leq 1.4 \cdot 10^{-10}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{1 - {\sin \phi_1}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -6.6e-7) (not (<= delta 1.4e-10)))
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))
   (+
    lambda1
    (atan2
     (* delta (* (sin theta) (cos phi1)))
     (- 1.0 (pow (sin phi1) 2.0))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -6.6e-7) || !(delta <= 1.4e-10)) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	} else {
		tmp = lambda1 + atan2((delta * (sin(theta) * cos(phi1))), (1.0 - pow(sin(phi1), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((delta <= (-6.6d-7)) .or. (.not. (delta <= 1.4d-10))) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    else
        tmp = lambda1 + atan2((delta * (sin(theta) * cos(phi1))), (1.0d0 - (sin(phi1) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -6.6e-7) || !(delta <= 1.4e-10)) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((delta * (Math.sin(theta) * Math.cos(phi1))), (1.0 - Math.pow(Math.sin(phi1), 2.0)));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -6.6e-7) or not (delta <= 1.4e-10):
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((delta * (math.sin(theta) * math.cos(phi1))), (1.0 - math.pow(math.sin(phi1), 2.0)))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -6.6e-7) || !(delta <= 1.4e-10))
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(delta * Float64(sin(theta) * cos(phi1))), Float64(1.0 - (sin(phi1) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -6.6e-7) || ~((delta <= 1.4e-10)))
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	else
		tmp = lambda1 + atan2((delta * (sin(theta) * cos(phi1))), (1.0 - (sin(phi1) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -6.6e-7], N[Not[LessEqual[delta, 1.4e-10]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(delta * N[(N[Sin[theta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Power[N[Sin[phi1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -6.6000000000000003e-7 or 1.40000000000000008e-10 < delta

   1. Initial program 99.7%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 82.1%

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

   6. Taylor expanded in theta around 0 82.1%

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

   if -6.6000000000000003e-7 < delta < 1.40000000000000008e-10

   1. Initial program 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in delta around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in delta around 0 99.7%

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

   9. Taylor expanded in phi1 around inf 99.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -6.6 \cdot 10^{-7} \lor \neg \left(delta \leq 1.4 \cdot 10^{-10}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{delta \cdot \left(\sin theta \cdot \cos \phi_1\right)}{1 - {\sin \phi_1}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -9.5 \cdot 10^{-5} \lor \neg \left(delta \leq 1.4 \cdot 10^{-10}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1 + \left(\frac{\cos \left(\phi_1 \cdot 2\right)}{2} - 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (or (<= delta -9.5e-5) (not (<= delta 1.4e-10)))
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) (sin delta))) (cos delta)))
   (+
    lambda1
    (atan2
     (* (cos phi1) (* (sin theta) delta))
     (+ 1.0 (- (/ (cos (* phi1 2.0)) 2.0) 0.5))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -9.5e-5) || !(delta <= 1.4e-10)) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0 + ((cos((phi1 * 2.0)) / 2.0) - 0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if ((delta <= (-9.5d-5)) .or. (.not. (delta <= 1.4d-10))) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta))
    else
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0d0 + ((cos((phi1 * 2.0d0)) / 2.0d0) - 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if ((delta <= -9.5e-5) || !(delta <= 1.4e-10)) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * Math.sin(delta))), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (1.0 + ((Math.cos((phi1 * 2.0)) / 2.0) - 0.5)));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if (delta <= -9.5e-5) or not (delta <= 1.4e-10):
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * math.sin(delta))), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (1.0 + ((math.cos((phi1 * 2.0)) / 2.0) - 0.5)))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if ((delta <= -9.5e-5) || !(delta <= 1.4e-10))
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * sin(delta))), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(1.0 + Float64(Float64(cos(Float64(phi1 * 2.0)) / 2.0) - 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if ((delta <= -9.5e-5) || ~((delta <= 1.4e-10)))
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * sin(delta))), cos(delta));
	else
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0 + ((cos((phi1 * 2.0)) / 2.0) - 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[Or[LessEqual[delta, -9.5e-5], N[Not[LessEqual[delta, 1.4e-10]], $MachinePrecision]], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if delta < -9.5000000000000005e-5 or 1.40000000000000008e-10 < delta

   1. Initial program 99.7%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 82.1%

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

   6. Taylor expanded in theta around 0 82.1%

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

   if -9.5000000000000005e-5 < delta < 1.40000000000000008e-10

   1. Initial program 99.7%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in delta around 0 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.7%

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

      3. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in delta around 0 99.7%

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

      1. unpow2 99.7%

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

      2. sin-mult 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. div-sub 99.7%

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

       2. +-inverses 99.7%

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right) \cdot \cos delta\right)} \]

       3. cos-0 99.7%

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

       4. metadata-eval 99.7%

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

       5. count-2 99.7%

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

       6. *-commutative 99.7%

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

   12. Simplified 99.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -9.5 \cdot 10^{-5} \lor \neg \left(delta \leq 1.4 \cdot 10^{-10}\right):\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot \sin delta\right)}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1 + \left(\frac{\cos \left(\phi_1 \cdot 2\right)}{2} - 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;delta \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.1 \cdot 10^{+30}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1 + \left(\frac{\cos \left(\phi_1 \cdot 2\right)}{2} - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= delta -3.1e-5)
   (+ lambda1 (atan2 (* (sin theta) (sin delta)) (cos delta)))
   (if (<= delta 4.1e+30)
     (+
      lambda1
      (atan2
       (* (cos phi1) (* (sin theta) delta))
       (+ 1.0 (- (/ (cos (* phi1 2.0)) 2.0) 0.5))))
     (+ lambda1 (atan2 (* theta (* (sin delta) (cos phi1))) (cos delta))))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -3.1e-5) {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	} else if (delta <= 4.1e+30) {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0 + ((cos((phi1 * 2.0)) / 2.0) - 0.5)));
	} else {
		tmp = lambda1 + atan2((theta * (sin(delta) * cos(phi1))), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (delta <= (-3.1d-5)) then
        tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta))
    else if (delta <= 4.1d+30) then
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0d0 + ((cos((phi1 * 2.0d0)) / 2.0d0) - 0.5d0)))
    else
        tmp = lambda1 + atan2((theta * (sin(delta) * cos(phi1))), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (delta <= -3.1e-5) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * Math.sin(delta)), Math.cos(delta));
	} else if (delta <= 4.1e+30) {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), (1.0 + ((Math.cos((phi1 * 2.0)) / 2.0) - 0.5)));
	} else {
		tmp = lambda1 + Math.atan2((theta * (Math.sin(delta) * Math.cos(phi1))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if delta <= -3.1e-5:
		tmp = lambda1 + math.atan2((math.sin(theta) * math.sin(delta)), math.cos(delta))
	elif delta <= 4.1e+30:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), (1.0 + ((math.cos((phi1 * 2.0)) / 2.0) - 0.5)))
	else:
		tmp = lambda1 + math.atan2((theta * (math.sin(delta) * math.cos(phi1))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (delta <= -3.1e-5)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), cos(delta)));
	elseif (delta <= 4.1e+30)
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), Float64(1.0 + Float64(Float64(cos(Float64(phi1 * 2.0)) / 2.0) - 0.5))));
	else
		tmp = Float64(lambda1 + atan(Float64(theta * Float64(sin(delta) * cos(phi1))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (delta <= -3.1e-5)
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	elseif (delta <= 4.1e+30)
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), (1.0 + ((cos((phi1 * 2.0)) / 2.0) - 0.5)));
	else
		tmp = lambda1 + atan2((theta * (sin(delta) * cos(phi1))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[delta, -3.1e-5], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[delta, 4.1e+30], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(N[Cos[N[(phi1 * 2.0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(theta * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if delta < -3.10000000000000014e-5

   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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi1 around 0 85.0%

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

   6. Taylor expanded in phi1 around 0 82.3%

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

   if -3.10000000000000014e-5 < delta < 4.10000000000000005e30

   1. Initial program 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in delta around 0 97.8%

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

      1. *-commutative 97.8%

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

      2. associate-*l* 97.8%

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

      3. *-commutative 97.8%

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

   7. Simplified 97.8%

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

   8. Taylor expanded in delta around 0 97.8%

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

      1. unpow2 99.7%

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

      2. sin-mult 99.7%

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

   10. Applied egg-rr 97.8%

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

       1. div-sub 99.7%

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

       2. +-inverses 99.7%

          \[\leadsto \lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta - \left(\left(\cos \phi_1 \cdot \left(\sin delta \cdot \cos theta\right)\right) \cdot \sin \phi_1 + \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(\phi_1 + \phi_1\right)}{2}\right) \cdot \cos delta\right)} \]

       3. cos-0 99.7%

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

       4. metadata-eval 99.7%

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

       5. count-2 99.7%

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

       6. *-commutative 99.7%

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

   12. Simplified 97.8%

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

   if 4.10000000000000005e30 < delta

   1. Initial program 99.6%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi1 around 0 80.2%

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

   6. Taylor expanded in theta around 0 70.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;delta \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{elif}\;delta \leq 4.1 \cdot 10^{+30}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1 + \left(\frac{\cos \left(\phi_1 \cdot 2\right)}{2} - 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq 4.4 \cdot 10^{-25}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= phi1 4.4e-25)
   (+ lambda1 (atan2 (* (sin theta) (sin delta)) (cos delta)))
   (+ lambda1 (atan2 (* theta (* (sin delta) (cos phi1))) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= 4.4e-25) {
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	} else {
		tmp = lambda1 + atan2((theta * (sin(delta) * cos(phi1))), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (phi1 <= 4.4d-25) then
        tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta))
    else
        tmp = lambda1 + atan2((theta * (sin(delta) * cos(phi1))), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (phi1 <= 4.4e-25) {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * Math.sin(delta)), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((theta * (Math.sin(delta) * Math.cos(phi1))), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if phi1 <= 4.4e-25:
		tmp = lambda1 + math.atan2((math.sin(theta) * math.sin(delta)), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((theta * (math.sin(delta) * math.cos(phi1))), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (phi1 <= 4.4e-25)
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * sin(delta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(theta * Float64(sin(delta) * cos(phi1))), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (phi1 <= 4.4e-25)
		tmp = lambda1 + atan2((sin(theta) * sin(delta)), cos(delta));
	else
		tmp = lambda1 + atan2((theta * (sin(delta) * cos(phi1))), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[phi1, 4.4e-25], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(theta * N[(N[Sin[delta], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 4.4000000000000004e-25

   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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 91.3%

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

   6. Taylor expanded in phi1 around 0 88.5%

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

   if 4.4000000000000004e-25 < phi1

   1. Initial program 99.5%

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

      1. associate-*l* 99.6%

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

      2. cos-neg 99.6%

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

      3. +-commutative 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi1 around 0 72.8%

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

   6. Taylor expanded in theta around 0 70.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq 4.4 \cdot 10^{-25}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \left(\sin delta \cdot \cos \phi_1\right)}{\cos delta}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 85.7% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.8%

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

5. Taylor expanded in phi1 around 0 86.0%

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

6. Taylor expanded in phi1 around 0 81.5%

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

Alternative 15: 76.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq 1.25 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= theta 1.25e+16)
   (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
   (+ lambda1 (atan2 (* (cos phi1) (* (sin theta) delta)) 1.0))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= 1.25e+16) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else {
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (theta <= 1.25d+16) then
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    else
        tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= 1.25e+16) {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.cos(phi1) * (Math.sin(theta) * delta)), 1.0);
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if theta <= 1.25e+16:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.cos(phi1) * (math.sin(theta) * delta)), 1.0)
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (theta <= 1.25e+16)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(cos(phi1) * Float64(sin(theta) * delta)), 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (theta <= 1.25e+16)
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	else
		tmp = lambda1 + atan2((cos(phi1) * (sin(theta) * delta)), 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[theta, 1.25e+16], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision]), $MachinePrecision] / 1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < 1.25e16

   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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 87.4%

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

   6. Taylor expanded in phi1 around 0 82.4%

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

   7. Taylor expanded in theta around 0 77.2%

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

   if 1.25e16 < theta

   1. Initial program 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in delta around 0 73.6%

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

      1. *-commutative 73.6%

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

      2. associate-*l* 73.6%

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

      3. *-commutative 73.6%

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

   7. Simplified 73.6%

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

   8. Taylor expanded in delta around 0 73.6%

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

   9. Taylor expanded in phi1 around 0 71.1%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;theta \leq 1.25 \cdot 10^{+16}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\cos \phi_1 \cdot \left(\sin theta \cdot delta\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 76.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq 0.095:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{\sin theta \cdot delta}{\cos delta}\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= theta 0.095)
   (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
   (+ lambda1 (atan2 (* (sin theta) delta) (cos delta)))))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= 0.095) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else {
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (theta <= 0.095d0) then
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    else
        tmp = lambda1 + atan2((sin(theta) * delta), cos(delta))
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= 0.095) {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	} else {
		tmp = lambda1 + Math.atan2((Math.sin(theta) * delta), Math.cos(delta));
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if theta <= 0.095:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	else:
		tmp = lambda1 + math.atan2((math.sin(theta) * delta), math.cos(delta))
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (theta <= 0.095)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	else
		tmp = Float64(lambda1 + atan(Float64(sin(theta) * delta), cos(delta)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (theta <= 0.095)
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	else
		tmp = lambda1 + atan2((sin(theta) * delta), cos(delta));
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[theta, 0.095], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(lambda1 + N[ArcTan[N[(N[Sin[theta], $MachinePrecision] * delta), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if theta < 0.095000000000000001

   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. Step-by-step derivation

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 87.8%

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

   6. Taylor expanded in phi1 around 0 82.8%

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

   7. Taylor expanded in theta around 0 77.9%

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

   if 0.095000000000000001 < theta

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 80.9%

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

   6. Taylor expanded in phi1 around 0 78.1%

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

   7. Taylor expanded in delta around 0 68.5%

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

Alternative 17: 74.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;theta \leq 9 \cdot 10^{+139}:\\ \;\;\;\;\lambda_1 + \tan^{-1}_* \frac{theta \cdot \sin delta}{\cos delta}\\ \mathbf{else}:\\ \;\;\;\;\lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta)
 :precision binary64
 (if (<= theta 9e+139)
   (+ lambda1 (atan2 (* theta (sin delta)) (cos delta)))
   lambda1))

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= 9e+139) {
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Fortran

real(8) function code(lambda1, phi1, phi2, delta, theta)
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8), intent (in) :: delta
    real(8), intent (in) :: theta
    real(8) :: tmp
    if (theta <= 9d+139) then
        tmp = lambda1 + atan2((theta * sin(delta)), cos(delta))
    else
        tmp = lambda1
    end if
    code = tmp
end function

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	double tmp;
	if (theta <= 9e+139) {
		tmp = lambda1 + Math.atan2((theta * Math.sin(delta)), Math.cos(delta));
	} else {
		tmp = lambda1;
	}
	return tmp;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	tmp = 0
	if theta <= 9e+139:
		tmp = lambda1 + math.atan2((theta * math.sin(delta)), math.cos(delta))
	else:
		tmp = lambda1
	return tmp

Julia

function code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0
	if (theta <= 9e+139)
		tmp = Float64(lambda1 + atan(Float64(theta * sin(delta)), cos(delta)));
	else
		tmp = lambda1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(lambda1, phi1, phi2, delta, theta)
	tmp = 0.0;
	if (theta <= 9e+139)
		tmp = lambda1 + atan2((theta * sin(delta)), cos(delta));
	else
		tmp = lambda1;
	end
	tmp_2 = tmp;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := If[LessEqual[theta, 9e+139], N[(lambda1 + N[ArcTan[N[(theta * N[Sin[delta], $MachinePrecision]), $MachinePrecision] / N[Cos[delta], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], lambda1]

Derivation

1. Split input into 2 regimes

2. if theta < 8.9999999999999999e139

   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. Step-by-step derivation

      1. associate-*l* 99.7%

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

      2. cos-neg 99.7%

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

      3. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi1 around 0 85.8%

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

   6. Taylor expanded in phi1 around 0 80.9%

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

   7. Taylor expanded in theta around 0 75.4%

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

   if 8.9999999999999999e139 < theta

   1. Initial program 99.8%

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

      1. associate-*l* 99.8%

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

      2. cos-neg 99.8%

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

      3. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi1 around 0 86.6%

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

   6. Taylor expanded in lambda1 around inf 69.0%

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

Alternative 18: 69.3% accurate, 1320.0× speedup

Math

\[\begin{array}{l} \\ \lambda_1 \end{array} \]

FPCore

(FPCore (lambda1 phi1 phi2 delta theta) :precision binary64 lambda1)

C

double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1;
}

Fortran

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

Java

public static double code(double lambda1, double phi1, double phi2, double delta, double theta) {
	return lambda1;
}

Python

def code(lambda1, phi1, phi2, delta, theta):
	return lambda1

Julia

function code(lambda1, phi1, phi2, delta, theta)
	return lambda1
end

MATLAB

function tmp = code(lambda1, phi1, phi2, delta, theta)
	tmp = lambda1;
end

Wolfram

code[lambda1_, phi1_, phi2_, delta_, theta_] := lambda1

Derivation

1. Initial program 99.7%

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

   1. associate-*l* 99.7%

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

   2. cos-neg 99.7%

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

   3. +-commutative 99.7%

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

3. Simplified 99.8%

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

5. Taylor expanded in phi1 around 0 86.0%

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

6. Taylor expanded in lambda1 around inf 68.9%

   \[\leadsto \color{blue}{\lambda_1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024086 
(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))))))))))